Categorical Properties
and Classifications of Several
Subcategories of Module Categories（加群圏の種々の部分圏の圏論的性質と分類について）

Haruhisa Enomoto
Student ID: 521801034

Graduate School of Mathematics
Nagoya University

Contents
Abstractv

Acknowledgementvi

Chapter 1. Introduction
1.1. Aim of this thesis
1.2. Summary of results

1
1
2

Chapter 2. Relations for Grothendieck groups and representation-finiteness2.1. Introduction
2.2. Preliminaries
2.3. General conditions for (a), (b) and (c) to be equivalent2.4. Applications
2.5. Finiteness of syzygies
2.A. Some properties of the category of effaceable functors

6
6
8
10
12
17
18

Chapter 3. The Jordan-Hölder property and Grothendieck monoids of exact categories
3.1. Introduction
3.2. Posets of admissible subobjects
3.3. Grothendieck monoid of an exact category
3.4. Characterization of (JHP)
3.5. (JHP) for extension-closed subcategories of module categories
3.6. Torsion-free classes of Type A and Bruhat inversion
3.7. Computations of the Grothendieck monoids
3.8. Counter-examples
3.9. Problems
3.A. Preliminaries on monoids

21
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24
32
35
39
44
52
56
60
61

Chapter 4.

Bruhat inversions in Weyl groups and torsion-free classes over preprojective
algebras
Introduction
Preliminaries on root system
Brick sequence of a torsion-free class
Torsion-free classes over preprojective algebras of Dynkin type
Torsion-free classes over path algebras of Dynkin type
Description and enumeration for type A

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68
75
80
87
89

Chapter 5. Monobrick, a uniform approach to torsion-free classes and wide subcategories
5.1. Introduction
5.2. Bijection between monobricks and left Schur subcategories
5.3. Maps to torsion-free classes and cofinally closed monobricks
5.4. Maps to semibricks and wide subcategories
5.5. Applications
5.6. Monobricks over Nakayama algebras
5.7. Examples of computations

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104
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111
117

4.1.
4.2.
4.3.
4.4.
4.5.
4.A.

iii

iv

CONTENTS

Chapter 6. Rigid modules and ICE-closed subcategories in quiver representations
6.1. Introduction
6.2. Basic definitions and the main result
6.3. Proof of the main theorem
6.4. Maps to torsion classes and wide subcategories
6.A. Enumerative results

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Bibliography136

Abstract
In this thesis, we study various subcategories of module categories in the representation theoryof algebras. There are several classes of important subcategories of module categories which
naturally arise in the representation theory and in connection with other areas of mathematics.
Almost all of such subcategories are extension-closed, thus can be regarded as Quillen’s exact
categories. In this thesis, we study categorical properties of subcategories and give classificationsof several kinds of subcategories, using exact categories.
This thesis consists of six chapters.
In Chapter 1, the introductory chapter of this thesis, we explain general ideas behind thisthesis, including several subcategories and exact categories.
In Chapter 2, we consider the Grothendieck group of an exact category, and consider thecriterion of the additive finiteness of an exact category via the Grothendieck group. This chapter
is based on the author’s published paper [Eno3].
In Chapter 3, we establish the theory of simple objects and the Jordan-Hölder property ofexact categories. This chapter is based on the author’s paper [Eno4].
In Chapter 4, motivated by the previous chapter, we give a classification of simple objects in
torsion-free classes over preprojective algebras and path algebras of Dynkin type. This chapter isbased on the author’s paper [Eno5].
In Chapter 5, we introduce and investigate the notion of a monobrick, which enables us to
study various subcategories in a uniform way by using simple objects. This chapter is based on
the author’s paper [Eno6].
In Chapter 6, we classify ICE-closed subcategories (subcategories closed under Images, Cokernels and Extensions) over path algebras by rigid modules. This chapter is based on the author’s
paper [Eno7].

v

Acknowledgement
First and foremost, I would like to thank my advisor Osamu Iyama, for his support and many
helpful comments over the past six years. I would also like to thank Hiroyuki Nakaoka for valuable
discussions.
I would like to express my gratitude to all the administrative staffs of Graduate School ofMathematics for the financial support and the resources.
In addition, I thank some of my colleagues and friends. I’m grateful to H. Inoue for providing
me with a lot of funny and relaxing time over the past nine(!) years since I entered this university.
I also thank Y. Gyoda and A. Sakai for valuable discussions on my research and pointing out sometypos and mistakes. There are more colleagues who support me, but I omit it for want of space.
Talking and discussion with them has certainly been stimulating and encouraging my interest inmathematics, and I owe it to them that I was able to study math (almost) continuously.
Last but not least, I thank my parents for their love and support. Also I thank VirtualCast, Inc. This company provides a VR live communication service Virtual Cast. I gave somemathematical (and non-mathematical) talks in a VR space using Virtual cast, and uploaded
recordings of them. Talking in a VR space largely has been encouraging my passion in mathematics
and my life.
This work is supported by JSPS KAKENHI Grant Number JP18J21556.

vi

CHAPTER 1

Introduction
Representation theory studies how an algebraic object acts on vector spaces. It dates backto Frobenius, who studied finite groups by using their representations. Since then, finite groups,
Lie groups and Lie algebras have played leading roles in representation theory. It was not until
around 1970 that the representation theory of algebras, the main topic of this thesis, had attracted
an attention.
Let Λ be a unital associative ring. The purpose of the representation theory of Λ is tounderstand the structure of the category of Λ-modules. To get interesting results, it is common
to impose some restriction on rings and modules we consider, such as noetherian and finitely
generated. Then for a noetherian ring Λ, the category mod Λ of finitely generated right modules
is a target of our investigation, which has a nice property such as being an abelian category.
1.1. Aim of this thesis
In the representation theory of Λ, we often consider full subcategories of mod Λ, instead ofmod Λ itself. For example, subcategories of mod Λ called torsion classes, torsion-free classes andwide subcategories of mod Λ have been main objects in the representation theory of finite dimensional algebra. As we shall see later, these subcategories have interesting relation to the algebraic
Lie theory via the root system, and actually have applications in various areas of mathematics, suchas Lusztig’s crystal basis of quantum groups, Schubert varieties or unipotent cells in Kac-Moodygroups, resolutions of Kleinian singularities, and so on.
There are two main themes of this thesis. Most of the past studies on these subcategories ofmod Λ aimed at classifying nice subcategories (such as torsion classes) by some methods. However,categorical structures of a given fixed subcategory have not been investigated in the literature. Thisis the first theme of this thesis.
Along with the study of categorical structures, the author noticed that there are more otherkinds of subcategories to study which have not been considered before. Most of the past studies
concerned with torsion classes, torsion-free classes and wide subcategories of module categories.
However, there are more subcategories having interesting properties. The second theme of thisthesis is to introduce new kinds of subcategories and give classifications of them.
Let us give more details. Most of the subcategories E of mod Λ which naturally arise in the
representation theory are closed under extensions: that is, for every short exact sequence
0

L

M

N

0

of Λ-modules, we have that L, N ∈ E implies M ∈ E.
Every extension-closed subcategory of mod Λ has an additional structure called an exact category in the sense of Quillen. An exact category is a generalization of an abelian category. Moreprecisely, it is an additive category together with a class of “short exact sequences” in that category,hence it has more information than its additive structure. We can endow every extension-closed
subcategory E of mod Λ with the natural structure of an exact category (“short exact sequences”
are just usual short exact sequences in mod Λ with all terms in E). We can (and should) regard E
as exact categories, not just as additive ones.
Using the exact structure, we can consider the several invariants and properties of E, which
we will investigate, and with which we will classify several subcategories, in this thesis:• The Grothendieck group K0 (E) of E.
• The Grothendieck monoid M(E) of E.
12

1. INTRODUCTION• Simple objects in E.
• The validity of the Jordan-Hölder type theorem in E.
• Projective objects in E.
We will explain these concepts in the next section.
1.2. Summary of results
In this section, we summarize main results in each chapter.
1.2.1. Chapter 2: Grothendieck group and additive finiteness. We say that an additive category E is additively finite if there are only finitely many indecomposable objects in E up toisomorphism. From the beginning of the representation theory of algebras, one of the importantquestions is: When is mod Λ additively finite? Such an algebra Λ is called a representation-finite
algebra, and there are lots of studies on it.
As for this, the following characterization using the Grothendieck group was shown by Butler
and Auslander. The Grothendieck group K0 (mod Λ) of mod Λ is an abelian group generated
by objects in mod Λ modulo relations M = L + N whenever we have a short exact sequence
0 → L → M → N → 0 in mod Λ.
Theorem 1.2.1 ([But, Aus3]). Let Λ be a finite-dimensional algebra. Then mod Λ is additively
finite if and only if the relation of the Grothendieck group K0 (mod Λ) is generated by so-called
almost split sequences.
Here almost split sequences are special kinds of short exact sequences which are minimal insome sense.
Since we are interested in an extension-closed subcategory E of mod Λ instead of mod Λ itself,
we want to study when E is additively finite. We can define the Grothendieck group K0 (E) of E ina similar way, and the main result in Chapter 2 is the following, which extends Butler-Auslander’sresult to a wide class of subcategories of mod Λ.
Theorem 1.2.2 (= Theorem A). Let Λ be a finite-dimensional algebra and E an extensionclosed subcategory of mod Λ which is covariantly finite and resolving (see Definitions 2.4.5 and2.4.6). Then E is additively finite if and only if the relation of the Grothendieck group K0 (E) is
generated by almost split sequences in E.
This extends and unifies several known results in the literature such as [MMP, PR].
1.2.2. Chapter 3: Characterization of the Jordan-Hölder property. As everyone majoring in algebra knows, every composition series of a given module with finite length is essentiallyunique up to a permutation of composition factors. This Jordan-Hölder theorem is very classicaland fundamental result in algebra. Since a composition series of a module M can be regarded as away to build up M from simple modules by iterated extensions, this theorem says that such ways
are essentially unique.
As we have said, we are interested in an extension-closed subcategory E of a module category,instead of the whole module category. Since E has a structure of an exact category, we have a
notion of short exact sequence, thus we can define a simple object in E as follows:
Definition 1.2.3. Let E be an exact category and M a non-zero object of E. Then M iscalled simple in E if there are no short exact sequence
0

L

M

N

0

in E such that L, N 6= 0.
Roughly speaking, simple objects in E are precisely objects which cannot be decomposed into
smaller pieces with respect to short exact sequences in E. As in the classical case, we can definecomposition series in E. We say that E satisfies the Jordan-Hölder property, often abbreviated by(JHP), if the precise analogue of the Jordan-Hölder theorem holds in E. It turns out that lotsof subcategories of mod Λ do not satisfy (JHP) for a finite-dimensional algebra Λ, and the main
result in Chapter 3 is the following characterization of (JHP).

1.2. SUMMARY OF RESULTS

3

Theorem 1.2.4 (= Theorems F, 3.5.10). Let E be an exact category. Then the following are
equivalent.
(1) E satisfies (JHP).
(2) The Grothendieck monoid M(E) of E is a free commutative monoid.
If E is a subcategory of mod Λ for a finite-dimensional algebra Λ of the form ⊥ U for a module U
with finite injective dimension (see Assumption 3.5.6), then the following is also equivalent:(3) The number of indecomposable projective objects in E is equal to the number of simple
objects in E.
Here the Grothendieck monoid M(E) of an exact category E is a natural monoid version of theGrothendieck group. The assumption on E above are satisfied for various classes of subcategories
such as functorially finite torsion(-free) classes and the category of Gorenstein projective modules,
see Example 3.5.13 for the detail.
1.2.3. Chapter 4: Torsion-free classes over preprojective algebras of Dynkin type.
The previous result suggests that it should be interesting to determine simple objects in a given
extension-closed subcategory. In Chapter 4, we do this for a subcategory called a torsion-free classof a module category of a preprojective algebra of Dynkin type.
Let Λ be a finite-dimensional algebra. Then a subcategory F of mod Λ is called a torsion-freeclass if it is closed under extensions and submodules. Dually, a subcategory T of mod Λ is called a
torsion class if it is closed under extensions and quotients. These two classes of subcategories havebeen playing central roles in the representation theory of algebras, and related to various areasof mathematics. However, properties of a fixed torsion(-free) class have not been investigated indetail, which we will study in Chapter 4 for the preprojective algebra case.
First, we recall about quiver representations and some results. Let k be a field and Q a quiver,that is, a directed graph. Then a representation M of Q consists of the following data:• To each vertex v ∈ Q, we associate a finite-dimensional k-vector space Mv .
• To each arrow a : v → v 0 ∈ Q, we associate a k-linear map Ma : Mv → Mv0 .
Then we can consider the category of representations of Q. It is known that for a finite acyclic
quiver Q, this category coincides with mod kQ for a so-called path algebra kQ of Q.
As for this, the Gabriel’s theorem established a connection of the representation theory ofalgebras and the Lie theory. To state this, we use the following notation. For a Dynkin quiver Q,
we denote by Φ, Φ+ and αv for a vertex v ∈ Q, the root system of type Q, the set of positive rootsin Φ and the simple root corresponding to the Dynkin vertex v respectively. For a representation
M of Q, we put
X(dim Mv )αv ,dim M :=v∈Qand call it the dimension vector of M . Then we have the following classification.
Theorem 1.2.5 (Gabriel’s theorem). Let Q be a Dynkin quiver. Then the assignment M 7→
dim M gives a bijection between the set of isomorphism classes of indecomposable representationsof Q and the set of positive roots Φ+ .
Roughly speaking, it says that representations of a Dynkin quiver give a categorification of theroot system. By this, to an element w of the Weyl group W of Φ, we can associate the subcategory
FQ (w) of mod kQ as follows:
FQ (w) := {M ∈ mod kQ | dim M ∈ inv(w)}.
Here inv(w) is called an inversion set of w, the set of positive roots in Φ which are sent to negativeroots by w−1 . This set is a very fundamental object in the Lie theory and root systems, forexample, the cardinality of inv(w) is known to be equal to the length of w.
To a Dynkin quiver Q, we can associate another algebra ΠQ called a preprojective algebra of
Q. This algebra has some common futures with the path algebra kQ, but has more symmetries
than kQ. More precisely, the representation theory of Q heavily depends on the choice of theorientations of the underlying Dynkin graph (for example, the categories of representations of

4

1. INTRODUCTION

1 ← 2 ← 3 and 1 → 2 ← 3 are not equivalent). However, ΠQ does not depend on the orientations.
Moreover, ΠQ is something like an enhancement of kQ, which means that the representation
theory of kQ are included in ΠQ . Thus the representation theory of ΠQ uniformly treats therepresentation theory of Dynkin quivers with all orientations at the same time.
Now we have the following Gabriel type theorem for preprojective algebras.
Proposition 1.2.6. Let Q be a Dynkin quiver and ΠQ a preprojective algebra. Then thedimension vector of every brick is a positive root of the corresponding root system.
Here a brick is a module whose endomorphism ring is a division ring. For example, every
indecomposable kQ-module is a brick. Although there exist infinitely many indecomposable ΠQ modules for almost all Dynkin quivers Q, there are only finitely many bricks, thus bricks in mod ΠQ
are more appropriate analogues of indecomposable kQ-modules.
Then [BIRS] defined a subcategory F(w) of mod ΠQ , which is an analogue of FQ (w) ⊂ mod kQ
above. In fact, dim B ∈ inv(w) holds for every brick B contained in F(w). Moreover, this category
also plays an important role in the theory of cluster algebras, since it categorifies the cluster
structure of the coordinate ring of a unipotent cell in the algebraic group by [GLS].
The following result of Mizuno is the starting point of Chapter 4.
Theorem 1.2.7 ([Miz, Theorem 2.30]). Let Q be a Dynkin quiver. Then the assignment
w 7→ F(w) gives a bijection between the following two sets:(1) The Weyl group W of the associated root system.
(2) The set of torsion-free classes in mod ΠQ .
Thus torsion-free classes in mod ΠQ are parametrized by elements of the Weyl group. Then
it is natural to expect that there is a relation between (exact-)categorical properties of F(w) andLie-theoretical property of w. In this direction, we obtain the following main result of Chapter 4.
Theorem 1.2.8 (= Theorems J, L). Let Q be a Dynkin quiver and w an element of thecorresponding Weyl group. Then by taking dimension vectors, we have a bijection between thefollowing two sets.
(1) The set of isomorphism classes of simple objects in F(w).
(2) The set of Bruhat inversions of w.
Moreover, F(w) satisfies (JHP) if and only if the number of supports of w is equal to the numberof Bruhat inversions of w. In addition, the same statement holds for FQ (w) ⊂ mod kQ instead ofF(w).
Here Bruhat inversions are inversions of w which cannot be written as a sum of smaller
inversions of w. This gives a complete classification of simple objects and the validity of (JHP) in
F(w) purely by using Lie theoretic language.
1.2.4. Chapter 5: Monobricks and new class of subcategories. As we mentioned in theprevious subsection, torsion classes and torsion-free classes have been central objects in the historyof the representation theory of algebras. Recently, Adachi-Iyama-Reiten gave a classification of
torsion-free classes using injective objects of them. More precisely, they showed that functoriallyfinite torsion-free classes in mod Λ for a finite-dimensional algebra Λ are in bijection with support
τ − -tilting Λ-modules. Here the functorial finiteness is some restriction on a torsion-free class F
which ensures that F has enough injective objects.
As we have seen, the set of simple objects of F, instead of projective or injective objects,
seems to be a new interesting object to study. Motivated by this idea, we succeed in classifying
all torsion-free classes in any length abelian category, which is one of the main results in Chapter
5:
Theorem 1.2.9 (= Theorem N). Let A be a length abelian category. Then taking simple
objects yields a bijection between the following two sets.
(1) The set of all torsion-free classes in A.
(2) The set of cofinally closed monobricks in A.

1.2. SUMMARY OF RESULTS

5

Here monobricks is a set M of bricks in A such that every non-zero morphism between objects
in M is an injection, and the cofinal closedness is a kind of poset-theoretical closedness condition,
see Definition 5.3.2.
Actually, we establish a bijection which includes the above bijection. A special well-knownclass of monobricks is a class of semibricks, a set of bricks such that every morphism between themis a zero morphism. It is a classical result that semibricks are in bijection with wide subcategories
of A, exact abelian subcategories closed under extensions.
We introduce the concept of left Schur subcategory of A, which unifies both torsion-free classes
and wide subcategories. Then we show the following bijection.
Theorem 1.2.10 (= Theorem O). Let A be a length abelian category. Then taking simple
objects yields a bijection between the following two sets:(1) The set of left Schur subcategories of A.
(2) The set of monobricks in A.
Moreover, this bijection restricts to the following bijections between• the set of wide subcategories of A and the set of semibricks in A, and• the set of torsion-free classes in A and the set of cofinally closed monobricks in A.
One of the benefits of this result is that the relation between torsion-free classes and wide
subcategories becomes more transparent by using monobricks. Each monobrick can be regarded
as a poset in a natural way, and this poset structure provides us a powerful tool to study thesetwo kinds of subcategories.
For example, it was shown in [MS] that if there are only finitely many torsion-free classes,
we have a bijection between the sets of torsion-free classes and wide subcategories. The proof in[MS] heavily depends on [AIR], and does not work in an length abelian category in general. In
contrast, we can quickly deduce it by using only some poset theoretical argument on monobricks(Theorem 5.5.4).
1.2.5. Chapter 6: ICE-closed subcategories in quiver representations. In the previous chapter, we introduced left Schur subcategories of an abelian category A. A typical example
is a subcategory closed under images, kernels and extensions. Motivated by this, in Chapter 6, wegive a classification of such subcategories in the category of representations of a Dynkin quiver Q,
or equivalently, mod kQ.
For the compatibility of [AIR] and [IT], we work with the dual notion ICE-closed subcategories of mod kQ, which are subcategories of mod kQ closed under taking Images, Cokernels andExtensions. In particular, torsion classes are ICE-closed, and Ingalls and Thomas classified themby support tilting modules in [IT] using projective objects in torsion classes. In Chapter 6, we
extend their classification to include all ICE-closed subcategories of mod kQ as follows:
Theorem 1.2.11 (= Theorem R). Let Q be a Dynkin quiver. Then taking projective objects
yields a bijection between the following two sets:(1) The set of ICE-closed subcategories of mod kQ.
(2) The set of isomorphism classes of rigid kQ-modules, that is, a kQ-module M satisfying
Ext1kQ (M, M ) = 0.
This bijection extends a bijection between the set of torsion classes in mod kQ and the set of
support tilting kQ-modules established by [IT].
Moreover, we show that the number of ICE-closed subcategories of mod kQ depends only on
the underlying Dynkin graph of Q, and does not depend on the choice of orientations of arrows.
We give an explicit formula of this number for each Dynkin type. In particular, for type A case,this number coincides with the combinatorial number known as the large Schröder number.
The representation theory of Dynkin quiver is the most basic, classical and fundamental object
which motivated research at the down of the representation theory of algebras, and rigid modules
are also quite fundamental and have appeared in many contexts. Therefore, it is a bit surprising
to the author that this result has not been observed in the literature.

CHAPTER 2

Relations for Grothendieck groups andrepresentation-finitenessThis chapter is based on the published paper [Eno3].
For an exact category E, we study the Butler’s condition “AR=Ex”: the relation of theGrothendieck group of E is generated by Auslander-Reiten conflations. Under some assumptions,
we show that AR=Ex is equivalent to that E has finitely many indecomposables. This can beapplied to functorially finite torsion(free) classes and contravariantly finite resolving subcategoriesof the module category of an artin algebra, and the category of Cohen-Macaulay modules over anorder which is Gorenstein or has finite global dimension. Also we showed that under some weakerassumption, AR=Ex implies that the category of syzygies in E has finitely many indecomposables.
2.1. Introduction
Let Λ be a finite-dimensional k-algebra over a field k. To the abelian category mod Λ of finitely
generated Λ-modules, we can associate an abelian group K0 (mod Λ) called the Grothendieck group.
This is the quotient group K0 (mod Λ, 0)/ Ex(mod Λ), where K0 (mod Λ, 0) denotes the free abeliangroup with the basis the set of isomorphism classes [X] of indecomposable objects X ∈ mod Λ,and Ex(mod Λ) the subgroup generated by [X] − [Y ] + [Z] for every exact sequence 0 → X → Y →Z → 0.
Auslander-Reiten sequences, or AR sequences for short, are special kind of short exact sequences in mod Λ, which are “minimal” in some sense. We denote by AR(mod Λ) the subgroup ofEx(mod Λ) generated by AR sequences. Then Butler and Auslander proved the following result.
Theorem 2.1.1 ([But, Aus3]). Let Λ be a finite-dimensional algebra. Then AR(mod Λ) =Ex(mod Λ) holds if and only if there are only finitely many indecomposable objects in mod Λ up toisomorphism.
Auslander conjectured that similar results hold for a more general class of categories otherthan mod Λ. In this paper, we study this conjecture in the context of Quillen’s exact categories.
For a Krull-Schmidt exact category E, the Grothendieck group K0 (E) = K0 (E, 0)/ Ex(E) of E isdefined in the same way. Also the notion of AR sequences is defined over E, which we call AR
conflations. Thus we have the subgroup AR(E) of Ex(E), and the aim of this paper is to study thefollowing question.
Question 2.1.2. For a Krull-Schmidt exact category E, when are the following equivalent?
(1) E is of finite type, that is, there exist only finitely many indecomposable objects in E up
to isomorphism.
(2) AR(E) = Ex(E) holds.
Let us mention some known results on this question. For the implication (1) ⇒ (2), we havethe following quite general result by the author:
Theorem 2.1.3 ([Eno2, Theorem 3.18], Corollary 2.4.4). Let R be a complete noetherian local
ring and E a Krull-Schmidt exact R-category such that E(X, Y ) is a finitely generated R-module
for each X, Y ∈ E. If E has enough projectives and is of finite type, then AR(E) = Ex(E) holds.
This can be applied to various situations in the representation theory of noetherian algebras.
We provide a proof of this in Corollary 2.4.4.
6

2.1. INTRODUCTION

7

On the other hand, the implication (2) ⇒ (1) is more subtle. Indeed, some counter-examples
are known, e.g. [MMP, Section 3]. However, there are affirmative results for this implication
in some concrete situations: E = CM Λ for an R-order Λ with finite global dimension [AR1,Proposition 2.3], and the category of modules with good filtrations over standardly stratified
algebras [MMP, PR]. Also some partial results are known ([Hir, Kob]).
In this paper, we first give a sufficient condition for (1) and (2) in Question 2.1.2 to beequivalent, by using a functorial method. For a Krull-Schmidt exact category E, we consider the
category Mod E of functors E op → Ab. There is an important subcategory eff E of Mod E, thecategory of effaceable functors (Definition 2.2.1), which nicely reflects the exact structure of E andis closely related to the group Ex(E). Then let us consider the following conditions:(a) E is of finite type.
(b) E is admissible [Eno2], that is, every object in eff E has finite length in Mod E.
(c) AR(E) = Ex(E) holds.
Simple modules contained in eff E bijectively correspond to AR conflations in E (Proposition 2.2.3).
Thus the condition (b) can be regarded as a functorial analogue of (c), hence it is somewhat closer
to (c) than (a) is. The relation of these conditions are summarized in Figure 1.
Secondly, we apply this condition to concrete situations and obtain results on Question 2.1.2.
For an artin algebra, we show the following result, which extends the results [MMP, Theorem 3]and [PR, Theorem 3.6].
Theorem A (= Theorem 2.4.11). Let Λ be an artin algebra and E a contravariantly finite
resolving subcategory of mod Λ. Then the following are equivalent.
(1) E is of finite type.
(2) AR(E) = Ex(E) holds.
When R is not artinian, we prove the following result. This extends [Hir, Kob] to the noncommutative case, and [AR1, Proposition 2.3] to Gorenstein orders. Also our method provides
another proof of [AR1, Proposition 2.3].
Theorem B (= Theorem 2.4.15, Corollary 2.5.3). Let R be a complete Cohen-Macaulay local
ring and Λ an R-order with at most an isolated singularity. Suppose that Ex(CM Λ) = AR(CM Λ)
holds. Then the following hold.
(1) ΩCM Λ, the category of syzygies of Cohen-Macaulay Λ-modules, is of finite type.
(2) If Λ has finite global dimension or Λ is a Gorenstein order, then CM Λ is of finite type.
Actually, most of the results above are deduced from the following relations between (a), (b)and (c):Theorem C (Proposition 2.3.3, Theorem 2.3.7). Let E be a Krull-Schmidt exact R-category
with a projective generator over a commutative noetherian ring R. Then the implications in Figure
1 hold.
E is Hom-finite(a) E is of finite type(b) E is admissible

E has a weak cogenerator(c) AR(E) = Ex(E)E satisfies (CF)

Figure 1. Summary of Section 2.3
Here we introduce the condition (CF) on exact categories concerning the Grothendieck groups(see Definition 2.3.5).
The paper is organized as follows. In Section 2.2, we introduce basic definitions and collectsome results about Grothendieck group and effaceable functors. In Section 2.3, we introduce someconditions on E and prove the relations indicated in Figure 1. In Section 2.4, we verify thatthe conditions are satisfied in some concrete cases. In Section 2.5, we study the finiteness of the

8

2. RELATIONS FOR GROTHENDIECK GROUPS AND REPRESENTATION-FINITENESS

category of syzygies. In Appendix A, we provide some interesting properties of the category of
effaceable functors.
2.1.1. Notation and conventions. Throughout this paper, we assume that all categories
and functors are additive and all subcategories are additive, full and closed under isomorphisms.
For an object G of an additive category E, we denote by add G the category consisting of all direct
summands of finite direct sums of G. We say that an additive category is of finite type if there
exists an object G ∈ E such that E = add G holds.
For a Krull-Schmidt category E, we denote by ind E the set of isomorphism classes of indecomposable objects in E. We refer the reader to [Krau] for the basics of Krull-Schmidt categories. We
say that an additive category E is an idempotent complete if every idempotent morphism splits.
It is well-known that a Krull-Schmidt category is idempotent complete.
As for exact categories, we use the terminologies inflations, deflations and conflations. We
refer the reader to [Büh] for the basics of exact categories. We say that an exact category E has
a projective generator P if E has enough projectives add P .
2.2. Preliminaries
First we introduce the basic definition about Auslander-Reiten theory in a Krull-Schmidt
category. Let E be a Krull-Schmidt category and J its Jacobson radical. A morphism g : Y → Z
in E is called right almost split if Z is indecomposable, g is in J and any morphism h : W → Z
in J factors through g. Dually we define left almost split.
fg
For a Krull-Schmidt exact category E, we say that a conflation 0 → X −→Y −→ Z → 0 in E isan AR conflation if f is left almost split and g is right almost split. We say that an indecomposable
object Z in E admits a right AR conflation if there exists an AR conflation ending at Z. We saythat E has right AR conflations if every indecomposable non-projective object admits a right AR
conflation. Left AR conflations are defined dually, and we say that E has AR conflations if it has
both right and left AR conflations.
Next we introduce some notation concerning the L
Grothendieck group. For a Krull-Schmidt
exact category E, let K0 (E, 0) be a free abelian group [X]∈ind E Z · [X] generated by the set ind E
of isomorphism classes of indecomposable objects in E, which we identify with the Grothendieckgroup of E with the split exact structure, that is, the exact structure where all conflations are split
exact. We denote by Ex(E) the subgroup of K0 (E, 0) generated by{[X] − [Y ] + [Z] | there exists a conflation X  Y  Z in E}.
We call the quotient group K0 (E) := K0 (E, 0)/ Ex(E) the Grothendieck group of E. We denote by
AR(E) the subgroup of Ex(E) generated by{[X] − [Y ] + [Z] | there exists an AR conflation X  Y  Z in E}.
Throughout this paper, we make use of functor categorical arguments. Let us recall the
related concepts. For an additive category E, a right E-module M is a contravariant additive
functor M : E op → Ab from E to the category of abelian groups Ab. We denote by Mod E thecategory of right E-modules, and morphisms are natural transformations between them. Thenthe category Mod E is a Grothendieck abelian category with enough projectives, and projective
objects are precisely direct summands of (possibly infinite) direct sums of representable functors.
We denote by mod E the category of finitely presented E-modules, that is, the modules M
such that there exists an exact sequence E(−, X) → E(−, Y ) → M → 0 for some X, Y in E. This
category mod E is not necessarily an abelian category, but mod E is closed under extensions in
Mod E by the horseshoe lemma. Thereby we always regard mod E as an exact category, which hasenough projectives in the sense of exact category. Moreover, if E is idempotent complete, thenprojectives in mod E are precisely representable functors.
Now let us introduce the notion of effaceability of E-modules, which plays an essential role
throughout this paper. This notion was originally introduced by Grothendieck [Gro, p. 141].

2.2. PRELIMINARIES

9Definition 2.2.1. Let E be an exact category. A right E-module M is called effaceable ifE(−,f )

there exists a deflation f : Y  Z in E such that E(−, Y ) −−−−→ E(−, Z) → M → 0 is exact. We
denote by eff E the subcategory of mod E consisting of effaceable functors.
The category eff E plays a quite essential role to study exact categories in a functorial method.
This definition for exact categories is due to [Fio], and the category eff E plays an importantrole in [Eno2] to classify exact structures on a given additive category. For the compatibility ofGrothendieck’s effaceability, see Proposition 2.A.1.
In the rest of this section, we collect basic properties about functor categories and effaceable
functors which we need in the sequel.
Proposition 2.2.2. Let E be a Krull-Schmidt category and J its Jacobson radical. Put
SZ := E(−, Z)/J (−, Z) for Z ∈ E. Then the following hold.
(1) The map Z 7→ SZ gives a bijection between ind E and the set of isomorphism classes ofsimple E-modules.
(2) A morphism g : Y → Z in E is right almost split if and only if Z is indecomposable andE(−,g)

E(−, Y ) −−−−→ E(−, Z) → SZ → 0 is exact.
Proof. It is well-known, see [Aus2] for example.

In particular, simple E-modules contained in mod E bijectively correspond to indecomposable
objects in E which admit right almost split maps. Moreover if E is an exact category, then one
can show more on eff E as follows.
Proposition 2.2.3. Let E be a Krull-Schmidt exact category and Z an indecomposable object
in E. Then the following are equivalent.
(1) Z admits a right AR conflation.
(2) SZ belongs to eff E.
(3) SZ is finitely presented and Z is non-projective.
In particular, simple E-modules contained in eff E bijectively correspond to indecomposables in E
which admit right AR conflations.
Proof. (1) ⇒ (2): We have a deflation g : Y  Z which is right almost split. Thus
SZ ∼= Coker E(−, g) belongs to eff E.
(2) ⇒ (3): Since SZ is in eff E, it is finitely presented. By [Eno2, Propositions 2.11], we havethat Z is non-projective since SZ (Z) 6= 0 holds.
(3) ⇒ (1): This is precisely [Eno2, Proposition A.3].

The following says that the category eff E has nice properties and controls the exact structureof E. Here we say that an E-module M is finitely generated if there exists a surjection from a
representable functor onto M .
Proposition 2.2.4. Let E be an idempotent complete exact category. Then the following
holds.
(1) Suppose that there exists an exact sequence 0 → M1 → M → M2 → 0 in Mod E and that
M1 is finitely generated. Then M is in eff E if and only if both M1 and M2 are in eff E.
fg(2) A complex X −→Y −→ Z in E is a conflation if and only if there exists an exact sequenceE(−,f )

E(−,g)

0 → E(−, X) −−−−→ E(−, Y ) −−−−→ E(−, Z) → M → 0
in Mod E with M in eff E.
Proof. We refer the reader to [Eno2, Proposition 2.10] for the proof of (1), and [Eno2,
Theorem 2.7] or the proof of Lemma 2.A.1 for (2).

The category eff E enjoys further nice properties, e.g. being abelian, and we refer the interested
reader to Appendix A.

10

2. RELATIONS FOR GROTHENDIECK GROUPS AND REPRESENTATION-FINITENESS

If E has enough projectives, then one has a simpler description of the category eff E. Let E
be an exact category with enough projectives and denote by P the subcategory of E consisting of
projective objects. We define the stable category E by the ideal quotient E/[P]. Then eff E can benaturally identified with the module category of E, which we omit the proof.
Lemma 2.2.5 ([Eno2, Lemma 2.13]). Let E be an exact category with enough projectives P.
Then the natural restriction functor Mod E → Mod E induces an equivalence mod E ' eff E.
2.3. General conditions for (a), (b) and (c) to be equivalent
In this section, we will provide some general results on the three conditions: (a) E is of finite
type, (b) E is admissible and (c) AR(E) = Ex(E). Here we say that E is admissible if every object in
eff E has finite length in Mod E ([Eno2]). See Proposition 2.A.3 for the equivalent condition of theadmissibility. Then our results in this section can be summarized in Figure 1 in the introduction.
To begin with, let us introduce some terminologies on R-categories needed in this paper.
Let R be a commutative noetherian ring and E an R-category. We say that an R-category E is
Hom-noetherian (resp. Hom-finite) if E(X, Y ) is finitely generated (resp. has finite length) as an
R-module for each X, Y in E.
To deal with admissibility, it is convenient to introduce the following criterion for deciding
whether the functor has finite length or not. Let E be a Krull-Schmidt category and M ∈ Mod E
a right E-module. Then the support supp M of M is defined bysupp M := {X ∈ ind E | M (X) 6= 0}.
The Hom-finiteness ensures that finiteness of modules can be checked only by considering theirsupport:Lemma 2.3.1. Let E be a Krull-Schmidt category and M ∈ mod E an E-module. If M has finite
length in Mod E, then supp M is a finite set. The converse holds if E is Hom-finite R-category.
Proof. See, for example, [Aus2, Theorem 2.12].



2.3.1. (a) Representation-finiteness and (b) admissibility. First, we consider the relation between: (a) E is of finite type, and (b) E is admissible.
Definition 2.3.2. Let C be an additive category. We say that an object C ∈ C is a weakcogenerator if C(X, C) = 0 implies that X is a zero object for every X ∈ C.
If we assume the existence of a weak cogenerator of the stable category, then (a) is almost
equivalent to (b).
Proposition 2.3.3. Let E be a Krull-Schmidt exact category with a projective generator.
Consider the following conditions:(1) E is admissible and E has a weak cogenerator.
(2) E is of finite type.
Then (1) ⇒ (2) holds. The converse holds if E is a Hom-finite R-category for a commutative
noetherian ring R.
Proof. (1) ⇒ (2): Since E has enough projectives, we may identify eff E with mod E by
Lemma 2.2.5. Let X be a weak cogenerator of E. By the assumption, we have that E(−, X) has
finite length. This implies that supp E(−, X) is finite by Lemma 2.3.1. Since every indecomposable
object W in E should satisfy E(W , X) 6= 0, it follows that E is of finite type. Now E itself is offinite type because we have a natural identification ind E = ind E t ind(add P ).
(2) ⇒ (1): Suppose that E is of finite type and E is Hom-finite over R. Then obviously the
direct sum of all indecomposables in E is a weak cogenerator of E. Moreover, since E is Hom-finite
and ind E is finite, every object in eff E = mod E has finite length by Lemma 2.3.1.
2.3. GENERAL CONDITIONS FOR (A), (B) AND (C) TO BE EQUIVALENT

11

2.3.2. (b) Admissibility and (c) AR=Ex. Now let us investigate the relation between
the condition (b) and (c). First we show that if (b) E is admissible, then (c) AR(E) = Ex(E) holds.
Next we will introduce the condition (CF) and show that the converse implication holds under it.
Proposition 2.3.4. Let E be a Krull-Schmidt exact category. Suppose that E is admissible.
Then Ex(E) = AR(E) holds.
Proof. This is proved in the similar way as [Eno2, Theorem 3.17]. We provide a proof here
for the convenience of the reader.
f1
g1
Let 0 → X1 −→ Y1 −→ Z1 → 0 be a conflation in E and we have to show [X1 ] − [Y1 ] + [Z1 ] ∈AR(E). Put M := Coker E(−, g1 ) in Mod E, then M is in eff E by the definition. Suppose thatf2
g2
there exists another conflation 0 → X2 −→ Y2 −→ Z2 → 0 in E such that M ∼= Coker E(−, g2 ).
Then we have exact sequencesE(−,fi )E(−,gi )0 → E(−, Xi ) −−−−−→ E(−, Yi ) −−−−−→ E(−, Zi ) → M → 0
in Mod E for i = 1, 2. Thus Schanuel’s lemma shows that E(−, X1 ) ⊕ E(−, Y2 ) ⊕ E(−, Z1 ) ∼=E(−, X2 ) ⊕ E(−, Y1 ) ⊕ E(−, Z2 ) in Mod E, hence X1 ⊕ Y2 ⊕ Z1 ∼= X2 ⊕ Y1 ⊕ Z2 in E by the Yoneda
lemma. This implies that [X1 ] − [Y1 ] + [Z1 ] = [X2 ] − [Y2 ] + [Z2 ] in K0 (E, 0). Thus it suffices toshow the following claim.
Claim: For any M ∈ eff E, there exists at least one exact sequence in Mod E
E(−,f )

E(−,g)

0 → E(−, X) −−−−→ E(−, Y ) −−−−→ E(−, Z) → M → 0
with [X] − [Y ] + [Z] ∈ AR(E). Note that such a complex 0 → X → Y → Z → 0 is automaticallya conflation by Proposition 2.2.4(2).
We will show this claim by induction on l(M ), the length of M as an E-module. Suppose thatl(M ) = 1, that is, M is a simple E-module. We may assume that M = SZ for an indecomposablefg

object Z ∈ E by Proposition 2.2.2, and there exists an AR conflation 0 → X −→ Y −→ Z → 0E(−,f )

in E by Proposition 2.2.3. This gives the desired projective resolution 0 → E(−, X) −−−−→
E(−,g)

E(−, Y ) −−−−→ E(−, Z) → SZ → 0 of SZ , and [X] − [Y ] + [Z] ∈ AR(E) holds.
Now suppose that l(M ) > 1. Take a simple E-submodule M1 of M . Then we have an exact
sequence 0 → M1 → M → M2 → 0 in Mod E. Since M1 is finitely generated, Proposition 2.2.4(1)shows that M1 and M2 are both in eff E. By induction hypothesis, we have an exact sequenceE(−,fi )E(−,gi )0 → E(−, Xi ) −−−−−→ E(−, Yi ) −−−−−→ E(−, Zi ) → Mi → 0 with [Xi ] − [Yi ] + [Zi ] ∈ AR(E) for
each i = 1, 2. By the horseshoe lemma, we obtain a projective resolutionE(−,f )

E(−,g)

0 → E(−, X1 ⊕ X2 ) −−−−→ E(−, Y1 ⊕ Y2 ) −−−−→ E(−, Z1 ⊕ Z2 ) → M → 0.
Then we have
[X1 ⊕ X2 ] − [Y1 ⊕ Y2 ] + [Z1 ⊕ Z2 ] =X([Xi ] − [Yi ] + [Zi ]) ∈ AR(E).

i=1,2

Thus the claim follows.



Next let us investigate when the converse of Proposition 2.3.4 holds. The author does notknow whether this always holds, but we show that this is the case for particular cases. To thispurpose, let us introduce the condition (CF), which stands for the Conservation of Finiteness.
Definition 2.3.5. For an exact category E, we say that E satisfies (CF) if it satisfies thefollowing:figi(CF) Let 0 → X −→ Y −→ Z → 0 be two conflations in E for i = 1, 2 and put Mi :=Coker E(−, gi ) ∈ eff E. If M1 has finite length in Mod E, then so does M2 .
For later purposes, the following equivalent condition is more useful.
Lemma 2.3.6. For an exact category E, the condition (CF) is equivalent to the following
condition.

12

2. RELATIONS FOR GROTHENDIECK GROUPS AND REPRESENTATION-FINITENESS
fi

gi(CF0 ) Let 0 → Xi −→ Yi −→ Zi → 0 be conflations in E for i = 1, 2 and put Mi := Coker E(−, gi )in eff E. Suppose that [X1 ] − [Y1 ] + [Z1 ] = [X2 ] − [Y2 ] + [Z2 ] holds in K0 (E, 0). If M1 has
finite length in Mod E, then so does M2 .
Proof. Obviously it suffices to show that (CF) implies (CF0 ). Since [X1 ] − [Y1 ] + [Z1 ] =[X2 ] − [Y2 ] + [Z2 ] holds in K0 (E, 0), we have that X2 ⊕ Y1 ⊕ Z2 ∼= X1 ⊕ Y2 ⊕ Z1 holds in E by the
Krull-Schmidtness of E. Now consider the following two complexes: 0 1
f1 0
0 0h

1 0 
0 f2
0 0h

0 g1 0
0 0 1

i

0 0 1
0 g2 0i

0 → X1 ⊕ X2 −−−−−→ X2 ⊕ Y1 ⊕ Z2 −−−−−−→ Z1 ⊕ Z2 → 0,
0 → X1 ⊕ X2 −−−−−→ X1 ⊕ Y2 ⊕ Z1 −−−−−−→ Z1 ⊕ Z2 → 0.
These are conflations since they are direct sums of conflations and split exact sequences. Applying(CF) to them, it is easy to check that (CF0 ) holds.
Now we can state the following main result in this section.
Theorem 2.3.7. Let E be a Krull-Schmidt exact category. Then the following are equivalent.
(1) E is admissible.
(2) Ex(E) = AR(E) holds and E satisfies (CF).
(3) Ex(E) ⊗Z Q = AR(E) ⊗Z Q holds and E satisfies (CF).
Proof. (1) ⇒ (2): If every object in eff E has finite length, then we have Ex(E) = AR(E) by
Proposition 2.3.4, and the condition (CF) is trivially satisfied.
(2) ⇒ (3): Obvious.
fg(3) ⇒ (1): Let M be in eff E and take a conflation 0 → X −→Y −→ Z → 0 in E such thatE(−,f )

E(−,g)

0 → E(−, X) −−−−→ E(−, Y ) −−−−→ E(−, Z) → M → 0 is exact. By AR(E) ⊗Z Q = Ex(E) ⊗Z Q,
gi
there exist a positive integer a, integer bi and an AR conflation 0 → Xi → Yi −→ Zi → 0 for eachi such that we have an equality in K0 (E, 0):nXbi ([Xi ] − [Yi ] + [Zi ]).
a([X] − [Y ] + [Z]) =i=1

We may assume that bi > 0 for i ≤ m and −ci := bi < 0 for i > m. Thus we have an equalitynmX
Xa([X] − [Y ] + [Z]) +ci ([Xi ] − [Yi ] + [Zi ]) =bi ([Xi ] − [Yi ] + [Zi ])(2.3.1)i=m+1

i=1

in K0 (E, 0) such that all the coefficients are positive integers. Thus each side of the equation (2.3.1)
comes from the single conflation in E. Put Si := Coker E(−, gi ) ∈ mod E for each i, which
Lnis simple
E-module contained in eff E. Then the left hand side in (2.3.1) corresponds to M a ⊕ i=m+1 SiciLmin eff E and the right hand side corresponds to i=1 Sibi in eff E. Since the latter has finite length,
the condition (CF0 ) implies that so does the former. Thus M has finite length.
2.4. Applications
In Section 2.3, we have shown the equivalence of (a), (b) and (c) under three technical assumptions: Hom-finiteness of E, the existence of a weak cogenerator of E, and the condition (CF)(see Figure 1 for a summary of these results). In this section, we show that these assumptions are
satisfied for some concrete cases.
Main applications we have in mind are the representation theory of noetherian algebras, where
the base ring is not necessarily artinian. For the convenience of the reader, we recall some related
definitions.
Definition 2.4.1. Let R be a noetherian local ring and Λ an R-algebra.
(1) Λ is called a noetherian R-algebra if Λ is finitely generated as an R-module. If in addition
R is artinian, then we say that Λ is an artin R-algebra.

2.4. APPLICATIONS

13(2) Suppose that R is Cohen-Macaulay. For a noetherian R-algebra, we denote by CM Λ
the category of finitely generated Λ-modules which are maximal Cohen-Macaulay as
R-modules.
(3) A noetherian R-algebra Λ is called an R-order if Λ ∈ CM Λ holds.
(4) We say that an R-order Λ has at most an isolated singularity if gl.dim Λp = ht p holds
for every non-maximal prime p ∈ Spec R.
(5) An R-order is called a Gorenstein order if CM Λ is a Frobenius exact category.
Let R be a complete Cohen-Macaulay local ring and Λ an R-order. Then CM Λ is closed under
extensions in mod Λ, thus is an exact category with a projective generator Λ. Moreover, CM Λ has
AR conflations if and only if Λ has at most an isolated singularity [Aus4].
2.4.1. Hom-finiteness of the stable category. First we consider the Hom-finiteness of E.
Recall that (a) ⇒ (b) holds if E is Hom-finite (Proposition 2.3.3). This condition is trivial if R isartinian, hence we obtain the following.
Corollary 2.4.2. Let E be a Hom-finite idempotent complete exact R-category over a commutative artinian ring R. Suppose that E is of finite type. Then E is admissible and AR(E) = Ex(E)holds.
For an artin algebra Λ, we can apply this result to any subcategory E of mod Λ which is closed
under extensions and direct summands.
If R is not artinian, we can use the following fact on Hom-finiteness.
Lemma 2.4.3 ([Eno2, Proposition A.5]). Let R be a complete noetherian local ring and E a
Hom-noetherian idempotent complete exact R-category. Suppose that E has enough projectives andconsider the following conditions:(1) E is of finite type.
(2) E has AR conflations.
(3) E is Hom-finite over R.
Then the implication (1) ⇒ (2) ⇒ (3) holds.
Thus we immediately obtain the following.
Corollary 2.4.4 ([Eno2, Corollary 3.18]). Let R be a complete noetherian local ring and E
a Hom-noetherian idempotent complete exact R-category. Suppose that E has enough projectives
and E is of finite type. Then E is admissible and AR(E) = Ex(E) holds.
Note that the assumption on E here is rather mild. In particular, this corollary can be applied
to E := CM Λ for an R-order Λ which is CM-finite, that is, CM Λ is of finite type. Thus thisprovides a further generalization of [Aus3, Proposition 2.2].
2.4.2. Existence of a weak cogenerator of the stable category. For the implication(b) ⇒ (a), we need a weak cogenerator of E (Proposition 2.3.3). To this purpose, let us recall the
notions of functorial finiteness and resolving subcategories.
Definition 2.4.5. Let A be an additive category and E an additive subcategory of A.
(1) A morphism f : EX → X in A is said to be a right E-approximation if EX is in E andevery morphism E → X with E ∈ D factors through f .
(2) E is said to be contravariantly finite if every object in A has a right E-approximation.
A left E-approximation and covariantly finiteness are defined dually. We say that E is functoriallyfinite if it is both contravariantly and covariantly finite.
Definition 2.4.6. Let Λ be a noetherian ring and E a subcategory of mod Λ. We say that E
is a resolving subcategory of mod Λ if it satisfies the following conditions:(1) E contains Λ.
(2) E is closed under extensions, that is, for each exact sequence 0 → X → Y → Z → 0, if
both X and Z are in E, then so is Y .

14

2. RELATIONS FOR GROTHENDIECK GROUPS AND REPRESENTATION-FINITENESS(3) E is closed under kernels of surjections, that is, for each exact sequence 0 → X → Y →Z → 0, if both Y and Z are in E, then so is X.
(4) E is closed under summands.
Since each resolving subcategory is an extension-closed subcategory of an abelian category, it
has the natural exact structure. Thereby we always regard it as an exact category.
The following result provides a rich source of exact categories such that its stable category
has a weak cogenerator.
Proposition 2.4.7. Let Λ be a semiperfect noetherian ring and E a contravariantly finite
resolving subcategory of mod Λ. Then E has a weak cogenerator, which can be chosen as a rightE-approximation of Λ/ rad Λ.
Proof. Since E is resolving in mod Λ, it is straightforward to see that E has a projective
generator Λ. Let us construct a weak cogenerator X of E (cf. [AR1, Lemma 2.4]). Since Λ issemiperfect, Λ/ rad Λ is a direct sum of all simple Λ-modules up to multiplicity. Take a rightE-approximation f : X → Λ/ rad Λ of Λ/ rad Λ, and we claim that X is a weak cogenerator of E.
Suppose that E(W , X) = 0. Then it follows from this that every morphism from W to finitely
generated semisimple modules factors through some projective module. We will show that W isprojective, that is, W = 0 in E.
For a Jacobson radical rad W of W , let π : W  W/ rad W be a natural projection and takea projective cover p : P  W/ rad W (this is possible since Λ is semiperfect). It follows that there
exists a projective cover ϕ : P  W such that the following diagram commutes:
ϕ

P
p

W
π

ψ

P
p

W/ rad W
Since W/ rad W is semisimple, π must factor through some projective module. Thus there exists
ψ : W → P which makes the above diagram commute. On the other hand, since p is right minimal,ψ ◦ ϕ must be an isomorphism. It follows that ϕ is an isomorphism, thus W is projective.

Now one immediately obtains the following result about the equivalence of (a) and (b) for acontravariantly finite resolving subcategories over noetherian algebras. We will treat the case ofartin algebras later (see Theorem 2.4.11) since one can prove more.
Corollary 2.4.8. Let R be a complete noetherian local ring, Λ a noetherian R-algebra andE a contravariantly finite resolving subcategory of mod Λ. Then E is of finite type if and only if E
is admissible.
Proof. There exists a weak cogenerator of E by Proposition 2.4.7 since Λ is semiperfect.
Thus E is of finite type if E is admissible by Proposition 2.3.3. The other implication follows from
Corollary 2.4.4.

Example 2.4.9. Let R be a complete Cohen-Macaulay local ring and Λ an R-order. Then it
is well-known that CM Λ is a contravariantly finite resolving subcategory of mod Λ (e.g. [AuBu]).
In particular, CM Λ has a weak cogenerator by Proposition 2.4.7 and Corollary 2.4.8 applies toCM Λ. Thus Corollary 2.4.8 generalizes the result [Aus3, Lemma 2.4 (b)] on the category CM Λ.
2.4.3. On the condition (CF). Recall that in Section 2.3.2, we have investigated the relation between (b) admissibility and (c) AR=Ex, and Theorem 2.3.7 shows that the condition (CF)ensures that (b) and (c) are equivalent. Now let us give some actual examples where E satisfies(CF). In this subsection, we fix a commutative noetherian ring R.
First we deal with exact categories which are Hom-finite.
Proposition 2.4.10. Let E be a Hom-finite exact R-category. Then E satisfies (CF).

2.4. APPLICATIONS
fi

15

gi

Proof. Let 0 → X −→ Y −→ Z → 0 be conflations in E for i = 1, 2, and put Mi :=Coker E(−, gi ) ∈ eff E. Then we have that [M1 ] = [E(−, X)] − [E(−, Y )] + [E(−, Z)] = [M2 ] holds
in K0 (mod E). We will show that if M1 has finite length, then so does M2 . It suffices to show that
supp M2 is finite by Lemma 2.3.1.
Let X be an indecomposable object in E. For each F in mod E, the assignment F 7→
lengthR F (X) makes sense since E is Hom-finite. It clearly extends to the group homomorphism
χX : K0 (mod E) → Z. Now we have that X ∈ supp M1 ⇔ χX [M1 ] 6= 0 ⇔ χX [M2 ] 6= 0 ⇔ X ∈supp M2 by [M1 ] = [M2 ]. Thus supp M1 = supp M2 holds, and this is finite since M1 has finite
length.

Now we immediately obtain the following general result on artin algebras.
Theorem 2.4.11. Let Λ be an artin algebra and E a contravariantly finite resolving subcategoryof mod Λ. Then the following are equivalent.
(1) E is of finite type.
(2) E is admissible.
(3) AR(E) = Ex(E) holds.
(4) AR(E) ⊗Z Q = Ex(E) ⊗Z Q holds.
Proof. Since Λ is artin algebra, Λ is semiperfect and E is Hom-finite. Thus Propositions 2.3.3
and 2.4.7 imply that (1) and (2) are equivalent. Furthermore, since E is Hom-finite, E satisfies(CF) by Proposition 2.4.10. Thus the other conditions are also equivalent by Theorem 2.3.7. Contravariantly finite resolving subcategories are closely related to so-called cotilting modules
by the famous result of [AR2]. For the convenience of the reader, we explain their result here.
Definition 2.4.12. Let Λ be an artin algebra and U a Λ-module in mod Λ. Then U is calleda cotilting Λ-module if it satisfies the following conditions:(1) the injective dimension of U is finite.
(2) Ext>0Λ (U, U ) = 0 holds.
(3) There exists an exact sequence 0 → Un → · · · → U0 → DΛ → 0 of Λ-modules with
Ui ∈ add U for each i, where D denotes the standard duality D : mod Λop → mod Λ.
For a module U in mod Λ, we denote by ⊥ U the full subcategory of mod Λ consisting of modules
X satisfying Ext>0Λ (X, U ) = 0.
Proposition 2.4.13 ([AR2, Theorem 5.5]). Let Λ be an artin algebra and U a cotilting Λmodule. Then ⊥ U is a contravariantly finite resolving subcategory of mod Λ.
Thus we can apply Theorem 2.4.11 to E := ⊥ U for a cotilting module U over an artin algebra.
This provides a rich source of examples:• Let E be a functorially finite torsionfree class of mod Λ for an artin algebra Λ (see [ASS]).
Then by factoring out the annihilator of E, we may assume that E is a faithful functoriallyfinite torsionfree class. It is well-known that such E is of the form ⊥ U for a cotilting
Λ-module with id U ≤ 1 (see e.g. [Sma, Theorem]), so Theorem 2.4.11 applies to E. By
duality, every functorially finite torsion class over artin algebras is also an example.
• An artin algebra Λ is called Iwanaga-Gorenstein if both id(ΛΛ ) and id(Λ Λ) are finite. In
this case, we write GP Λ := ⊥ Λ and call it the category of Gorenstein-projective modules.
It is immediate that Λ itself is a cotilting Λ-module, so we can apply Theorem 2.4.11 to
E := GP Λ.
Next we will consider the case dim R > 0. Although E is rarely Hom-finite, we can conclude(CF) by using the Hom-finiteness of the stable category E in some cases. Let us introduce someterminologies to state our result.
Let E be an exact category with enough projectives. In this case, we have the syzygy functorΩ : E → E. We denote by X in E for the image of X in E under the natural projection E  E.
For an object X in E, we say that X has finite projective dimension if there exists some integern ≥ 0 such that Ωn X = 0 holds in E.

16

2. RELATIONS FOR GROTHENDIECK GROUPS AND REPRESENTATION-FINITENESS

Recall that an exact category E is called Frobenius if E has enough projectives and enoughinjectives, and the classes of projectives and injectives coincide.
Proposition 2.4.14. Let R be a complete noetherian local ring and E a Hom-noetherian
idempotent complete exact R-category with AR conflations. Suppose that E satisfies either one of
the following conditions:(1) E has enough projectives and every object in E has finite projective dimension, or(2) E is Frobenius.
Then E satisfies (CF).
fi

gi

Proof. Let 0 → X −→ Y −→ Z → 0 be conflations in E for i = 1, 2. Put Mi :=Coker E(−, gi ) ∈ eff E for each i and suppose that M1 has finite length. We will show that sodoes M2 . First note that Lemma 2.2.5 applies to this situation since E has enough projectives, so
we identify eff E with mod E naturally. Then Mi has finite length as an E-module if and only if it
does so as an E-module. Moreover, since E is Hom-finite by Lemma 2.4.3, this occurs if and only
if supp Mi is finite, where the support is considered inside ind E.
(1) First we prove that [M1 ] = [M2 ] holds in K0 (mod E). Choose an integer n ≥ 0 such that
Ωn Z = 0 holds in E. Then it is classical that we have an exact sequence in mod E for each i:
0 → E(−, Ωn−1 X) → E(−, Ωn−1 Y ) → E(−, Ωn−1 Z) → · · ·→ E(−, ΩX) → E(−, ΩY ) → E(−, ΩZ) → E(−, X) → E(−, Y ) → E(−, Z) → Mi → 0.
Therefore it immediately follows that [M1 ] = [M2 ] holds in K0 (mod E).
Let X be an indecomposable object of E. Since E is a Hom-finite R-category by Lemma
2.4.3, we have the characteristic map χX : K0 (mod E) → Z which sends F to lengthR F (X). Then
we have X ∈ supp M1 ⇔ χX [M1 ] 6= 0 ⇔ χX [M2 ] 6= 0 ⇔ X ∈ supp M2 , where the support isconsidered inside ind E. Thus Proposition 2.3.1 implies that M2 has finite length.
(2) Put Ni := Ker E(−, fi ) for each i = 1, 2. Then we have exact sequences in mod E as in (1):E(−,fi )E(−,gi )0 → Ni → E(−, X) −−−−−→ E(−, Y ) −−−−−→ E(−, Z) → Mi → 0,E(−,Ωgi )E(−, ΩY ) −−−−−−→ E(−, ΩZ) → Ni → 0.
It follows from the first exact sequence that [M1 ⊕ N1 ] = [M1 ] + [N1 ] = [M2 ] + [N2 ] = [M2 ⊕ N2 ]holds in K0 (mod E). We will show that N1 has finite length. By the same argument as in (1), we
have that supp(M1 ⊕ N1 ) = supp(M2 ⊕ N2 ) holds in ind E. Now Ω : E → E is an equivalence since
E is Frobenius, and let us denote by Ω− : E → E the quasi-inverse of Ω. Then it is easily checkedthat X ∈ supp N1 ⇔ Ω− X ∈ supp M1 ⇔ X ∈ Ω(supp M1 ) holds for an indecomposable object
X ∈ E, hence supp N1 = Ω(supp M1 ). Therefore supp N1 is finite, and so is supp(M1 ⊕ N1 ) =supp(M2 ⊕ N2 ). Hence supp M2 is finite, which implies that M2 has finite length.
For an R-order Λ with at most an isolated singularity, Proposition 2.4.14 shows that CM Λ
satisfies (CF) if either Λ has finite global dimension or Λ is a Gorenstein order. Thus we obtainthe following result on the category CM Λ.
Theorem 2.4.15. Let R be a complete Cohen-Macaulay local ring and Λ an R-order with at
most an isolated singularity. Put E := CM Λ. Then the following are equivalent.
(1) E is of finite type.
(2) E is admissible.
Assume in addition that either Λ has finite global dimension or Λ is a Gorenstein order. Then thefollowing are also equivalent.
(3) AR(E) = Ex(E) holds.
(4) AR(E) ⊗Z Q = Ex(E) ⊗Z Q holds.
Proof. This follows from Example 2.4.9 and Proposition 2.4.14.

2.5. FINITENESS OF SYZYGIES

17

Remark 2.4.16. Theorem 2.4.15 for the case Λ has finite global dimension was shown in[AR1, Proposition 2.3], but their proof relies on higher algebraic K-theory. Also Theorem 2.4.15
generalizes [Hir] where Λ is assumed to be commutative and Gorenstein. The author does notknow whether all the conditions above are equivalent without any assumption on an R-order Λ,even if Λ is commutative.
2.5. Finiteness of syzygies
Let R be a complete Cohen-Macaulay local ring and suppose that AR(CM R) = Ex(CM R)
holds. Although we do not know whether CM R is of finite type, it is shown in [Kob] that
ΩCM R, the category of syzygies of CM R is so. In this section, we will extend this result to a
non-commutative order.
Let E be an exact category with enough projectives. In this section, we denote by ΩE thesubcategory of E consisting of objects X such that there exists an inflation X  P in E for someprojective object P . If E is Krull-Schmidt, then so is ΩE since ΩE is closed under direct sums andsummands. The essential image of ΩE under the natural projection E  E coincides with ΩE, theessential image of the syzygy functor Ω : E → E. We begin with the following property about ΩE,
which is of interest in itself.
Lemma 2.5.1. Let R be a commutative noetherian ring, and let E be a Hom-noetherian idempotent complete exact R-category with a projective generator P . Then the syzygy functor Ω : E → ΩE
has a fully faithful left adjoint Ω− : ΩE → E.
Proof. For each object W ∈ ΩE, take a left (add P )-approximation f : W → P W (this ispossible since E is Hom-noetherian). Since W is in ΩE, there exists an inflation f 0 : W  P 0 withP 0 ∈ add P . Thus f 0 factors through f , which implies that f is an inflation by the idempotent
completeness of E ([Büh, Proposition 7.6]). Now we define Ω− W ∈ E by the following conflation:
f

g

0→W −
→ PW −→ Ω− W → 0.
Next suppose that we have a map ϕ : W1 → W2 in ΩE. This induces the commutative diagram
0

W1

f1

W2

g1

f2

P W2

Ω− W 1

0

−

ψ

ϕ

0

P W1

Ω ϕ
g2

Ω− W 20in E where both rows are conflations, since f2 ◦ ϕ must factor through f1 . A simple diagram chaseshows that this defines a well-defined isomorphism Ω− : E(W1 , W2 ) → E(Ω− W1 , Ω− W2 ), with its
inverse induced by the syzygy functor Ω : E → ΩE. Thus we obtain a functor Ω− : ΩE → E, which
is fully faithful and satisfies ΩΩ− ∼= idΩE .
Finally we show that Ω− : ΩE → E is a left adjoint of Ω : E → ΩE. Let W be in ΩE and X in
E. Then the syzygy functor induces a morphism E(Ω− W, X) → E(W, ΩX) by ΩΩ− W ∼= W in E.
It is easy to show that this map is bijective, by considering the following diagram
0

W

0

ΩX

f

PW
PX

g

Ω− W

0

X

0

where PX is projective and f is a left add P -approximation. The details are left to the reader. 
Theorem 2.5.2. Let R be a complete noetherian local ring and E a Hom-noetherian idempotent complete exact R-category with a projective generator P and AR conflations. Suppose that
AR(E) ⊗Z Q = Ex(E) ⊗Z Q holds. Then the following holds.
(1) For every object M in eff E, we have that supp M ∩ ind(ΩE) is finite, where the support
is considered inside ind E.
(2) Assume in addition that E has a weak cogenerator. Then ΩE is of finite type.

18

2. RELATIONS FOR GROTHENDIECK GROUPS AND REPRESENTATION-FINITENESS

Proof. Throughout this proof, supports of functors are always considered inside ind E.
(1) First recall that the stable category E is Hom-finite over R by Lemma 2.4.3. By the similarargument as in Proposition 2.3.7, it suffices to prove the following weaker version of (CF).
fi

gi(CF)Ω Let 0 → X −→ Y −→ Z → 0 be conflations in E for i = 1, 2 and put Mi := Coker E(−, gi ) ∈eff E. If supp M1 is finite, then supp M2 ∩ ind(ΩE) is finite.
Assume the above situation and put Ni := Ker E(−, fi ) for each i = 1, 2. Then we have exact
sequences in mod E for i = 1, 2, as in the proof of Proposition 2.4.14(2):E(−,fi )E(−,gi )0 → Ni → E(−, X) −−−−−→ E(−, Y ) −−−−−→ E(−, Z) → Mi → 0,E(−,Ωgi )E(−, ΩY ) −−−−−−→ E(−, ΩZ) → Ni → 0.
As in the proof of Proposition 2.4.14(2), it is enough to show that supp N1 ∩ ind(ΩE) is finite.
Actually we will show that every element of supp N1 ∩ ind(ΩE) is isomorphic to ΩA for someA ∈ supp M1 , which obviously implies the desired claim.
Let W be an element of supp N1 ∩ ind(ΩE), so E(W , Ωg1 ) : E(W , ΩY1 ) → E(W , ΩZ1 ) is notsurjective. By Lemma 2.5.1, this is equivalent to that E(Ω− W , g1 ) : E(Ω− W , Y1 ) → E(Ω− W , Z1 )is not surjective. On the other hand, since Ω− : ΩE → E is fully faithful by Lemma 2.5.1, we havethat Ω− W is indecomposable. Thus Ω− W belongs to supp M1 . Now W ∼= ΩΩ− W in E holds,
which completes the proof.
(2) Let X be a weak cogenerator of E. By (1), we have that that ind(ΩE) ∩ supp E(−, X)
is finite. Since every indecomposable object W in E should satisfy E(W , X) 6= 0, it follows that
ΩE is of finite type. Now ΩE itself is of finite type because we have a natural identification
ind(ΩE) = ind(ΩE) t ind(add P ).

Now we apply this theorem to the category of Cohen-Macaulay modules. The obtained result
extends [Hir, Kob], where Λ was assumed to be commutative.
Corollary 2.5.3. Let R be a complete Cohen-Macaulay local ring and Λ an R-order with at
most an isolated singularity. If AR(CM Λ) ⊗Z Q = Ex(CM Λ) ⊗Z Q holds, then ΩCM Λ is of finite
type.
Proof. We can apply Theorem 2.5.2 to E := CM Λ since E has a weak cogenerator by Example
2.4.9.

2.A. Some properties of the category of effaceable functors
First we show that our notion of effaceability is equivalent to that of Grothendieck (see [Gro,p. 148] or [Wei, Exercise 2.4.5] for the standard definition).
Proposition 2.A.1. Let E be an idempotent complete exact category and M ∈ Mod E a right
E-module. Then the following are equivalent.
(1) M is in eff E.
(2) M is finitely presented, and for every W ∈ E and w ∈ M (W ), there exists a deflation
ψ : E  W in E such that (M ψ)(w) = 0.
g

Proof. (1) ⇒ (2): Suppose that M is in eff E and take a conflation 0 → X → Y −→Z→0in E such that M ∼= Coker E(−, g). Denote by π : E(−, Z)  M the natural projection. Clearly
M is finitely presented. Let W ∈ E and w ∈ M (W ). By the Yoneda lemma, we have a morphismw∗ : E(−, W ) → M . By the projectivity of E(−, W ) and the Yoneda lemma, we have a map
ϕ : W → Z such that π ◦ E(−, ϕ) = w∗ . Now let us take the pullback of g along ϕ:
0

X

E

ψ

W

0

ϕ

0

X

Y

g

Z

0

2.A. SOME PROPERTIES OF THE CATEGORY OF EFFACEABLE FUNCTORS

19

Note that ψ is a deflation. This yields the following commutative diagram in Mod E:
E(−, E)E(−,ψ)

E(−, W )w∗

E(−,ϕ)

0

E(−, X)

E(−, Y )E(−, Z)

E(−,g)

Mπ0

By applying these functors to E, it is straightforward to check that (M ψ)(w) = 0.
(2) ⇒ (1): Suppose that M satisfies the condition of (2). Since M is finitely presented, we
have a morphism g : Y → Z in E such that E(−, Y ) → E(−, Z) → M → 0 is exact. We have anelement z ∈ M (Z) which corresponds to E(−, Z) → M by the Yoneda lemma. By the assumption,
there exists a deflation ψ : E  Z such that (M ψ)(z) = 0. The Yoneda lemma implies that thecomposition E(−, E) → E(−, Z) → M is a zero map, hence the map E(−, ψ) lifts to E(−, Y ). Thus
we have a map h : E → Y such that g ◦ h = ψ. Since ψ is a deflation, we can conclude that so is
g by the idempotent completeness of E. Therefore M is in eff E.
Next we prove that eff E is an abelian subcategory of Mod E.
Theorem 2.A.2. Let E be an exact category. Then eff E is closed under kernels and cokernels
in Mod E. In particular, eff E is an abelian category such that the inclusion eff E → Mod E is exact.
Moreover, eff E is closed under extensions in Mod E.
Proof. Let ϕ : M1 → M2 be a morphism in eff E. For each i = 1, 2, we have the correspondingfi
gi
conflations 0 → Xi −→ Yi −→ Zi → 0 in E satisfying Mi = Coker E(−, gi ). We will show thatthe image, kernel and cokernel of ϕ in Mod E are contained in eff E. Sine ϕ induces a morphism
between the projective resolutions of M1 and M2 , we have the following commutative diagram in
E by the Yoneda lemma:
0

f1

X1
a

0

g1

Y1

Z1
c

b
f2

X2

g2

Y2

0

Z2

0

Take the pushout E of the top conflation along a : X1 → X2 . By the universal property, we obtainthe following commutative diagram
t

0

X1

[−a,f1 ]X2 ⊕ Y1[f,b1 ]

0

X1
a

0

g1

Y1

X2

f2

X2

g

E

t

E

0

Z1

0

[b2 ,g](2.A.1)

c
g2

Y2
t

f

0

Z1

b1
f

b2

0

0

g

[0,1]
f1

E

Z2

0

Z2

0

[1,0]

Y2 ⊕ Z1[g2 ,−c]in E such that b2 ◦ b1 = b holds. It is straightforward to see that diagram (2.A.1) commutes.
Moreover, all the rows are conflations, and the top-right square and the bottom-left square are
pullback-pushout squares, see [Büh, Proposition 2.12]
By the Yoneda embedding, we obtain the following commutative diagram in Mod E corresponding to (2.A.1), where all the rows are exact.

20

2. RELATIONS FOR GROTHENDIECK GROUPS AND REPRESENTATION-FINITENESS

0

E(−, X1 )E(−, X2 ⊕ Y1 )E(−, E)

K

0

ι

0

E(−, X1 )E(−, Y1 )E(−, Z1 )

M1

0

ϕ1

0

E(−, X2 )E(−, E)

E(−, Z1 )

M

0

ϕ2

0

E(−, X2 )E(−, Y2 )E(−, Z2 )

M2

0

π

0

E(−, E)

E(−, Y2 ⊕ Z1 )E(−, Z2 )C

0

We have that K, M and N belong to eff E. We can check that ϕ1 is surjective and ϕ2 is injective,
hence M is a image of ϕ. Moreover, one can show that ι is a kernel of ϕ1 (or equivalently, ϕ) and
π is a cokernel of ϕ2 (or equivalently ϕ). We leave the details to the reader.
For the extension-closedness of eff E, we refer the reader to [Eno2, Proposition 2.10].
In this paper, we often deal with the finiteness of length of effaceable E-modules. Since eff E
and Mod E are both abelian, the length of effaceable modules seems to depend on the ambient
category we adopt. This is not the case by the following.
Proposition 2.A.3. Let E be an exact category and M an object in eff E.
(1) M has finite length in eff E if and only if it does so in Mod E.
(2) Suppose that mod E is abelian. Then M has finite length in Mod E if and only if it does
so in mod E.
Moreover, the length and the composition factors of M in Mod E coincide with those in eff E andin mod E.
This follows from the following elementary observation.
Lemma 2.A.4. Let E be an additive category and A an abelian subcategory of Mod E such thatthe inclusion A → Mod E is exact. Suppose that A is closed under finitely generated submodules.
Then the following are equivalent for every object M in A.
(1) M has finite length in Mod E.
(2) M has finite length in A.
Moreover, the length and the composition factors of M in Mod E coincides with those in A.
Proof. (1) ⇒ (2): This is clear since the lattice of subobjects of M in A is a subposet ofthat in Mod E.
(2) ⇒ (1): It suffices to show that every simple object in A is also simple in Mod E. Let M bea simple object in A and N a non-zero submodule of M in Mod E. By using the Yoneda lemma, it
is easily checked that there is a non-zero finitely generated submodule N 0 of N in Mod E. Since N 0
is a finitely generated submodule of M , we have that N 0 belongs to A by the assumption. Thus
N 0 = M holds by N 0 ≤ M in A and N 0 6= 0.
The remaining assertions are clear from the above proof and the Jordan-Hölder theorem. 
Proof of Proposition 2.A.3. (1) The category eff E satisfies the conditions in Lemma
2.A.4 by Propositions 2.A.2 and 2.2.4(1).
(2) Suppose that mod E is abelian. Then it is well-known that the embedding mod E → Mod E
is exact and that mod E is closed under finitely generated submodules, see [Aus1]. Thus Lemma
2.A.4 applies.
CHAPTER 3

The Jordan-Hölder property and Grothendieck monoids ofexact categories
This chapter is based on [Eno4].
We investigate the Jordan-Hölder property (JHP) in exact categories. First we introduce
Grothendieck monoids of exact categories, and show that (JHP) holds if and only if the Grothendieckmonoid is free. Moreover, we give a criterion for this which only uses the Grothendieck group andthe number of simple objects. Next we apply these results to the representation theory of artin
algebras. For a large class of exact categories including functorially finite torsion(-free) classes,(JHP) holds precisely when the number of indecomposable projectives is equal to that of simples.
We study torsion-free classes in a quiver of type A in detail using the combinatorics of symmetric groups. In particular, we show that simples correspond to Bruhat inversions of a c-sortableelement, and give the combinatorial criterion for (JHP).
3.1. Introduction
To begin with, let us recall the Jordan-Hölder theorem for modules. This classical theorem
says that the ways in which M can be built up from simple modules are essentially unique.
Theorem (the Jordan-Hölder theorem). Let Λ be a ring and M a Λ-module of finite length.
Then any composition series of M are equivalent. In particular, composition factors of M together
with their multiplicities are uniquely determined by M .
Let us express this situation as follows: the category of Λ-modules of finite length satisfies the
Jordan-Hölder property, abbreviated by (JHP). The aim of this paper is to investigate to what
extent this property is valid for various settings, especially those arising in the representation
theory of algebras. It is known that an abelian category satisfies (JHP) if every object has finite
length (e.g. [Ste, p.92]), so the abelian case is rather trivial. In this paper, we study (JHP) in
the context of Quillen’s exact categories, which generalize abelian categories and serve as a useful
categorical framework for studying various subcategories of module categories.
As in the case of module categories, we can define a poset of admissible subobjects of an objectof an exact category, and classical notions like simple objects, composition series and (JHP) makesense in this setting (see Section 2). Typical examples of exact categories are extension-closed
subcategories of mod Λ for an artin algebra Λ, and in this case, all objects have at least one
composition series. However, it turns out that there exist many categories which do not satisfy(JHP), as well as those which do. Here are some examples (we refer to Section 3.8.2 for more
examples).
Example D (c.f. [BHLR, Example 6.9]). Let k be a field and define E as the category offinite-dimensional k-vector spaces whose dimensions are not equal to 1. Then obviously E is closed
under extensions in mod k, thus an exact category, and k 2 and k 3 are simple objects in E. However,
we have the following two different decompositions of k 6 into simples:k3 ⊕ k3 = k6 = k2 ⊕ k2 ⊕ k2 .
Thus composition series (and lengths) are not unique.
The next examples are torsion-free classes of a module category, which have been studied in
the representation theory of algebras. See Section 3.6 for the detailed explanation of this example.
21

22

3. (JHP) AND GROTHENDIECK MONOIDS OF EXACT CATEGORIESExample E (c.f. Example 3.6.17 (1)). Let Q be a quiver 1 → 2 ← 3 and consider the path
algebra kQ. Then the Auslander-Reiten quiver of mod kQ is given as follows:
P3
S2

S1

: E2

I2
P1

: E1

S3

Consider the subcategory E1 := add{S2 , P1 , P3 , I2 , S3 } of mod Λ. Then it is checked that S2 , P1 , S3
are simples of E1 , and E1 satisfies (JHP). For example, P3 decomposes into S2 and S3 , and I2
decomposes into P1 and S3 .
On the other hand, consider E2 := add{S2 , P1 , P3 , I2 }. Then (JHP) fails in E2 : in fact, all four
indecomposables are simple in E2 , but we have two exact sequences
and

0 → P 1 → P1 ⊕ P3 → P3 → 0(a split sequence),

0 → S2 → P1 ⊕ P3 → I2 → 0(an almost split sequence),

which implies that P1 ⊕ P3 has two different decomposition: one into P1 and P3 and the otherinto S2 and I2 . Thus composition factors of P1 ⊕ P3 are not unique.
To explorer (JHP), we make use of a lesser-known invariant of exact categories, Grothendieckmonoids. First let us make some observation on the Grothendieck group K0 (E) of E, which is morefamous. If E satisfies (JHP), then we can easily check that K0 (E) is a free abelian group, since
simple objects form a free basis of K0 (E) by (JHP). Thus (JHP) implies the freeness of K0 (E).
However, the converse is not true: K0 (Ei ) ∼= Z3 holds for i = 1, 2 in Example E but (JHP) fails
in E2 . Therefore, we need a more sophisticated invariant of exact categories than Grothendieckgroups. To this purpose, we propose to study a Grothendieck monoid M(E) of an exact category
E, which is a monoid defined by the same universal property as the Grothendieck group. In the
representation theory of algebras, this monoid is closely related to the monoid of dimension vectorsof modules (see Corollary 3.5.9).
By using M(E), we obtain the following simple characterization of (JHP):
Theorem F (= Theorem 3.4.12). Let E be a skeletally small exact category. Then the following are equivalent:(1) E satisfies (JHP).
(2) M(E) is a free monoid.
(3) K0 (E) is a free abelian group, and the images of non-isomorphic simple objects in K0 (E)form a basis of K0 (E).
By using this theorem, we obtain the following easier criterion for (JHP) which can be usefulin actual situations.
Theorem G (= Theorem 3.4.13). Let E be a skeletally small exact category. Suppose that
K0 (E) is finitely generated. Then the following are equivalent:(1) E satisfies (JHP).
(2) K0 (E) is a free abelian group, and rank K0 (E) is equal to the number of non-isomorphic
simple objects in E.
We apply these results to the context of the representation theory of artin algebras. In manycases, the Grothendieck group turns out to be free of finite rank, whose rank is equal to the numberof non-isomorphic indecomposable projective objects (Proposition 3.5.8). Thus Theorem G tells
us that all we have to do to check (JHP) is to count the number of simple objects in E, and thento compare it to that of projectives. In particular, we can give a criterion whether a functoriallyfinite torsion(-free) class satisfies (JHP) using the language of τ -tilting theory (Corollary 3.5.15).
As an easy application of this theorem, we show the following result on Nakayama algebras.
Corollary H (= Corollary 3.5.19). Let Λ be a Nakayama algebra. Then every torsion-freeclass of mod Λ satisfies (JHP).

3.1. INTRODUCTION

23

Finally, we discuss simple objects in torsion-free classes of the category mod kQ for a type An
quiver Q. Torsion-free classes of mod kQ are classified in [IT, AIRT, Tho]: they are in bijection
with cQ -sortable elements in the symmetric group Sn+1 . Let F(w) be the torsion-free class ofmod kQ corresponding to a cQ -sortable element w. Then it is natural to expect that we candescribe simples in F(w) and the validity of (JHP) by using the combinatorics of Sn+1 . We obtainthe following result along this line.
Theorem I (= Theorem 3.6.14, Corollary 3.6.15). Let Q be a quiver of type An and w ∈ Sn+1
a cQ -sortable element. Consider the corresponding torsion-free class F(w) of mod kQ. Then we
have the following:(1) Simple objects in F(w) are in bijection with Bruhat inversions of w.
(2) F(w) satisfies (JHP) if and only if the number of supports of w is equal to that of Bruhat
inversions of w.
Here Bruhat inversions are inversions of w which give rise to covering relations in the Bruhat
order of Sn+1 (see Definition 3.6.2). In the paper [Eno5], it is shown that this holds for otherDynkin types (see Remark 3.6.19).
3.1.1. Organization. The content of each section is as follows. In Section 2, we study posetsof admissible subobjects of objects in exact categories, and give several basic definitions which we
use throughout this paper, such as (JHP). In the latter part of Section 2, we give some categoricalconditions which ensure that subobject posets are (modular) lattices. In Section 3, we define theGrothendieck monoid of an exact category, and study its basic properties as a monoid. In Section
4, we give characterizations of (JHP) in terms of Grothendieck monoids or groups (Theorems F
and G).
In Section 5, we apply the general results to the representation theory of artin algebras. We
mainly consider good extension-closed subcategories of module categories (see Assumption 3.5.6),
and study (JHP) for this class of categories. In Section 6, we consider torsion-free classes ofmod kQ where Q is a quiver of type A and prove Theorem I. In Section 7, we give examples ofcomputations of the Grothendieck monoids. In Section 8, we collect counter-examples on various
conditions which we have investigated. In Section 9, some open problems are discussed. In
Appendix A, we collect basic properties on monoids which we need in the body part.
3.1.2. Conventions and notation. Throughout this paper, we assume that all categories
are skeletally small, that is, the isomorphism classes of objects form a set. In addition, all subcategories are assumed to be full and closed under isomorphisms. For a category E, we denote by
Iso E the set of all isomorphism classes of objects in E. For an object X in E, the isomorphismclass of X is denoted by [X] ∈ Iso E.
For a set of object C of an additive category E, we denote by add C the subcategory of E
consisting of direct summands of finite direct sums of objects in C.
For a Krull-Schmidt category E, we denote by ind E the set of isomorphism classes of indecomposable objects in E. We denote by |X| the number of non-isomorphic indecomposable direct
summands of X.
By a module we always mean right modules unless otherwise stated. For a noetherian ringΛ, we denote by mod Λ (resp. by proj Λ) the category of finitely generated right Λ-modules (resp.
finitely generated projective right Λ-modules).
As for exact categories, we use the terminologies inflations, deflations and conflations. We
refer the reader to [Büh] for the basics of exact categories. For an inflation A  X in an exact
category, we denote by X  X/A the cokernel of A  X. We say that an exact category E has aprogenerator P (resp. an injective cogenerator I) if P is projective (resp. I is injective) and every
object in E admits a deflation from a finite direct sum of P (resp. an inflation into a finite direct
sum of I). For an exact category E, we denote by P(E) (resp. I(E)) the category consisting of all
projective (resp. injective) objects in E.
Let A be an abelian category and E a subcategory of A. We say that E is extension-closed
if for every short exact sequence 0 → X → Y → Z → 0 in A, we have that Y belongs to E

24

3. (JHP) AND GROTHENDIECK MONOIDS OF EXACT CATEGORIES

whenever both X and Z belong to E. In such a case, unless otherwise stated, we regard E as anexact category with the natural exact structure; conflations are precisely short exact sequences of
A whose terms all belong to E.
For a poset P and two elements a, b ∈ P with a ≤ b, we denote by [a, b] the interval poset[a, b] := {x ∈ P | a ≤ x ≤ b} with the obvious partial order.
By a monoid M we mean a commutative semigroup with a unit, and we always use anadditive notation: the operation is denoted by +, and the unit of addition is always denoted by
0. A homomorphism between monoids is a map which preserves the addition and 0.
We denote by N the monoid of non-negative integers: N = {0, 1, 2, · · · } with the addition +.
For a set A, we denote by #A the cardinality of A.
3.2. Posets of admissible subobjects
To study the Jordan-Hölder property (JHP) on exact categories, one has to define what
(JHP) exactly means, which we will do in this section. The contents of this section are naturalgeneralizations of the corresponding ones in module categories or abelian categories.
3.2.1. Basic properties. First we collect some basic definitions on the poset of admissible
subobjects, which we use throughout this paper.
Definition 3.2.1. Let E be a skeletally small exact category and X an object of E.
• We call an inflation A  X an admissible subobject of X (c.f. [BHLR, Definition 3.1]).
We often call A an admissible subobject of X in this case.
∼• Two subobjects A, B of X are called equivalent if there exists an isomorphism A −→Bwhich makes the following diagram commutes:
X

∼

A
B

We denote by P(X) the equivalence class of admissible subobjects of X. When we wantto emphasize the ambient category E, we write PE (X).
• We define a partial order on P(X) as follows: We write A ≤ B for A, B ∈ P(X) if there
exists an inflation A  B which makes the following diagram commutes:
A

X

B
It is easily checked that this relation actually gives a partial order on P(X).
We remark that P(X) always has the greatest element X and the smallest element 0.
Example 3.2.2. Let A be an abelian category and E an extension-closed subcategory of A.
Then for an object X ∈ E, we have that
PE (X) = {A | A is a subobject of X in A which satisfies A, X/A ∈ E}.
Although we assume that there exists an inflation A  B in order to have A ≤ B, if E isweakly idempotent complete, then this condition is not necessary. We refer to [Büh, Section 7] forweakly idempotent completeness.
Lemma 3.2.3. Let E be a weakly idempotent complete exact category. Then for an object X
and A, B ∈ P(X) of X, the following are equivalent:(1) A ≤ B holds in P(X).

3.2. POSETS OF ADMISSIBLE SUBOBJECTS

25(2) There exists a morphism ϕ : A → B which makes the following diagram commutes:
A

X

ϕ

ι

B
Proof. (1) ⇒ (2): Obvious.
(2) ⇒ (1): It is enough to check that ϕ is an inflation. This follows from that ιϕ is an inflationand E is weakly idempotent complete, see [Büh, Proposition 7.6] for example.

Example 3.2.4. Let A be an abelian category and E an extension-closed subcategory of A.
Suppose that E is closed under direct summands, that is, if A ⊕ B belongs to E then so do A and
B. Then E is idempotent complete, thus weakly idempotent complete. Many important exact
categories investigated in the representation theory of algebras arise in this way.
The next proposition ensures that for any interval [A, B] in P(X), we have an isomorphism ofposets [A, B] ∼= P(B/A), as in the case of abelian categories.
Proposition 3.2.5. Let E be a skeletally small exact category and X an object of E. Thenthe following hold, where all the intervals are considered in P(X).
∼ P(A).
(1) For A ∈ P(X), we have an isomorphism of posets [0, A] =∼ P(X/A). Moreover,(2) For A ∈ P(X), we have an isomorphism of posets (−)/A : [A, X] =X/B ∼= (X/A)/(B/A) holds for any B in [A, X].
(3) For A, B ∈ P(X) with A ≤ B, we have an isomorphism of posets (−)/A : [A, B] ∼=P(B/A). Moreover, for any X1 , X2 in P(X) with A ≤ X1 ≤ X2 ≤ B, we have that
X2 /X1 ∼= (X2 /A)/(X1 /A).
Proof. (1) For an element B ∈ [0, A], there exists an inflation B  A since B ≤ A holds.
Thus B ∈ P(A) holds. On the other hand, let B ∈ P(A). Then we have an inflation B  A.
Since inflations are closed under compositions, it follows that the composition B  A  X isan inflation, hence B is an admissible subobject of X. Consequently we have B ∈ P(X), andclearly B ∈ [0, A] holds. These assignments are easily shown to be mutually inverse isomorphismsof posets between [0, A] and P(A).
(2) For an element B ∈ [A, X], we have inflations A  B  X. Then we obtain the following
commutative diagram
A

B

B/A

A

X

X/A

X/B

X/B

in E (see [Büh, Lemma 3.5]). Thus the assignment B 7→ B/A gives a morphism of poset(−)/A : [A, X] → P(X/A). Moreover, by the above sequence, X/B ∼= (X/A)/(B/A) holds.
Conversely, let M  X/A be an admissible subobject of X/A. By taking the pullback of
X  X/A along M  X/A, we obtain the following diagram
A

B

M

A

X

X/A

X/B

X/B

26

3. (JHP) AND GROTHENDIECK MONOIDS OF EXACT CATEGORIES

in E where all rows and columns are conflations (see [Büh, Proposition 2.15]). Since A ≤ B holdsin P(X), the assignment M 7→ B gives a morphism of poset P(X/A) → [A, X]. These maps areeasily seen to be mutually inverse to each other.
(3) This follows from (1) and (2), since we have the following chain of isomorphisms of posets
P(A)

P(B)

P(B/A)∼=[A, B]∼=∼=[0, B][A, B]

where the middle interval is considered in P(B) and the two right intervals are considered in P(A).
The isomorphism X2 /X1 ∼= (X1 /A)/(X2 /A) is obtained by applying (2) to A ∈ P(X2 ).
Now we introduce simple objects in exact categories, that is, objects which cannot be decomposed into smaller pieces with respect to conflations. This objects plays a central role throughout
this paper, and to determine simple objects in a given exact category is an essential (and difficult)task when we check (JHP).
Definition 3.2.6. Let E be an exact category and X an object of X. We say that X is simple
if the poset P(X) of admissible subobjects of X has exactly two distinct elements 0 and X. Thisis equivalent to that X is not zero and there exists no conflation of the form L  X  N in E
with L, N 6= 0. We denote by sim E the set of isomorphism classes of simple objects in E.
We remark that the notion of simple objects in exact categories was also defined in [BHLR,
Definition 3.2] and [BeGr, Definition 5.2].
3.2.2. Basic definitions on inflation series, composition series and (JHP). In module
categories, submodule series (a chain of submodules) serves as a basic tool when we consider
composition series and the Jordan-Hölder theorem. It is natural to introduce the corresponding
notion, inflation series, in the context of exact categories. Here are the definitions of it and related
notions, including (JHP). Note that some of the definitions were also given in [BeGr, BHLR, HR].
First recall some notions from poset theory. A chain of a poset P is a totally ordered subsetof P . A chain T of P is called maximal if there exists no chain of P which properly contains T .
A chain T is called finite if T is a finite set. Such a chain is of the form x0 < x1 < · · · < xn , andwe say that this chain has length n in this case.
Let E be a skeletally small exact category.
• For X in E, an inflation series of X is a finite sequence of inflations 0 = X0  X1 
· · ·  Xn = X. We often identify it with a weakly increasing sequence 0 = X0 ≤ X1 ≤· · · ≤ Xn = X in P(X).
• We say that an inflation series 0 = X0  X1  · · ·  Xn = X of X is a proper inflation
series if none of Xi  Xi+1 is an isomorphism, or equivalently, X0 < X1 < · · · < Xnholds in P(X). In this case, we say that this inflation series has length n. We often
identify proper inflation series with finite chains of P(X) which contain 0 and X.
• Let X and Y be objects in E, and let 0 = X0  X1  · · ·  Xn = X and 0 =Y0  Y1  · · ·  Ym = Y be two inflation series of X and Y respectively. Then these
inflation series are called isomorphic if m = n and there exists a permutation σ of theset {1, 2, · · · n} such that Xi /Xi−1 and Yσ(i) /Yσ(i)−1 are isomorphic in E for all i. In thiscase, we say that X and Y have isomorphic inflation series.
• An inflation series 0 = X0  X1  · · ·  Xn = X of X is called a composition series
if X0 < X1 < · · · < Xn is a maximal chain in P(X), or equivalently, each quotient
Xi /Xi−1 is a simple object for all i (this follows from Proposition 3.2.5). We often
identify composition series with finite maximal chains of P(X).

3.2. POSETS OF ADMISSIBLE SUBOBJECTS27• E is called a length exact category if every object X in E, the set of lengths of proper
inflation series of X has an upper bound, or equivalently, the set of lengths of finite
chains of P(X) has an upper bound.
• A length exact category E satisfies the unique length property if any composition seriesof X have the same length for every object X in E.
• A length exact category E satisfies the Jordan-Hölder property, abbreviated by (JHP), if
any composition series of X are isomorphic to each other for every object X in E.
First we prove some properties of length exact categories which easily follow from definitions.
Actually the proof only uses the general theory of posets.
Proposition 3.2.7. For a length exact category E and an object X in E, the following holds:(1) Every chain of P(X) is finite.
(2) Composition series of X are precisely maximal chains of P(X).
(3) Every proper inflation series of X can be refined to a composition series.
(4) X has at least one composition series.
Proof. (1) Suppose that P(X) has a chain T consisting of infinitely many elements. Then bychoosing a finite subset of T , we can obtain a finite chain of P(X) with an arbitrary large length,
which is a contradiction.
(2) This follows from (1) and the definition of compositions series.
(3) Suppose that a proper chain 0 = X0 < X1 < · · · < Xn = X of P(X) is given. If this chain is
maximal, then we have nothing to do. Suppose that this is not the case. If [Xi−1 , Xi ] = {Xi−1 , Xi }for every i, then this chain is clearly maximal. Thus there exist some i and Y ∈ P(X) with
Xi−1 < Y < Xi . Consequently, we obtain a chain X0 < X1 < · · · < Xi−1 < Y < Xi < · · · < Xnof P(X) with length n + 1, which is a refinement of the original chain. By iterating this process,we will eventually obtain a finite maximal chain, since lengths of finite chains of P(X) have an
upper bound.
(4) We may assume that X 6= 0. Then (4) follows by applying (3) to a chain 0 < X ofP(X).
Remark 3.2.8. Let E be an exact category and suppose that for every X ∈ E there exists atleast one composition series of X. Even so, E may not be a length exact category, since lengthsof finite chains of P(X) may be arbitrary large (unfortunately the author does not know such an
example). However, if E is an abelian category (or more generally, an exact category such thatevery subobject poset is a modular lattice), then E is a length exact category. This follows from
Theorem 3.2.18.
Remark 3.2.9. In the recent paper [HR], it was shown that nice exact categories, which are
pre-abelian categories satisfying the nice axiom, satisfy (JHP).
By definition, (JHP) implies the unique length property for length exact categories. For
concrete (counter-)examples concerning (JHP) and the unique length property, we refer the reader
to Section 3.8.2.
In the remaining part of this section, we give a sufficient condition for the unique length
property by using the theory of modular lattices.
3.2.3. Quasi-abelian implies lattice. Recall that a lattice L is a poset such that for every
two elements a, b ∈ L, there exist the meet a ∧ b, the greatest lower bound, and the join a ∨ b, theleast upper bound. In abelian categories, it is classical that the subobject poset P(X) is always alattice by considering sum and intersection of subobjects. However, this does not hold in general,
even if we consider pre-abelian categories (see Examples 3.8.1 and 3.8.2). In this subsection, we
will show that P(X) is a lattice when we consider torsion(-free) classes of abelian categories, ormore generally, quasi-abelian categories.
First we recall the definition of quasi-abelian categories.
Definition 3.2.10. Let E be an additive category.

28

3. (JHP) AND GROTHENDIECK MONOIDS OF EXACT CATEGORIES(1) E is pre-abelian if every morphism in E has a kernel and a cokernel. It is well-known thatpre-abelian category has all pullbacks and pushouts.
(2) A pre-abelian category E is quasi-abelian if the class of kernel morphisms is closed underpushouts and the class of cokernel morphisms is closed under pullbacks.
Quasi-abelian categories in the above sense are sometimes called almost abelian [Rum1, Rum2],
or semi-abelian [Raı̆]. See Historical remark in [Rum3, Section 2] for the somewhat involved historyof quasi-abelian categories.
Every quasi-abelian category has the natural greatest exact structure, and we always consider
this greatest exact structure in what follows. More precisely, the following proposition holds. We
refer to [Büh, Proposition 4.4] for the proof.
Proposition 3.2.11. Let E be a quasi-abelian category. Then E has the greatest exact structure, in which conflations, inflations and deflations are precisely kernel-cokernel pairs, kernel
morphisms and cokernel morphisms respectively.
By this proposition, for an object X in a quasi-abelian category E, an inflation A  X is
nothing but a kernel morphism in E. Moreover, for two admissible subobjects ιA : A  X andιB : B  X, we have that A ≤ B holds in P(X) if and only if ιA factors through ιB . This follows
from Proposition 3.2.3 since every pre-abelian category is idempotent complete.
A typical example of quasi-abelian categories are torsion(-free) classes of abelian categories.
Definition 3.2.12. Let A be an abelian category.
(1) Let T and F be subcategories of A. We say that a pair (T , F) is a torsion pair in A if
it satisfies the following conditions:(a) A(T , F) = 0 holds, that is, A(T, F ) = 0 holds for every T ∈ T and F ∈ F.
(b) For every object X ∈ A, there exists a short exact sequences
0 → TX → X → FX → 0
in A with T X ∈ T and F X ∈ F.
In this case, we say that T is a torsion class and F is a torsion-free class.
(2) We say that a torsion pair (T , F) is hereditary if T is closed under subobjects in A. In
this case, we say that F is a hereditary torsion-free class.
For a torsion pair (T , F) on an abelian category, it is easily checked that T is closed underquotients and extensions, and that F is closed under subobjects and extensions. Hence we canregard T and F as exact categories.
The next proposition is classical, e.g. [Rum2, Theorem 2]. For the convenience of the reader,
we give a simple proof which makes use of the exact structure.
Proposition 3.2.13. Let (T , F) be a torsion pair in an abelian category A. Then T and F
are both quasi-abelian.
Proof. We only show that F is quasi-abelian. Since F is closed under extensions in A, wehave that F has the natural exact structure, where conflations are short exact sequences with allterms in F. By the axiom of exact categories, inflations are closed under pushouts and deflations
are closed under pullbacks. Thus it suffices to show that E is pre-abelian, that every kernel
morphism in F is an inflation and that every cokernel morphism in F is a deflation.
First we show that every morphism has a kernel, and every kernel morphism is an inflation.
Let f : X → Y be an arbitrary morphism in F. Then we have the following commutative diagram
in A with exact rows:
f
ι
0
K
X
Y
0

K

X

Im f

0.

Since F is closed under subobjects in A, we have K, Im f ∈ F. Therefore ι : K → X is a kernel off in F, and this is actually an inflation in F since the bottom row is a conflation in F.

3.2. POSETS OF ADMISSIBLE SUBOBJECTS

29

Next we show that every morphism in F has a cokernel and every cokernel morphism is a
deflation in F. Let f : X → Y be an arbitrary morphism in F and denote by C the cokernel of f
in A. Then we have the following commutative diagram in A with exact rows and columns
X
f

0

K

Y

0

TC

C

π

FC

0

FC

0

0
where T C ∈ T and F C ∈ F. Then K ∈ F holds since F is closed under subobjects, and it isstraightforward to see that π is a cokernel of f in F. It follows that the middle row is a conflation
in F and that π is actually a deflation in F.
We omit the following lemma, which can be checked straightforwardly.
Lemma 3.2.14. Let E be an additive category and suppose that we have a commutative diagram
i0

E
0

fg

W

p.b.

g

i

X

K

f

Y
Y

such that the left square is pullback and i is a kernel of f . Then i0 is a kernel of f g. In particular,kernel morphisms are closed under pullbacks if exists.
Now we can prove that quasi-abelian implies lattice property.
Proposition 3.2.15. Let E be a quasi-abelian category with the greatest exact structure and
X an object of E. Then P(X) is a lattice.
Proof. Let A, B ∈ P(X) be two admissible subobjects of X.
We first construct the meet A ∩ B. Note that since E is pre-abelian, E has pullbacks. Now wetake the following pullback:
A∩B

A
p.b.

B

X

Then by Lemma 3.2.14, the above morphisms A ∩ B → A and A ∩ B → B are kernel morphisms,
hence inflations. Thus the composition A ∩ B  A  X is an inflation, so it is an admissible
subobject of X. Hence we have A ∩ B ∈ P(X) and that A ∩ B ≤ A, B. Suppose that W ∈ P(X)satisfies W ≤ A and W ≤ B. Then by the universal property of the pullback, we have a morphism
W → A∩B such that the composition W → A∩B  X is an inflation. Note that E is idempotent
complete because E is pre-abelian. Then Lemma 3.2.3 shows that W ≤ A ∩ B holds in P(X).
Therefore A ∩ B is the meet of A and B in the poset P(X).
Next we will show the existence of the join A+B ∈ P(X). Here we shall give two constructions.
(First Construction): This construction is in fact the dual of the previous one, but weinclude it here for the completeness. Since E is pre-abelian, it has pushouts. Now we take thefollowing pushout:
X

X/A
p.o.

X/B

C

30

3. (JHP) AND GROTHENDIECK MONOIDS OF EXACT CATEGORIESThen by the dual of Lemma 3.2.14, we have that the above morphism X/A → C is a cokernel
morphism, hence a deflation. It follows that the composition X  X/A  C is a deflation, so C
can be written as C = X/(A + B) for some A + B ∈ P(X). Now we leave it the reader to verify
that A + B is actually a join of A and B in P(X).
(Second Construction): The second construction is similar to the usual construction of thejoin in abelian categories: the join A + B is obtained by taking the image of the map A ⊕ B → X.
Moreover, we will use this construction later to show the modular property of the lattice. To state
the construction, we use the following basic property on quasi-abelian categories.
Lemma 3.2.16 ([Büh, Proposition 4.8]). Every morphism f in a quasi-abelian category admitsa factorization f = ιp such that ι is a kernel morphism and p is an epimorphism.
Now for admissible subobjects ιA : A  X and ιB : B  Y , consider the induced morphismp[ιA , ιB ] : A ⊕ B → X. By the above lemma, there exist an object A + B and morphisms A ⊕ B −
→
ι
A+B →
− X such that [ιA , ιB ] = ιp holds, p is an epimorphism and ι is a kernel morphism. We
claim that ι : A + B  X is a join of A and B.
Clearly ιA and ιB factors through ι, so A, B ≤ A + B holds by Proposition 3.2.3. On the other
hand, suppose that ιC : C  X satisfies A, B ≤ C. Then since ιA , ιB factors through ιC , theuniversal property of A ⊕ B yield a morphism ϕ : A ⊕ B → C which makes the following diagram
commutes, where the bottom row is a conflation.
A⊕B

p

ϕ

0

C

A+B
ι

ιC

X

πC

X/C

0

It follows that πC ιp = 0, and since p is an epimorphism, we have that πC ι = 0. Thus ι factors
through ιC , so A + B ≤ C holds in P(X) by Proposition 3.2.3.

Remark 3.2.17. Proposition 3.2.15 can be proved directly in the case of a torsion-free class,as follows. Let (T , F) be a torsion pair in an abelian category A and X an object of F. We givea construction of a meet and a join of A, B ∈ PF (X).
• A meet A ∩ B is the usual meet of subobjects in an abelian category A. It suffices toobserve that A ∩ B is actually admissible, that is, X/(A ∩ B) belongs to F. We have thefollowing exact sequence in A:
0 → X/(A ∩ B) → X/A ⊕ X/B
Since F is closed under direct sums and subobjects, it follows that X/(A ∩ B) belongs
to F, that is, A ∩ B is an admissible subobject of X.
• A join of A and B in PF (X) is in general different from the usual join A+B of subobjects
in an abelian category A. The problem is that A + B is not necessarily admissible, that
is, X/(A + B) may not belong to F. However, we have the following exact sequence
0 → T (X/(A + B)) → X/(A + B) → F (X/(A + B)) → 0.
with the left term in T and the right term in F, and let us define A + B ∈ PF (X) by
the isomorphism X/(A + B) ∼= F (X/(A + B)). We leave it to the reader to check thatA + B is indeed a join of A and B in PF (X).
3.2.4. Integral quasi-abelian implies modularity and the unique length property.
It is known that a submodule lattice, or more generally, a subobject lattice in an abelian category,is a modular lattice (see e.g. [Ste, Proposition IV.5.3]). First, let us see what modularity ofsubobject lattices implies in our context.
Let us recall the definition of modularity. A lattice L is called modular if for every a, b, x in
L with a ≤ b, we have (x ∨ a) ∧ b = (x ∧ b) ∨ a. Modularity is an useful tool to study composition
series, since we have the following Jordan-Hölder theorem for modular lattices. For the proof, werefer the reader to [Ste, Corollary 3.2, Proposition 3.3].

3.2. POSETS OF ADMISSIBLE SUBOBJECTS

31

Theorem 3.2.18 (The Jordan-Hölder theorem for modular lattices). Let L be a modular
lattice with the least element 0 and the greatest element 1. Suppose that L has at least one finite
maximal chain. Then the following holds:(1) Every chain of L must be finite.
(2) Every maximal chain of L has the same length.
(3) Every chain can be refined to some maximal chain, thus the set of lengths of chains of L
has an upper bound.
Moreover, we have some kind of uniqueness of composition series: for two maximal chains0 = x0 < x1 < · · · < xl = 1 and 0 = y0 < y1 < · · · < yl = 1, each intervals [xi−1 , xi ] and [yj−1 , yj ]are “isomorphic” up to permutations (isomorphic here means projective in lattice theory, see e.g.
[Ste, III. Section 2] for the detail). Actually, the Jordan-Hölder theorem for abelian categories canbe proved by using the above lattice-theoretic one (see [Ste, p.92] for example).
For exact categories, modularity of subobject lattices does not imply (JHP) in general because“isomorphic” above may not induce an isomorphism. However, modularity clearly does imply theunique length property, so the following is an immediate corollary of Theorem 3.2.18.
Corollary 3.2.19. Let E be an exact category. Suppose that for every object X ∈ E, theposet P(X) is a modular lattice and that at least one composition series of X exists. Then E is alength exact category and satisfies the unique length property.
In the rest of this section, we will show that integral quasi-abelian categories are exact categories in which this situation occurs.
Definition 3.2.20. A quasi-abelian category E is called integral quasi-abelian if the class of
epimorphisms is closed under pullbacks and the class of monomorphisms is closed under pushouts.
The typical example is a hereditary torsion-free class of abelian categories (Definition 3.2.12),
see e.g. [Rum1, Lemma 6] for the proof.
To show modularity, we need the following lemma of integral quasi-abelian categories, whichasserts that a regular morphism is an essential epimorphism:
Lemma 3.2.21. Let E be an integral quasi-abelian category. Suppose that we have morphismsg
A−→B−→ C in E which satisfy the following conditions:(1) g is a monomorphism and an epimorphism (that is, g is regular).
(2) gf is an epimorphism.
Then f is an epimorphism.
f

Proof. Consider the following commutative diagram in E:
A

f

B

π

g

p.o.

C

π0

Coker f
g

0.

0

W

Then we have π 0 gf = g 0 πf = 0, and since gf is an epimorphism, π 0 = 0. Thus g 0 π = 0 holds. On
the other hand, since monomorphisms are stable under pushouts and g is a monomorphism, so is
g 0 . Therefore π = 0 holds, so f is an epimorphism.
To prove modularity, the following criterion is quite useful.
Lemma 3.2.22 ([Ste, Proposition III.2.3]). Let L be a lattice. Then the following are equivalent:(1) L is a modular lattice.
(2) Take any a, b ∈ L with a ≤ b, and take x, c, c0 ∈ [a, b]. Suppose that x ∨ c = b = x ∨ c0
and x ∧ c = a = x ∧ c0 hold (that is, c and c0 are complements of x in [a, b]), and thatc ≤ c0 holds. Then we have c = c0 .
Proposition 3.2.23. Let E be an integral quasi-abelian category with the greatest exact structure and X an object of E. Then P(X) is a modular lattice.

32

3. (JHP) AND GROTHENDIECK MONOIDS OF EXACT CATEGORIESProof. For admissible subobjects A and B of X, we denote by A ∩ B (resp. A + B) the meet
(resp. join) of them constructed in Proposition 3.2.15.
We make use of Lemma 3.2.22. By Proposition 3.2.5, every interval of P(X) is isomorphic toP(Y ) for some object Y of E. Thus it suffices to show the following claim:(Claim): Let A, B1 , B2 be admissible subobjects of X. Suppose that A ∩ B1 = A ∩ B2 = 0,
A + B1 = A + B2 = X and B1 ≤ B2 hold in P(X). Then B1 = B2 holds.
(Proof of Claim). Let ιA , ιB1 and ιB2 be the inflations corresponding to A, B1 and B2 respectively. By the construction of the meet given by Proposition 3.2.15, we have the following pullback
diagram for each i = 1, 2.
0

Bi
p.b.

A

ιBi

ιA

X

It follows that 0 → A ⊕ Bi is a kernel of [ιA , ιBi ] : A ⊕ Bi → X, so [ιA , ιBi ] is a monomorphism
for each i = 1, 2.
On the other hand, by the second construction of the join given in Proposition 3.2.15, we havethe following factorization on the join A + Bi for each i:ri[ιA , ιBi ] : A ⊕ Bi −→A + Bi  Xsuch that ri is an epimorphism. Now since A + Bi = X, we have ri = [ιA , ιBi ] : A ⊕ Bi → X.
We have shown that ri is both a monomorphism and an epimorphism for each i. Let ι : B1 
B2 be the inflation corresponding to B1 ≤ B2 . Then we have the following commutative diagram:A ⊕ B1

r1

X[ 10 0ι ]A ⊕ B2

r2

X

Since r2 is a monomorphism and an epimorphism and r1 is an epimorphism, it follows that[ 10 0ι ] : A⊕B1 → A⊕B2 is an epimorphism by Lemma 3.2.21. However, [ 10 0ι ] is a kernel morphism,
so it must be an isomorphism. Thus ι also is an isomorphism by the standard matrix calculation.

Corollary 3.2.24. Let E be an integral quasi-abelian category which is length as an exact
category (for example, a hereditary torsion-free class of length abelian category). Then E satisfies
the unique length property.
Proof. This immediately follows from Proposition 3.2.23 and Corollary 3.2.19.



We refer the reader to Example 3.8.5 for examples which satisfy the unique length property.
3.3. Grothendieck monoid of an exact category
To an exact category E, we can associate the Grothendieck group K0 (E), which is the abeliangroup defined by the universal property with respect to conflations. The investigation and calculation of this group is a very classical topic in the various area of mathematics. However, muchless attention has been paid to the monoid version of the Grothendieck group, the Grothendieckmonoid M(E), which we introduce in this section.
Remark 3.3.1. Let us mention some related works on Grothendieck monoids. The notion ofGrothendieck monoids was originally introduced and investigated in [BeGr] to study Hall algebrasof exact categories. Actually there is some overlap between this paper and [BeGr], and the author
is grateful to J. Greenstein for pointing out it.
Also Brookfield studied the structure of the Grothendieck monoid of the category of finitely
generated modules over noetherian rings in [Bro1, Bro2, Bro3].

3.3. GROTHENDIECK MONOID OF AN EXACT CATEGORY

33

3.3.1. Construction and basic properties of the Grothendieck monoid. Recall thatmonoids are assumed to be commutative in our convention. First we give the definition ofGrothendieck monoids and study their basic properties as monoids. We refer to Appendix A
for some unexplained notions on monoids.
Definition 3.3.2. Let E be a skeletally small exact category.
(1) A map f : Iso E → M to a monoid M is said to respect conflations if it satisfies thefollowing conditions:(a) f [0] = 0 holds.
(b) For every conflation
0→X→Y →Z→0in E, we have that f [Y ] = f [X] + f [Z] holds in M .
(2) A Grothendieck monoid M(E) is a monoid M(E) together with a map π : Iso E → M(E)which satisfies the following universal property:(a) π respects conflations in E.
(b) Every map f : Iso E → M to an monoid M which respects conflations in E uniquely
factors through π, that is, there exists a unique monoid homomorphism f : M(E) →M which satisfies f = f π.
Iso E

f

M

π
f

M(E)
By abuse of notation, we often write π[X] = [X] to represent an element in M(E).
First of all, we must show that the Grothendieck monoid actually exists.
Proposition 3.3.3. For a skeletally small exact category E, its Grothendieck monoid M(E)exists.
Proof. Define the operation + on Iso E by [A] + [B] := [A ⊕ B], then clearly Iso E is a monoid
with an additive unit [0].
Now let us define a monoid congruence ∼ on Iso E which is generated by the following relations(see Proposition 3.A.6): For every conflation X  Y  Z in E, we impose [Y ] ∼ [X] + [Z]. Using
this, we obtain a monoid M(E) := Iso E/ ∼. Now it is clear from the construction that M(E)enjoys the required universal property of the Grothendieck monoid.
This construction was used to define Grothendieck monoids in [BeGr, Definition 2.6]. The
following is a more direct characterization whether [X] ∼ [Y ] holds in M(E). Note that for an
inflation series 0 = X0  X1  · · ·  Xn = X of X, we can easily show inductively that[X] = [X1 ] + [X2 /X1 ] + · · · + [Xn /Xn−1 ] holds in M(E). Thus if X and Y have isomorphic
inflation series, then [X] = [Y ] holds in M(E).
Proposition 3.3.4. Let E be a skeletally small exact category and X, Y two objects of E.
Then [X] = [Y ] holds in M(E) if and only if there exist a sequence of objects X = X0 , X1 , · · · ,
Xm = Y in E such that Xi and Xi−1 have isomorphic inflation series for each i.
Proof. We freely use the construction of M(E) in Proposition 3.3.3. For two elements[X], [Y ] in Iso E, we write [X] ≈ [Y ] if there exists objects X = X0 , X1 , · · · , Xm = Y such that Xi
and Xi−1 have isomorphic inflation series for each i. It is clear that ≈ is an equivalence relation
on Iso E, and it suffices to show that ≈ and ∼ coincides.
First we show that ≈ is a monoid congruence on Iso E. It is enough to show that for every
object A in E, if X and Y have isomorphic inflation series, then so do A ⊕ X and A ⊕ Y . Let
0 = X0  X1  · · ·  Xn = X and 0 = Y0  Y1  · · ·  Yn = Y be two isomorphic
inflation series. Then it is obvious that two inflation series X0  · · ·  Xn  Xn ⊕ A andY0  Y1  · · ·  Yn  Yn ⊕ A are isomorphic, where the last inflations are the inclusion into
direct summands. Therefore ≈ is a monoid congruence on Iso E.

34

3. (JHP) AND GROTHENDIECK MONOIDS OF EXACT CATEGORIESNext we will prove that ≈ coincides with ∼. From the argument above this proposition, we
have that [X] ≈ [Y ] implies [X] ∼ [Y ] since [X] = [X0 ] ∼ [X1 ] ∼ · · · ∼ [Xm ] = [Y ]. To show
the converse implication, it suffices to show that for every conflation X  Y  Z, we have that
[Y ] ≈ [X] + [Z] = [X ⊕ Z]. This is clear since we have inflation series 0  X  Y of Y and
0  X  X ⊕ Z of X ⊕ Z which are isomorphic since Y /X ∼= Z.
Using this description of the Grothendieck monoid, we can prove some properties of it.
Proposition 3.3.5 (c.f. [BeGr, Lemma 2.9]). Let E be a skeletally small exact category. Thenthe following hold.
(1) For an object X in E, we have that [X] = 0 in M(E) if and only if X ∼= 0.
(2) M(E) is reduced.
Proof. (1) This follows from Proposition 3.3.4, since if 0 and X has isomorphic inflation
series, then clearly X ∼= 0 holds.
(2) Suppose that [X] + [Y ] = 0 holds in M(E). Then [X ⊕ Y ] = 0 holds, hence X ⊕ Y ∼= 0 by(1). Therefore both X and Y must be isomorphic to 0 in E. Thus M(E) is reduced.

We can prove that non-isomorphic simples in E are distinct in M(E), and in fact they are in
bijection with atoms of M(E). This property is remarkable compared to the Grothendieck group,
since non-isomorphic simples may represent the same element in the Grothendieck group K0 (E)(see Section 3.8.3.1). We refer the reader to Definition 3.A.8 for the notion of atoms in monoids.
Proposition 3.3.6 (c.f. [BeGr, Lemma 5.3]). Let E be a skeletally small exact category. Thenthe following hold.
(1) Let S and X be two objects in E. Suppose that [S] = [X] holds in M(E) and that S issimple. Then S ∼= X holds in E.
(2) Let S1 and S2 be two simple objects in E. Then [S1 ] = [S2 ] holds in M(E) if and only if
S1 ∼= S2 holds in E.
'(3) The assignment S 7→ [S] from Ob E to M(E) induces a bijection sim E −→ Atom M(E).
Proof. (1) Suppose that S and X have isomorphic inflation series for X ∈ E and a simple
object S in E. Then it is clear that S ∼= X holds, since possible inflation series of S is of the form
0  · · ·  0  S = · · · = S. Now the assertion of (1) inductively follows from Proposition 3.3.4.
(2) This is a special case of (1).
(3) We first show that [S] is an atom in the monoid M(E). Suppose that [S] = [X] + [Y ].
Then we have that [S] = [X ⊕ Y ], so (1) implies that S ∼= X ⊕ Y . Since simple objects are∼
indecomposable, it follows that X ∼0orY0holds,
thatis, [X] = 0 or [Y ] = 0. Therefore [S]==is an atom.
Conversely, suppose that X is not simple. We may assume that X 6= 0, so we have a nontrivial conflation X1  X  X2 with X1 , X2 6= 0. Then [X] = [X1 ] + [X2 ] holds in M(E) and
we have [X1 ], [X2 ] 6= 0 by Proposition 3.3.5. Thus [X] is not an atom.
Now we have proved that atoms in M(E) are precisely elements which come from simple
objects. Then the remaining claim follows immediately from (2).
For later use, we show another necessary condition for [X] = [Y ] in M(E). Recall that asubcategory D of an exact category E is called Serre if for any conflation 0 → X → Y → Z → 0
in E, we have that Y ∈ D holds if and only if both X ∈ D and Z ∈ D hold.
Proposition 3.3.7. Let E be a skeletally small exact category and X, Y two objects of E.
Suppose that X belongs to a Serre subcategory D of E and that [X] = [Y ] holds in M(E). Then Y
also belongs to D.
Proof. By Proposition 3.3.4, it suffices to show the assertion when X and Y have isomorphic
inflation series. Take inflation series 0 = X0  X1  · · ·  Xn = X and 0 = Y0  Y1  · · · 
Yn = Y and a permutation σ of {1, · · · , n} such that Xi /Xi−1 ∼= Yσ(i) /Yσ(i)−1 for each i. Since
X ∈ D and D is closed under subquotients, Xi /Xi−1 belongs to D, thus so does Yi /Yi−1 for eachi. Therefore, it inductively follows that Y ∈ D holds, since D is extension-closed and Y can beobtained from Yi /Yi−1 ’s by extensions.
3.4. CHARACTERIZATION OF (JHP)

35

3.3.2. Grothendieck monoid and positive part of Grothendieck group. The Grothendieckgroup of an exact category is a classical invariant, which is defined by the similar universal property
as the Grothendieck monoid. In this section, we will recall the Grothendieck group and discuss
its relation to the Grothendieck monoid.
Definition 3.3.8. Let E be a skeletally small exact category.
(1) A map f : Iso E → G to an abelian group G is said to respect conflations if it satisfies
f [Y ] = f [X] + f [Z] in G for every conflation X  Y  Z in E.
(2) A Grothendieck group K0 (E) of an exact category E is an abelian group K0 (E) together
with a map π : Iso E → K0 (E) which satisfies the following universal property:(a) π respects conflations in E.
(b) Every map f : Iso E → G to an abelian group G which respects conflations in E
uniquely factors through π, that is, there exists a unique group homomorphism
f : K0 (E) → G which satisfies f = f π.
Iso E

f

G

π
f

K0 (E)
By abuse of notation, we often write π[X] = [X] to represent an element of K0 (E).
By the defining properties of the Grothendieck monoid and the Grothendieck group, it isclear that the latter is obtained from the former by taking the group completion, as the following
proposition claims. We refer the reader to Appendix A for this notion.
Proposition 3.3.9. Let E be a skeletally small exact category. Then the Grothendieck group
K0 (E) exists, and it is realized as the group completion gp M(E) of the Grothendieck monoid M(E).
In what follows, we always identify gp M(E) with K0 (E) and we denote by ι : M(E) → K0 (E)the natural map which satisfies ι[X] = [X] for every X in E.
Definition 3.3.10. Let E be a skeletally small exact category. We denote by K+
0 (E) the image+of the natural map ι : M(E) → K0 (E), that is, K+
(E)
:=
{[X]
|
X
∈
E}.
We
call
K
(E)
the positive
00part of the Grothendieck group.
It follows from Proposition 3.A.15 that ι : M(E) → K0 (E) induces an isomorphism of monoids
M(E)can ∼= K+
0 (E), where M(E)can is the largest cancellative quotient of M(E). Thus the positive
part has lost information on non-cancellative part compared to the Grothendieck monoid. Evenso, K+
0 (E) is a more sophisticated invariant of E than K0 (E). Note that we have many examples
in which the Grothendieck monoids are not cancellative, see Section 3.8.3.
We shall see later in Corollary 3.5.9 that for a large class of exact categories E arising in the
representation theory of artin algebras, including functorially finite torsion(-free) classes, we canidentify K+
0 (E) with the monoid of dimension vectors of modules in E.
3.4. Characterization of (JHP)
In this section, We will show the basic relationship between structures of exact categories andcombinatorial properties of the Grothendieck monoids.
3.4.1. Length-like functions and length exact categories. First we will introduce ananalogue of dimension or length of modules, which characterizes length exact categories.
Definition 3.4.1. Let E be a skeletally small exact category. We say that a map ν : Iso E → N
is a weakly length-like function if it satisfies the following conditions:(1) For every conflation X  Y  Z in E, we have ν[Y ] ≥ ν[X] + ν[Z].
(2) ν[X] = 0 implies X ∼= 0 for every X ∈ E.
We say that ν is a length-like function if it satisfies (2) and the following:(10 ) For every conflation X  Y  Z in E, we have ν[Y ] = ν[X] + ν[Z].

36

3. (JHP) AND GROTHENDIECK MONOIDS OF EXACT CATEGORIES

Remark 3.4.2. Obviously, a length-like function is weakly length-like. It is clear that we canidentify a length-like function with a length-like function on the monoid M(E) by Proposition
3.3.5 (1), see Definition 3.A.16.
Note that (weakly) length-like functions are far from being unique (for example, any positive
multiple of a length-like function is a length-like function).
A weakly length-like function gives an upper bound for all possible lengths of X, as follows.
Lemma 3.4.3. Let E be a skeletally small exact category with a weakly length-like function
ν. For every X ∈ E and proper inflation series 0 = X0 < X1 < · · · < Xn = X of X, we haven ≤ ν[X]. In particular, E is a length exact category.
Proof. Since ν is weakly length-like, we have inductively
ν[X] ≥ ν[X1 ] + ν[X/X1 ]≥ ν[X1 ] + ν[X2 /X1 ] + ν[X/X2 ]
···
≥

n
X

ν[Xi /Xi−1 ].

i=1Since Xi−1 6= Xi for each i, we have Xi /Xi−1 6= 0. Therefore ν[Xi /Xi−1 ] > 0, hence ν[Xi /Xi−1 ] ≥1, since ν is weakly length-like. It follows that ν[X] ≥ n.
This means that the set of lengths of finite chains of P(X) has an upper bound ν[X] for every
X ∈ E, thus E is a length exact category.

Actually, the existence of a weakly length-like function is equivalent to the length-ness:
Theorem 3.4.4. Let E be a skeletally small exact category. Then the following are equivalent:(1) E has a weakly length-like function.
(2) E is a length exact category.
Proof. (1) ⇒ (2): This is Lemma 3.4.3.
(2) ⇒ (1): For an object X ∈ E, the lengths of chains of P(X) has an upper bound, thus wecan define ν[X] by the following:
ν[X] := max{ l | there exists a chain of P(X) of length l }.
We will show that ν : Iso E → N is a weakly length-like function. If ν[X] = 0, then X ∼= 0 holds
since otherwise we have a chain 0 < X of length one.
Suppose that we have a conflation X  Y  Z in E and put n := ν[X] and m := ν[Z].
We have chains 0 = X0 < X1 < · · · < Xn = X of P(X) and 0 = Z0 < Z1 < · · · < Zm = Z
of P(Z). By Proposition 3.2.5, we have an isomorphism of posets [X, Y ] ∼= P(Z), so we have achain X = Z0 < Z1 < · · · < Zm = Y of P(Y ) which corresponds to Zi ’s. Thus we obtain a chain
0 = X0 < · · · < Xn (= X = Z0 ) < Z1 < · · · < Zm = Y of length m + n. Therefore ν[Y ] ≥ m + nfollows by the definition of ν[Y ].
Remark 3.4.5. The weakly length-like function constructed in the proof of (2) ⇒ (1) is thesame as the length given in [BHLR, Section 6].
Using this theorem, we obtain numerous examples of length exact categories.
Example 3.4.6. The typical example in the representation theory of artin algebra is given
as follows. Let Λ be an artin algebra and E an extension-closed subcategory of mod Λ. The
assignment X 7→ l(X), where l(X) denotes the (usual) length of X as a Λ-module, gives a lengthlike function on E. It is clear that the same argument holds for any extension-closed subcategoryof a length abelian category.
Remark 3.4.7. It can be showed that the category of maximal Cohen-Macaulay modules
CM R over a commutative Cohen-Macaulay ring R (or more generally, CM Λ for an R-order Λ)
has a length-like function. It is an interesting problem to investigate when these categories satisfy(JHP).

3.4. CHARACTERIZATION OF (JHP)

37

The following gives the criterion for the Grothendieck monoid to be finitely generated.
Proposition 3.4.8. Let E be a skeletally small exact category. Suppose that E is a length
exact category. Then M(E) is atomic, and the following are equivalent:(1) sim E is a finite set.
(2) Atom M(E) is a finite set.
(3) M(E) is a finitely generated.
Proof. Since E is a length exact category, each object X of E admits a composition series
0 = X0  X1  · · ·  Xn , and Xi /Xi−1 is simple for each i. It follows that [X] = [X1 /X0 ] +[X2 /X1 ] + · · · + [Xn /Xn−1 ] holds in M(E), and Proposition 3.3.6 implies that each [Xi /Xi−1 ] is
an atom in M(E). Thus M(E) is atomic.
Proposition 3.3.6 shows that (1) is equivalent to (2). On the other hand, Proposition 3.A.9(2) shows that (2) is equivalent to (3).

3.4.2. Freeness of monoids and (JHP). Now we can prove the characterization of (JHP)
in terms of its Grothendieck monoid.
Theorem 3.4.9. Let E be a skeletally small exact category. Then the following are equivalent:(1) E is a length exact category and satisfies (JHP).
(2) M(E) is a free monoid, or equivalently, a factorial monoid.
In this case, M(E) is a free monoid with basis Atom M(E) = {[S] | S ∈ sim E}, and K0 (E) is a freeabelian group with basis {[S] | S ∈ sim E}.
Proof. Recall that we have an equality Atom M(E) = {[S] | S ∈ sim E} by Proposition 3.3.6.
Also note that the conditions in (2) are equivalent by Proposition 3.A.9 (5) and Proposition 3.3.5.
(1) ⇒ (2): Let us denote by F the free monoid with basis Atom M(E). Then we have a natural
monoid morphism ϕ : F → M(E) defined by ϕ[S] = [S] for [S] in Atom M(E). We shall construct
the inverse of this map.
For [X] in Iso E, take a composition series 0 = X0 < X1 < · · · < Xn = X of X. Define themap Iso E → F by [X] 7→ [X1 /X0 ] + [X2 /X1 ] + · · · + [Xi /Xi−1 ] ∈ F . This map does not dependon the choice of composition series by (JHP). We show that this map respects conflations in E.
Suppose that we have a conflation X  Y  Z in E and that we have composition series
0 = X0 < X1 < · · · < Xn = X in P(X) and 0 = Z0 < Z1 < · · · < Zm = Z in P(Z). By
Proposition 3.2.5 (2), we have an isomorphism of posets [X, Y ] ∼= P(Z), so we have a chain
X = Z0 < Z1 < · · · < Zm = Y in P(Y ) which corresponds to the chosen composition series of
Z. For each i, we have Zi /Zi−1 ∼= Zi /Zi−1 by Proposition 3.2.5 (2), so Zi /Zi−1 is simple. Thus
0 = X0 < · · · < Xn (= X = Z0 ) < Z1 < · · · < Zm = Y is a composition series of Y . Now it isobvious that the map Iso E → F constructed above respects conflations. Thus we obtain a monoid
homomorphism ψ : M(E) → F .
We claim that ϕ and ψ are mutually inverse to each other.
(ψϕ = idF ): Since F is generated by [S] with S ∈ sim E, it suffices to show that ψϕ[S] = [S]holds. This is obvious since ϕ[S] = [S], and S has the composition series 0 < S.
(ϕψ = idM(E) ): Let X be an object in E with a composition series 0 = X0 < X1 < · · · <
Xn = X. Then ϕψ[X] = [X1 /X0 ] + [X2 /X1 ] + · · · + [Xn /Xn−1 ]. On the other hand, inductively
we have [X] = [X1 /X0 ] + [X2 /X1 ] + · · · + [Xn /Xn−1 ] in M(E), thus ϕψ[X] = [X] holds.
(2) ⇒ (1): First we show that E is a length exact category. By Lemma 3.A.18, we have that
M(E) has a length-like function. Thus E has a length-like function, therefore it is a length exact
category by Proposition 3.4.4.
Next we will show that E satisfies (JHP). Let X be an object of E and let 0 = X0 < X1 <· · · < Xm = X and 0 = Y0 < Y1 < · · · < Yn = X be two composition series of X. Then we have[X] = [X1 /X0 ] + [X2 /X1 ] + · · · + [Xm /Xm−1 ] = [Y1 /Y0 ] + [Y2 /Y1 ] + · · · + [Yn /Yn−1 ]in M(E), where [Xi /Xi−1 ] and [Yj /Yj−1 ] belong to Atom M(E) for each i. Then since M(E) is
free on Atom M(E) by Proposition 3.A.9 (3), it follows that m = n holds, and that there exists
a permutation σ of the set {1, 2, · · · , n} such that [Xi /Xi−1 ] = [Yσ(i) /Yσ(i)−1 ] holds in M(E)

38

3. (JHP) AND GROTHENDIECK MONOIDS OF EXACT CATEGORIES

for each i. This implies that Xi /Xi−1 ∼= Yσ(i) /Yσ(i)−1 by Proposition 3.3.6 (2). Thus these twocomposition series are isomorphic.
The remaining assertions follow from Proposition 3.A.9 (3) and Example 3.A.11.

The most classical example in which (JHP) holds is, as stated in the introduction, a lengthabelian category. Recall that an abelian category is called length if every object has a composition
series, which implies that it is length as an exact category (see Remark 3.2.8).
Corollary 3.4.10. Let A be a length abelian category. Then M(A) is a free monoid withbasis {[S] | S ∈ sim A}. In particular, K0 (A) is a free abelian group with the same basis.
Proof. This follows from Theorem 3.4.9 and the Jordan-Hölder theorem for length abeliancategories (see e.g. [Ste, p.92]).

3.4.3. Grothendieck groups and (JHP). We have shown that (JHP) can be checked by
the freeness of the Grothendieck monoid, but the computation of it is rather hard. In this subsection, we give a criterion for (JHP) in terms of the Grothendieck group, which is easier to computethan the Grothendieck monoid. All contents of this subsection is just a formal consequence of thegeneral theory of monoids, that is, the general criterion for a given monoid to be free (Theorem
3.A.20 and Corollary 3.A.21).
Before we state this, we will make an inequality concerning the Grothendieck groups.
Proposition 3.4.11. Let E be a skeletally small length exact category. Suppose that K0 (E) isa free abelian group. Then the following inequality holds:
rank K0 (E) ≤ # Atom M(E) = # sim E.
Proof. This follows from the corresponding statement on monoids, Proposition 3.A.19.



Now let us state our main characterizations of (JHP) in terms of the Grothendieck group.
Theorem 3.4.12. Let E be a skeletally small exact category. Then the following are equivalent:(1) E is a length exact category and satisfies (JHP).
(2) M(E) is free.
(3) E is a length exact category, and K0 (E) is a free abelian group of basis {[S] | S ∈ sim E}.
(4) E is a length exact category, and all elements [S] ∈ K0 (E) with S ∈ sim E are linearly
independent over Z in K0 (E).
Proof. This immediately follows from Theorems 3.4.9 and 3.A.20 once we observe the following:• Atom M(E) = {[S] | S ∈ sim E} holds (Proposition 3.3.6).
• E has a length-like function if and only if M(E) has a length-like function (Definition
3.4.1).
• M(E) is reduced (Proposition 3.3.5).
• If E is a length exact category, then M(E) is atomic (Proposition 3.4.8).

If the Grothendieck group is known to be free of finite rank, then we have a more convenientcharacterization: we only have to count the number of simples and compare it to the rank.
Theorem 3.4.13. Let E be a skeletally small exact category. Then the following are equivalent:(1) E is a length exact category satisfying (JHP) and # sim E is finite.
(2) M(E) is free and # sim E is finite.
(3) M(E) is a finitely generated free monoid.
(4) E is a length exact category, # sim E is finite and K0 (E) is a free abelian group with basis{[S] | S ∈ sim E}.
(5) The following conditions hold:(a) E is a length exact category.
(b) K0 (E) is a free abelian group of finite rank.

3.5. (JHP) FOR EXTENSION-CLOSED SUBCATEGORIES OF MODULE CATEGORIES

39(c) # sim E = rank K0 (E) holds.
Proof. This immediately follows from Theorems 3.4.9 and 3.A.20, as in the proof of Theorem
3.4.12.
This characterization has lots of applications later, since Grothendieck groups of various exact
categories arising in the representation theory of algebras are turned out to be free of finite rank(see Proposition 3.5.12).
3.4.4. Half-factoriality of monoids and the unique length property. In this subsection, we will give a characterization of the unique length property in terms of Grothendieckmonoids. As we have seen in Theorem 3.4.9, (JHP) corresponds to freeness, or equivalently, factoriality of monoids. We will see that the unique length property corresponds to half-factoriality.
This is natural since both two properties are about the uniqueness of length of factorizations. We
refer the reader to Definition 3.A.8 for the notion of half-factorial monoids.
Theorem 3.4.14. Let E be a skeletally small exact category. Then the following are equivalent:(1) E is a length exact category and satisfies the unique length property.
(2) E has a length-like function l satisfying l[S] = 1 for every simple object S in E.
(3) M(E) is half-factorial.
Proof. The proof is similar to that of Theorem 3.4.9, and actually is easier.
(1) ⇒ (2): We will construct a length-like function l : Iso E → N. Let X be an object of
E. Since E satisfies the unique length property, X has at least one composition series, and every
composition series of X has the same length n. We define l[X] := n. Then l is a length-likefunction by the similar argument to the proof of Theorem 3.4.4. Moreover, clearly l[S] = 1 holds
for every simple object S in E.
(2) ⇔ (3): By Remark 3.4.2, we can identify a length-like function on E with that of M(E).
Moreover, we have an equality Atom M(E) = {[S] | S ∈ sim E} by Proposition 3.3.6. Thus thisequivalence follows from Lemma 3.A.18.
(2) ⇒ (1): By Proposition 3.4.4, our category E is a length exact category. Let X be an objectof E. Then it is easily checked that the length of any composition series of X is equal to l[X].
Thus E satisfies the unique length property.

3.5. (JHP) for extension-closed subcategories of module categories
In this section, we investigate (JHP) for extension-closed subcategories of module categories
of artin algebras. In the rest of this paper, we fix a commutative artinian ring R. An R-algebraΛ is called an artin R-algebra if Λ is finitely generated as an R-module. We often omit the basering R and call Λ an artin algebra. For an artin R-algebra Λ, we denote by D : mod Λ → mod Λop
the standard Matlis duality.
3.5.1. Basic properties on Grothendieck monoids. First we collect some basic properties on the Grothendieck monoid, which immediately follows from the general observations wehave made. In particular, our exact category always becomes a length exact category.
Proposition 3.5.1. Let Λ be an artin algebra and E an extension-closed subcategory of mod Λ.
Then the following hold.
(1) The assignment X 7→ l(X), where l(X) denotes the usual length of X as a Λ-module,
induces a length-like function on E.
(2) E is a length exact category.
(3) M(E) is atomic.
(4) M(E) is finitely generated if and only if sim E is a finite set.
Proof. These follow from Example 3.4.6 (1), Theorem 3.4.4 and Proposition 3.4.8.

The simplest example is E = mod Λ, and we can compute M(mod Λ) and K0 (mod Λ) as follows.

40

3. (JHP) AND GROTHENDIECK MONOIDS OF EXACT CATEGORIES

Proposition 3.5.2. Let Λ be an artin algebra, and let S1 , · · · , Sn be a complete set of simple
right Λ-modules up to isomorphism. Then M(mod Λ) is a free monoid with basis [S1 ], · · · , [Sn ]. In
particular, K0 (mod Λ) is a free abelian group with the same basis. Moreover, |Λ| = # ind P(mod Λ) =# sim(mod Λ) = n holds.
Proof. All the assertions except the last follow form Corollary 3.4.10. The last one holds
because there is a bijection between non-isomorphic indecomposable projective Λ-modules andnon-isomorphic simple Λ-modules.

By this, the Grothendieck group K0 (mod Λ) is often identified with Zn , where n is the numberof simple Λ-modules. This is what is usually called the dimension vectors of modules.
Definition 3.5.3. Let Λ be an artin algebra, and fix a complete set of non-isomorphic simple
Λ-modules {[S1 ], · · · , [Sn ]}. Then we denote by dim : mod Λ → Zn the assignment which sends X
to (a1 , · · · , an ), where ai is the Jordan-Hölder multiplicity of Si in X. This induces an isomorphism∼
K0 (mod Λ) −→ Zn . For a Λ-module X, we call dim X the dimension vector of X.
A typical example of extension-closed subcategories is an Ext-perpendicular category withrespect to a given module.
Definition 3.5.4. Let Λ be an artin algebra and U ∈ mod Λ a Λ-module. We denote by ⊥ U
the subcategory of mod Λ which consists of X ∈ mod Λ such that Ext>0Λ (X, U ) = 0 holds, that is,ExtiΛ (X, U ) = 0 for all i > 0.
Note that ⊥ U is an extension-closed subcategory of mod Λ, and we always regard it as anexact category from now on. Exact categories arising in this way have nice homological properties
as follows. These can be checked directly, so we omit the proofs.
Proposition 3.5.5. Let Λ be an artin algebra and U ∈ mod Λ a Λ-module, and put E := ⊥ U .
Then the following hold.
(1) For any short exact sequence 0 → X → Y → Z → 0 in mod Λ, if Y and Z belong to E,
then so does X.
(2) E is closed under direct summands, thus it is idempotent complete.
(3) proj Λ ⊂ E holds.
(4) E is an exact category with a progenerator Λ.
3.5.2. Ext-perpendicular categories of modules with finite injective dimension. To
check whether (JHP) holds or not, it is easier to deal with the case where the Grothendieck groupis free of finite rank, since we can use Theorem 3.4.13. Let us introduce an assumption which
ensures this.
Assumption 3.5.6. There exists an artin algebra Λ and a Λ-module U ∈ mod Λ with finite
injective dimension such that E is exact equivalent to ⊥ U .
We give examples of such E later in Example 3.5.13.
Lemma 3.5.7. Let E, Λ and U be as in Assumption 3.5.6. Then the following hold.
(1) Put n := id(UΛ ). For every module X ∈ mod Λ, there exists an exact sequence
0 → Ωn X → Pn−1 → · · · → P1 → P0 → X → 0(3.5.1)n

in mod Λ such that each Pi is finitely generated projective and Ω X belongs to E.
(2) The natural inclusion E ,→ mod Λ induces an isomorphism of the Grothendieck groups∼
K0 (E) −→ K0 (mod Λ).
Proof. (1) By taking a projective resolution of X, obviously there exists a short exact sequence of the form (3.5.1) such that each Pi is finitely generated projective. Thus it suffices to>nnshow that Ωn X ∈ ⊥ U . This is because Ext>0Λ (Ω X, U ) = ExtΛ (X, U ) = 0 since id U = n.
(2) This follows from (1) and Quillen’s Resolution Theorem [Qui, §4] on algebraic K-theory.
We give an elementary proof here, which is similar as given in [Yos, Lemma 13.2].

3.5. (JHP) FOR EXTENSION-CLOSED SUBCATEGORIES OF MODULE CATEGORIES

41

We shall construct the inverse of the natural homomorphism K0 (E) → K0 (mod Λ). For amodulePX ∈ mod Λ, take an exact sequence of the form (3.5.1). Then consider the assignment
X 7→ 0≤i<n (−1)i [Pi ] + (−1)n [Ωn X]. This assignment does not depend on the choice of exact
sequences of the form (3.5.1) by the Schanuel lemma, and it respects short exact sequences by theHorseshoe lemma. Thus we obtain the map K0 (mod Λ) → K0 (E). This is the desired inverse, andwe leave it the reader to check the details.
To sum up, our exact category has the following nice properties.
Proposition 3.5.8. Let E be as in Assumption 3.5.6. Then the following hold.
(1) E is a length exact category with a progenerator.
(2) K0 (E) is free of finite rank.
(3) rank K0 (E) = # ind P(E) = |P | holds, where P is a progenerator of E.
Proof. (1) follows from Propositions 3.5.1 (2) and 3.5.5 (4). (2) and (3) follows directly from
Lemma 3.5.7 and Proposition 3.5.2.
In this situation, the positive part of the Grothendieck group of E can be identified with theset of dimension vectors of modules which belong to E.
Corollary 3.5.9. Let Λ and E be as in Assumption 3.5.6, and fix a complete set of simple
Λ-modules {[S1 ], · · · , [Sn ]}. Then the natural map M(E) → K0 (E) → K0 (mod Λ) ∼= Zn induces anisomorphism of monoids between M(E)can and the monoid of dimension vectors of modules in E:
dim E = { dim X | X ∈ E } ⊂ Nn .
Proof. This is immediate from Proposition 3.A.15, the definition of K+
0 (E), an isomorphism∼
K0 (E) −→ K0 (mod Λ) shown in Proposition 3.5.5 and the definition of the dimension vector.

Now our previous characterization of (JHP) has rather simple consequence in this situation.
Theorem 3.5.10. Let E be an exact category which satisfies Assumption 3.5.6. Then E satisfies (JHP) if and only if # sim E = # ind P(E) holds, that is, the number of non-isomorphic
simples in E is equal to that of non-isomorphic indecomposable projectives in E.
Proof. Immediately follow from Proposition 3.5.8 and Theorem 3.4.13.

Actually most exact categories which have been investigated in the representation theory ofartin algebras satisfy Assumption 3.5.6. Among them, those arising from cotilting modules havebeen attracted an attention.
Definition 3.5.11. Let Λ be an artin algebra. We say that a Λ-module U ∈ mod Λ is cotilting
if it satisfies the following conditions:(1) The injective dimension of U is finite.
(2) Ext>0Λ (U, U ) = 0 holds.
(3) There exists an exact sequence
0 → Un → · · · → U1 → U0 → DΛ → 0
in mod Λ for some n such that Ui ∈ add U for each i.
The simplest example of cotilting modules is DΛ. In this case, the perpendicular category(DΛ) coincides with the module category mod Λ. For a general cotilting module, although ⊥ U
is not abelian, it has nice properties.
⊥

Proposition 3.5.12. Let Λ be an artin algebra and U a cotilting Λ-module. Put E := ⊥ U .
Then the following hold:(1) E has a progenerator Λ and an injective cogenerator U .
(2) K0 (E) is a free abelian group of finite rank.
(3) rank K0 (E) = # ind P(E) = # ind I(E) holds.

42

3. (JHP) AND GROTHENDIECK MONOIDS OF EXACT CATEGORIESProof. (1) By Proposition 3.5.5, the exact category E has a progenerator Λ. We refer to[AR2, Theorem 5.4] for the proof of the fact that U is an injective cogenerator of E.
(2) This follows from Propositions 3.5.5 and 3.5.2.
(3) By Propositions 3.5.5 and 3.5.2, we only have to show that # ind I(E) = |U | is equal to# ind P(E) = |Λ|. This follows from tilting theory, e.g. [Hap].
In particular, it says the number of indecomposable projectives and injectives in E coincide.
In the case of mod Λ, this number is also equal to the number of simples. Thus Theorem 3.5.10
says that the violation of this coincidence is nothing but an obstruction for (JHP).
Example 3.5.13. The following exact categories E satisfy Assumption 3.5.6, hence Theorem
3.5.10 holds for them.
(1) E := ⊥ U for a cotilting Λ-module U over an artin algebra Λ. Note that # ind P(E) =# ind I(E) holds in this case, by Proposition 3.5.12.
(2) E = GP Λ for an Iwanaga-Gorenstein artin algebra Λ. Here Λ is called IwanagaGorenstein if id ΛΛ and id Λ Λ are both finite, and in this case, we denote by GP Λ thecategory ⊥ Λ, called the category of Gorenstein-projective modules.
(3) Functorially finite torsion-classes and torsion-free classes of mod Λ for an artin algebra
Λ. This classes of categories has been attracted an attention, since τ -tilting theory gives
a powerful combinatorial tool to investigate them, e.g. [AIR].
Proof. (1) and (2) follow from definition. (3) is well-known to experts (e.g. see [Iya, Proposition 1.2.1]), but we give a sketch here for the convenience of the reader.
We show that a functorially finite torsion-free class F is a special case of (1). By factoring out
the annihilator, we may assume that F is faithful torsion-free class of Λ, that is, the intersectionof annihilators of modules in F is zero. Then it can be shown that ΛΛ ∈ F holds. The well-known
characterizations on classical (co)-tilting modules tells us that F = ⊥ U holds for a cotilting module
U with id U ≤ 1 (see e.g. [ASS, Theorem VI.6.5]).
For a functorially finite torsion class T , the similar argument shows that T comes from theclassical tilting module. Now the Brenner-Butler theorem ([ASS, Theorem VI.3.8]) tells us that T
is exact equivalent to the torsion-free class over another algebra which is induced by some classicalcotilting module. Thus T is also a special case of (1).
For those familiar with τ -tilting theory, we will state a consequence of our result in τ -tiltingtheoretical language. We omit the related definitions and notation, see [AIR] for the detail.
Corollary 3.5.14. Let Λ be an artin algebra and F = Sub U a functorially finite torsion-freeclass of mod Λ where U is a support τ − -tilting module. Then the following are equivalent:(1) F satisfies (JHP).
(2) The number of non-isomorphic simple objects is equal to |U |.
Dually we obtain the τ -tilting and torsion-class version:
Corollary 3.5.15. Let Λ be an artin algebra and T = Fac T a functorially finite torsion
class of mod Λ where T is a support τ -tilting module. Then the following are equivalent:(1) F satisfies (JHP).
(2) The number of non-isomorphic simple objects is equal to |T |.
In the case of functorially finite torsion-free classes, we obtain the following finiteness resulton the positive cone M(F)can ∼= K+
0 (F). Note that M(F) itself is not in general finitely generated(see Section 3.8.3.1 for example).
Proposition 3.5.16. Let Λ be an artin algebra and F a torsion-free class of mod Λ such that F
is the smallest torsion-free class which contains some Λ-module U (for example, F is functoriallynfinite). Then K+
0 (F) is isomorphic to a finitely generated submonoid of N for some n.
Proof. This follows from the following fact: F is described as the category of modules whichhave finite filtrations such that each successive quotient is a submodule of U (see [MS, Lemma3.5. (JHP) FOR EXTENSION-CLOSED SUBCATEGORIES OF MODULE CATEGORIES

433.1]). It follows that M(F) is generated by {[V ] | V is a submodule of U }. By Corollary 3.5.9, we
may identify K+
0 (F) with the monoid of dimension vectors of modules in F, which is generated by
the set of dimension vectors of submodules of U . This set is obviously finite, so K+
0 (F) is finitely
generated.

3.5.3. Torsion-free classes over Nakayama algebras. In this subsection, we investigate
torsion-free classes over Nakayama algebras, and show that any such categories satisfy (JHP).
First we recall the notion of Nakayama algebras. For an artin algebra Λ, we say that a
Λ-module M is uniserial if the set of submodules of M is totally ordered by inclusion. An
artin algebra Λ is called Nakayama if every indecomposable right and left projective Λ-module
is uniserial. We will use the following description of indecomposable modules over Nakayama
algebras (see e.g. [ASS, Chapter V] or [ARS, Section VI.2] for the detail).
Proposition 3.5.17. Let Λ be a Nakayama algebra. Then the following hold.
(1) Every indecomposable modules in mod Λ is uniserial.
(2) M ∈ ind(mod Λ) is uniquely determined by the following data:(a) The simple module S := top M = M/ rad M .
(b) m := l(M ), the length of M as a Λ-module.
In particular, Λ is of finite representation type. We denote this module by S (m) .
(3) If two indecomposable modules M and N in mod Λ satisfy top M ∼= top N and l(M ) ≥l(N ), then there exists a surjection M  N .
We will investigate simple objects in a torsion-free class of mod Λ for a Nakayama algebra.
Note that since mod Λ has finitely many indecomposables, every torsion-free class of mod Λ isfunctorially finite, so it satisfies Assumption 3.5.6 by Example 3.5.13 (3).
Theorem 3.5.18. Let Λ be a Nakayama algebra and F a torsion-free class of mod Λ. Then
there exists bijections between the following three sets:(1) top F, the set of isomorphism classes of simple modules top M for M ∈ ind F.
(2) sim F, the set of isomorphism classes of simple objects in F.
(3) ind P(F), the set of isomorphism classes of indecomposable projective objects in F.
The maps from (2) and (3) to (1) are given by M 7→ top M . On the other hand, for a simple
module S ∈ top F in (1), the corresponding objects are given by S (m) in (2) and S (n) in (3), wherem := min{i | S (i) ∈ F},
andn := max{i | S (i) ∈ F}.

Proof. Let us denote by top : ind F → top F the map which sends M to top M . This induces
maps sim F → top F and ind P(F) → top F. We will show that these two maps are bijections.
(top : sim F → top F is a bijection): First we will show that this map is a surjection. For
S ∈ top F, put m := min{i | S (i) ∈ F}. We claim that S (m) is a simple object in F. Suppose thisis not the case. Then there exists a short exact sequence
0 → K → S (m) → M → 0
in F with K, M 6= 0. Since M is a quotient of S (m) , it has top S. But this contradicts theminimality of m because 0 < l(M ) < l(S (m) ) = m holds. Thus S (m) belongs to sim F. Hence themap sim F → top F is a surjection.
Next we will show that this map is an injection. Suppose that this is not the case. Then there
exist a simple module S and 0 < i < j such that both S (i) and S (j) are simple in F. Then we
have a short exact sequence
0 → K → S (j) → S (i) → 0
in mod Λ by Proposition 3.5.17 (3), and K 6= 0 since i 6= j. However, K belongs to F since F
is closed under submodules. Thus the above is a conflation in F, which shows that S (j) is not a
simple object in F. This is a contradiction, so sim F → top F is an injection.
It is clear from the above argument that the inverse of the map top : sim F → top F is given
by S 7→ S (m) as claimed.

44

3. (JHP) AND GROTHENDIECK MONOIDS OF EXACT CATEGORIES(top : ind P(F) → top F is a bijection): First we will show that this map is a surjection. Let
M be an indecomposable object in F and put S := top M . Since F satisfies Assumption 3.5.6, it
has enough projectives by Proposition 3.5.8 1. Thus there exists a deflation
P1 ⊕ P2 ⊕ · · · ⊕ Pl  M
in F, where Pi ∈ ind P(F) for each i. This map is a surjection in mod Λ, so it induces a surjection
top P1 ⊕ top P2 · · · ⊕ top Pl  top M = S.
It follows that top Pi = S holds for some i. This means that the map ind P(F) → top F issurjective.
Next we will show that this map is an injection. Suppose that this is not the case. Then there
exists a simple module S and 0 < i < j such that both S (i) and S (j) are projective objects in F.
Now Proposition 3.5.17 (3) implies that there exists a short exact sequence
0 → K → S (j) → S (i) → 0
in mod Λ. Since F is closed under submodules, this is a conflation in F, and since i < j, wehave K 6= 0. However, the projectivity of S (i) implies that the above sequence splits, which is acontradiction since S (j) is indecomposable. Thus top : ind P(F) → top F is an injection.
Finally, we shall describe the inverse of top : ind P(F) → top F. For an object S ∈ top F, put
n := max{i | S (i) ∈ F }, and we claim that the map S 7→ S (n) is the inverse. Take any 1 ≤ i < nwith S (i) ∈ F. Then as in the proof of injectivity, we have the following conflation in F
0 → K → S (n) → S (i) → 0
with K 6= 0, and this sequence does not split since S (n) is indecomposable. Thus S (i) cannot be
projective in F. By this fact and the fact that top : ind P(F) → top F is surjective, we must havethat the inverse of this map is given by S 7→ S (n) .
As an immediate corollary, we obtain the following result.
Corollary 3.5.19. Let Λ be a Nakayama algebra and F a torsion-free class of mod Λ. ThenF satisfies (JHP).
Proof. By Theorem 3.5.18, we have an equality # sim F = # ind P(F). Thus the assertion
follows from Theorem 3.5.10.

3.6. Torsion-free classes of Type A and Bruhat inversion
In this section, we investigate simple objects in a torsion-free class of the category of representation of a quiver of type A by using the combinatorics of the symmetric group. For a quiverQ, we denote by kQ the path algebra of Q over a field k. As usual, we identify representations of
Q with right kQ-modules.
For an acyclic quiver Q, a classification of torsion-free classes of mod kQ with finitely many
indecomposables is known: they are in bijection with so-called c-sortable elements of the Coxeter
group of Q ([IT] for the Dynkin case and [AIRT, Tho] for the general case). For a quiver of type A,
the corresponding Coxeter group is just a symmetric group, and we can describe all indecomposable
kQ-modules in a quite explicit way. In this section, we freely use these description particular totype A, but in a paper [Eno5], we will show that the results are valid in other Dynkin types, ormore generally, preprojective algebras of Dynkin types (see Remark 3.6.19).
1Another way to show this is to use [Eno2, Corollary 3.15]: Every Hom-finite Krull-Schmidt exact k-category
has enough projectives if it has only finitely many indecomposables.

3.6. TORSION-FREE CLASSES OF TYPE A AND BRUHAT INVERSION

45

3.6.1. Bruhat inversions of elements in the symmetric group. First we recall combinatorial notions on the symmetric groups we need later. These notions are well-studied in thecontext of Coxeter groups, but we will give an explicit description for type A case here for theconvenience of the reader. The standard reference is [BB].
We denote by Sn+1 the symmetric group which acts on the set {1, 2, · · · , n, n + 1} fromleft. We often use the one-line notation to represent elements of Sn+1 , that is, we write w =w(1)w(2) · · · w(n + 1) for w ∈ Sn+1 . We denote by (i j) for 1 ≤ i, j ≤ n + 1 the transposition of
the letters i and j, and write T for the set of all transpositions in Sn+1 . Then Sn+1 is generated by
the simple reflections si := (i i + 1) for 1 ≤ i ≤ n. We write S for the set of all simple reflections.
For example, we have s2 s1 s3 s2 = 3412 in S4 . Note that for a transposition t = (i j) and w ∈ Sn+1 ,the element tw is obtained by interchanging two letters i and j in the one-line notation for w, e.g.
we have (3 4) · 3412 = 4312.
Each element w ∈ Sn+1 can be written as a product of simple reflections:w = si1 si2 · · · silThis expression of w is called reduced if l is the minimal among all such expressions. In this case,
we call l the length of w and write `(w) := l.
For an element w ∈ Sn+1 , a transposition t ∈ T is called an inversion of w if `(tw) < `(w)
holds, and we denote by inv(w) the set of all inversions of w. It is known that a transposition (i j)with i < j is an inversion of w if and only if j precedes i in the one-line notation for w, that is,
w−1 (i) > w−1 (j). It is also known that `(w) = # inv(w) holds.
Example 3.6.1. The following are examples of inversions in S5 .
(1) For w1 = s1 s3 s2 s4 s1 s3 s2 s4 = 45231 we have:inv(w1 ) = {(1 2), (1 3), (1 4), (1 5), (2 4), (2 5), (3 4), (3 5)}.
(2) For w2 = s1 s3 s2 s4 s1 s3 s2 s1 = 54213 we have:inv(w2 ) = {(1 2), (1 4), (1 5), (2 4), (2 5), (3 4), (3 5), (4 5)}
The following class of inversions plays an important role in this paper, since we shall see that
this corresponds to simple objects in a torsion-free class.
Definition 3.6.2. We say that an inversion t of an element w ∈ Sn+1 is a Bruhat inversion
of w if it satisfies `(tw) = `(w) − 1. We denote by Binv(w) the set of Bruhat inversions of w.
We can interpret Bruhat inversions in terms of the cover relation of the Bruhat order. Recall
that the Bruhat order on Sn+1 is a partial order ≤ generated by the following relation: for everyt ∈ T and w ∈ Sn+1 with `(tw) < `(w), we have that tw < w holds. In what follows, we always
denote by ≤ (and <) the Bruhat order on Sn+1 .
Lemma 3.6.3 ([BB, Lemma 2.1.4]). For a transposition t = (i j) ∈ T with i < j and anelement w ∈ Sn+1 , the following are equivalent:(1) t is a Bruhat inversion of w.
(2) tw is covered by w in the Bruhat order, that is, tw < w holds and there exists no elementu ∈ Sn+1 satisfying tw < u < w.
(3) In the one-line notation for w, the letter j precedes i, and there exists no l with i < l < jsuch that the letter l appears between j and i.
(4) (i j) ∈ inv(w) and there exists no l with i < l < j satisfying (i l), (l j) ∈ inv(w).
Example 3.6.4. The following are Bruhat inversions of elements in Example 3.6.1.
(1) For w1 = 45231 we have:Binv(w1 ) = {(1 2), (1 3), (2 4), (2 5), (3 4), (3 5)}.
(2) For w2 = 54213 we have:Binv(w2 ) = {(1 2), (2 4), (3 4), (4 5)}Note that # inv(w1 ) = # inv(w2 ) = 8 but Binv(w1 ) = 6 > 4 = # Binv(w2 ).

46

3. (JHP) AND GROTHENDIECK MONOIDS OF EXACT CATEGORIES

We will use the notion of support of elements in Sn+1 .
Definition 3.6.5. Let w be an element of Sn+1 . Then i ∈ {1, · · · , n} is called a support of w
if there exists some reduced expression of w which contains si . We denote by supp(w) the set of
all supports of w. We say that w has full support if supp(w) = {1, 2, · · · , n}.
In fact, if i is in supp(w), then any reduced expression of w contains si ([BB, Corollary 1.4.8]).
We will use the following characterization of the support later.
Lemma 3.6.6. For w ∈ Sn+1 and i ∈ {1, · · · , n}, the following are equivalent:(1) i ∈ supp(w).
(2) There exists some j with 1 ≤ j ≤ i such that w−1 (j) > i, that is, j does not appear in
the initial segment of length i in the one-line notation for w.
(3) There exists some j with 1 ≤ j ≤ i such that w(j) > i.
(4) There exists some l with i < l such that l appears in the initial segment of length i in the
one-line notation for w.
For example, by Using this criterion, we can easily check that supp(45231) = {1, 2, 3, 4, 5},supp(21543) = {1, 3, 4}, and supp(12543) = {3, 4}.
3.6.2. Coxeter element, c-sortable elements and torsion-free classes. We will describe the Ingalls-Thomas bijection [IT] between sortable elements and torsion-free classes for aquiver of type A, following [Tho]. In what follows, Q is a quiver of type A with n vertices, whose2···
n . As usual, we identify right kQ-modules overunderlying graph is 1
the path algebra Q and representations of Q. For a kQ-module M and a vertex i of Q, we denoteby Mi the vector space attached to i.
A Coxeter element of Sn+1 is an element c ∈ Sn+1 which is obtained as the product of all
simple reflections s1 , · · · , sn ∈ S in some order, or equivalently, an element with length n which
has full support. We say that a Coxeter element c is associated to Q if si appears before sj in cwhenever there exists an arrow i ← j in Q. It is known that Coxeter elements are in bijection
with orientations of edges in the underlying graph of Q.
Example 3.6.7. We give an example of the correspondence between Coxeter elements in S4
and orientations of A3 quiver.
Orientations of A3
1←2←3
1←2→3
1→2←3
1→2→3

Coxeter elements in S4
s1 s2 s3 = 2341
s1 s3 s2 = s3 s1 s2 = 2413
s2 s1 s3 = s2 s3 s1 = 3142
s3 s2 s1 = 4123

In what follows, we adopt the following convention:
Assumption 3.6.8. Q is a quiver of type A with n vertices, whose underlying graph is givenby2···
and c is the Coxeter element of Sn+1 associated with Q.
1nTorsion-free classes of mod kQ are classified by the combinatorial notion called c-sortableelements, which was introduced by Reading [Rea1].
Definition 3.6.9. Let c be a Coxeter element of Sn+1 . We say that an element w of Sn+1 is
c-sortable if there exists a reduced expression of the form w = c(0) c(1) · · · c(m) such that each c(i)is a subword of c satisfying supp(c(0) ) ⊃ supp(c(1) ) ⊃ · · · ⊃ supp(c(m) ).
Now we can state the correspondence between c-sortable elements and torsion-free classes. A
support of a module M ∈ mod kQ is a vertex i ∈ Q with Mi 6= 0, and we denote by [i, j) the setof elements l ∈ {1, 2, · · · , n} with i ≤ l < j. The following is just a restatement of the well-knownclassification of indecomposable representations of Q.

3.6. TORSION-FREE CLASSES OF TYPE A AND BRUHAT INVERSION

47

Proposition 3.6.10. Let Q be a quiver as in Assumption 3.6.8. For a transposition (i j) ∈ Twith i < j, there exists a unique indecomposable module M := M[i,j) in mod kQ whose support is[i, j). This module is a representation of Q defined by the following:• For vertices, Ml = k if l ∈ [i, j) and Ml = 0 otherwise.
• For arrows i → j in Q, we put idk : k → k if Mi = Mj = k, and 0 otherwise.
∼∼

Moreover, this give a bijection M[ ) : T −→ ind(mod kQ) which restricts to S −→ sim(mod Λ).
Example 3.6.11. Let Q be the quiver Q = 1 → 2 ← 3. Then the Auslander-Reiten quiver ofmod kQ is as follows, where indecomposables are labelled according to Proposition 3.6.10.
M[2,4)
M[2,3)

M[1,2)

M[1,4)

M[1,3)

M[3,4)

Now torsion-free classes of mod kQ are classified as follows. Here a support of a subcategory
F of mod kQ is the union of supports of modules in F.
Theorem 3.6.12 ([Tho, Theorem 4.2]). Let Q and c be as in Assumption 3.6.8. For w ∈ Sn+1 ,define the subcategory F(w) of mod kQ by
F(w) := add{M[i,j) | (i j) ∈ inv(w) with i < j}.
then the following hold.
∼(1) If w is c-sortable, then F(w) is a torsion-free class of mod kQ, and the bijection T −→∼ind(mod kQ) given by (i j) 7→ M[i,j) restricts a bijection inv(w) −→ ind F(w).
(2) The map w 7→ F(w) gives a bijection between c-sortable elements of Sn+1 and torsion-freeclasses of mod kQ.
(3) Let w be a c-sortable element. Then supp(w) coincides with the support of F(w), andthe equality # supp(w) = # ind P(F(w)) = # ind I(F(w)) holds.
Proof. The proof is essentially contained in other references such as [Tho], but we will clarify
the relation between our convention and others, since others often use the language of root systems.
We freely use basics of the root system. Let Φ be a root system of type An whose Dynkin graphis as in Assumption 3.6.8, and let e1 , · · · , en be the set of simple roots corresponding to vertices
1, · · · , n. We identify Sn+1 with the Weyl group W of Φ as usual. Under this identification, atransposition (i j) of Sn+1 corresponds to a reflection with respect to a positive root α[i,j) :=Proots.
l∈[i,j) el . By this, the set of transpositions are in bijection with the set of positiveP
For a kQ-module M , we define the dimension vector dim M by dim M := l dimk (Ml )el .
Then M[i,j) in Proposition 3.6.10 satisfies dim M[i,j) = α[i,j) .
The following description of inv(w) is well-known, see e.g. [BB, Proposition 4.4.6]:
Lemma 3.6.13. Let t ∈ Sn+1 = W be a transposition, α ∈ Φ the positive root corresponding
to t and w = si1 si2 · · · sir a reduced expression of arbitrary w ∈ W . Then t ∈ inv(w) if and only
if α = si1 si2 · · · sil−1 (eil ) for some 1 ≤ l ≤ r.
Now return to our proof. (1) and (2) follow from [Tho, Theorem 4.2] and our above identification. Let us prove (3).
Let w be a c-sortable element. By Example 3.5.13, the equality # ind P(F(w)) = # ind I(F(w))
holds. On the other hand, the number of supports of F(w) is equal to # ind I(F(w)) by the general theory of support (τ -)tilting theory, see e.g. [IT, Proposition 2.5(4), Lemma 2.9] or [AIR,Theorem 2.7]. Thus it suffices to show that the number of supports of F(w) is equal to # supp(w).
Take a reduced expression w = c(0) c(1) · · · c(m) such that c(i) is a subword of c and satisfies
supp(c(0) ) ⊃ supp(c(1) ) ⊃ · · · ⊃ supp(c(m) ). First suppose that l ∈ supp(w), which means that sl
appears in the word c(0) . Then we can write the reduced expression of c(0) as c(0) = c0 sl c00 suchthat c0 (and c00 ) does not contain the letter sl . Let (i j) be a transposition which corresponds to48

3. (JHP) AND GROTHENDIECK MONOIDS OF EXACT CATEGORIES

a positive root α := c0 (el ), then it belongs to inv(w) by Lemma 3.6.13. Clearly the el -componentof α is equal to 1, thus l is a support of M[i,j) since dim M[i,j) = α.
Conversely, suppose that l is a support of M := M[i,j) for some (i j) ∈ inv(w). Take a positiveroot which corresponds to (i j). Then we have dim M = α, thus the el -component of α is strictly
positive. On the other hand, Lemma 3.6.13 shows that l must be appear in a reduced expressionof w, thus l ∈ supp(w).

3.6.3. Bruhat inversions and simples. Our main result in this section is the following,
which establishes a bijection between simples in F(w) and Bruhat inversions of w.
Theorem 3.6.14. Let Q and c be as in Assumption 3.6.8 and w a c-sortable element of Sn+1 .
∼Then the natural bijection inv(w) −→ ind F(w) restricts to a bijection Binv(w) → sim F(w). In
other words, for (i j) ∈ inv(w) with i < j, the object M[i,j) is simple in F(w) if and only if (i j)
is a Bruhat inversion of w.
Before we give a proof, let us state the immediate consequence of this, which characterizes(JHP) in a purely combinatorial way. See Example 3.6.17 for the actual example of Theorem
3.6.14 and Corollary 3.6.15.
Corollary 3.6.15. Let Q and c be as in Assumption 3.6.8 and w a c-sortable element of
Sn+1 . Then F(w) satisfies (JHP) if and only if # supp(w) = # Binv(w) holds.
Proof. Theorem 3.6.14 implies # sim F(w) = # Binv(w). On the other hand, Theorem 3.6.12
implies # ind P(F(w)) = # supp(w). Thus our assertion follows from Theorem 3.5.10.

To prove Theorem 3.6.14, we use the following explicit exact sequences in mod kQ.
Lemma 3.6.16. Let (i j) be a transposition in Sn+1 with i < j and consider the kQ-module
M[i,j) given in Proposition 3.6.10. Then the following hold.
(1) For each l with i < l < j, one of the following two exact sequences exists:
either0 → M[i,l) → M[i,j) → M[l,j) → 0,or

0 → M[l,j) → M[i,j) → M[i,l) → 0.

(3.6.1)(3.6.2)0(2) Suppose that we have a monomorphism M[l,l0 ) ,→ M[i,j) for some l < l . Then we have[l, l0 ) ⊂ [i, j), and there exists an exact sequence
0 → M[l,l0 ) → M[i,j) → M[i,l) ⊕ M[l0 ,j) → 0.

(3.6.3)Moreover, there exist the following two exact sequences:and

0 → M[i,l0 ) → M[i,j) → M[l0 ,j) → 0,

(3.6.4)0 → M[l,j) → M[i,j) → M[i,l) → 0.

(3.6.5)Here we put M[a,a) = 0 for a ∈ {1, · · · , n + 1}.
Proof. (1) Consider the orientation of the edge between l − 1 and l in Q. Suppose that
we have l − 1 ← l in Q. A subspace M[i,l) of M[i,j) is closed under the action of Q in M[i,j)because there exists no path which starts from one of [i, l) and ends at one of [l, j). Thus M[i,l) is
a submodule of M[i,j) , and we obtain (3.6.1). By the same reason, if we have l − 1 → l in Q, thenwe have (3.6.2).
(2) Since M[l,l0 ) must be a subspace of M[i,j) , it is clear that [l, l0 ) ⊂ [i, j) holds. The supportof the quotient M[i,j) /M[l,l0 ) is a disjoint union [i, l) t [l0 , j), and it is easily checked that the actionof kQ on this quotient coincides with that on M[i,l) ⊕ M[l0 ,j) . Thus we obtain (3.6.3). Moreover,since M[l,l0 ) is closed under actions of kQ in M[i,j) , we must have that either i = l or l − 1 → l in
Q holds, and that either l0 = j or l0 − 1 ← l0 in Q holds. Thus the existence of two exact sequences(3.6.4) and (3.6.5) follows from the proof of (1).
Proof of Theorem 3.6.14. Let (i j) be an inversion of w with i < j. Since any simple
object is indecomposable, it suffices to show that (i j) is a Bruhat inversion of w if and only if
M[i,j) is simple in F(w).

3.6. TORSION-FREE CLASSES OF TYPE A AND BRUHAT INVERSION

49

First, suppose that (i j) is not a Bruhat inversion of w. Then by Proposition 3.6.3, there
exists some l with i < l < j such that both (i l) and (l j) belong to inv(w). Thus both M[i,l) andM[l,j) belong to F(w). Now we have an exact sequence (3.6.1) or (3.6.2) by Lemma 3.6.16. In
either case, this gives a conflation in F(w), and since M[i,l) and M[l,j) are non-zero, M[i,j) is nota simple object in F(w).
Conversely, suppose that M[i,j) is not a simple object in F(w). Then we have a non-isomorphic
inflation N  M[i,j) in F(w) for some object 0 6= N of F(w). Take an indecomposable direct
summand M[l,l0 ) of N . Then the composition M[l,l0 ) ,→ N  M[i,j) is a non-isomorphic inflation in
F(w), because the section M[l,l0 ) ,→ N is an inflation and inflations are closed under compositions.
Now Lemma 3.6.16 (2) tells us that [l, l0 ) ⊂ [i, j) and that M[i,l) ⊕ M[l0 ,j) belongs to F(w), hence
both M[i,l) and M[l0 ,j) belong to F(w) since F(w) is closed under direct summands.
Since M[l,l0 ) ,→ M[i,j) is not an isomorphism, we have i < l or l0 < j. Suppose that the former
holds. Then since M[i,l) ∈ F(w), the transposition (i l) is an inversion of w. On the other hand,the exact sequence (3.6.5) of Lemma 3.6.16 implies that M[l,j) belongs to F(w), since F(w) is
closed under submodules. Thus (l j) is also an inversion of w. This implies that (i j) is not a
Bruhat inversion by Lemma 3.6.3. The case i < l0 < j is completely similar, except we use (3.6.4)
instead of (3.6.5).

Example 3.6.17. Let us look at several examples.
(1) Let Q be the quiver Q = 1 → 2 ← 3. Then the Coxeter element associated with
Q is c = s2 s1 s3 = 3142 ∈ S4 . In Table 1, we list all c-sortable elements and thecorresponding inversions, Bruhat inversions and torsion-free classes. The black vertices
indicate simple objects in F(w) and the white ones indicate the rest of indecomposablesin F(w). According to this table, we conclude that F(w) satisfies (JHP) except w = 3412.
(2) Let Q be the quiver Q = 1 ← 2 → 3 ← 4. The associated Coxeter element is c =s1 s3 s2 s4 = 24153 ∈ S5 . The Auslander-Reiten quiver of mod kQ is as follows:M[3,5)
M[3,4)

M[1,5)

M[1,4)
M[1,2)

M[1,3)
M[2,3)

M[2,5)

M[2,4)

M[4,5)In Table 2, we list all faithful torsion-free classes of mod kQ, which corresponds to csortable elements with full support. Here as in (1), the black vertices indicate simple
objects in F(w) and the white the rest. From this table, for example, we can check thatthe number of faithful torsion-free classes satisfying (JHP) is 8. By using the computer
program, we can calculate that the number of all torsion-free classes is 42, which is theCatalan number, and 34 ones among them satisfy (JHP).
As an application of Corollary 3.6.15, we obtain the following result on the linearly oriented
case. The proof is purely combinatorial. Note that this also follows from Corollary 3.5.19, sincekQ is a Nakayama algebra in this case.
Corollary 3.6.18. Let Q be a linearly oriented quiver of type A. Then every torsion-freeclasses F of mod kQ satisfies (JHP).
Proof. We may assume that Q is 1 → 2 → · · · → n, so c = sn · · · s2 s1 . We use the
combinatorial characterization of the c-sortable-ness given in [Rea1, Theorem 4.12]: an elementw ∈ Sn+1 is c-sortable if it is 231-avoiding, that is, there exists no i < l < j such that l appears
before j and j appears before i in the one-line notation for w.
By Theorem 3.6.12, there exists a c-sortable element w ∈ Sn+1 satisfying F = F(w). Then
according to Corollary 3.6.15, the assertion amounts to the following purely combinatorial lemma(or equivalently, this lemma follows from two categorical statements: Corollaries 3.6.15 and 3.5.19).

50

3. (JHP) AND GROTHENDIECK MONOIDS OF EXACT CATEGORIES

Table 1. Example of Theorem 3.6.14 for Q = 1 → 2 ← 3F(w)c-sortable elements w

supp(w)

inv(w)

Binv(w)e = 1234∅∅∅

s2 = 1324

2

(2 3)

(2 3)

s1 = 2134

1

(1 2)

(1 2)

s3 = 1243

3

(3 4)

(3 4)

s2 s1 = 3124

1, 2(1 3), (2 3)(1 3), (2 3)s2 s3 = 1342

2, 3(2 3), (2 4)(2 3), (2 4)s1 s3 = 2143

1, 3(1 2), (3 4)(1 2), (3 4)s2 s1 s2 = 3214

1, 2(1 2), (1 3), (2 3)(1 2), (2 3)s2 s3 s2 = 1432

2, 3(2 3), (2 4), (3 4)(2 3), (3 4)s2 s3 s1 = 3142

1, 2, 3

(1 3), (2 3), (2 4)(1 3), (2 3), (2 4)s2 s3 s1 s2 = 3412

1, 2, 3

(1 3), (1 4),(2 3), (2 4)(1 3), (1 4),(2 3), (2 4)s2 s3 s1 s2 s1 = 4312

1, 2, 3

(1 3), (1 4),(2 3), (2 4), (3 4)(1 3), (2 3), (3 4)s2 s3 s1 s2 s3 = 3421

1, 2, 3

(1 2), (1 3), (1 4),(2 3), (2 4)(1 2), (2 3), (2 4)s2 s3 s1 s2 s3 s1 = 4321

1, 2, 3

(1 2), (1 3), (1 4),(2 3), (2 4), (3 4)(1 2), (2 3), (3 4)Lemma. For every 231-avoiding element w of Sn+1 , we have # Binv(w) = # supp(w).
∼To prove this lemma, we will construct an explicit bijection Binv(w) −→ supp(w). In what
follows, we will work on the one-line notation for w.
Let (i j) be a Bruhat inversion of w with i < j. We claim that i ∈ supp(w) holds. If this is
not the case, then the initial segment of length i contains exactly 1, 2, · · · , i by Lemma 3.6.6. Thisis a contradiction since j (> i) appears before i by (i j) ∈ inv(w). Thus i ∈ supp(w) holds, andwe obtain a map Binv(w) → supp(w) by (i j) 7→ i.
We will show that this map is an injection. Suppose (i j1 ) and (i j2 ) are two different Bruhat
inversion of w with i < j1 < j2 , so j1 and j2 appears before i. Then j1 must appear before

3.6. TORSION-FREE CLASSES OF TYPE A AND BRUHAT INVERSION

51

Table 2. Example of Theorem 3.6.14 for faithful torsion-free classes over Q =1 ← 2 → 3 ← 4.
inv(w)

c-sortable elements w
c = s1 s3 s2 s4= 24153

inv(c) =(1 2), (1 4)(3 4), (3 5)Binv(w)

F(w)# sim F(w)inv(c)

4

cs1 = 42153

inv(c), (2 4)(1 2), (2 4),(3 4), (3 5)4cs3 = 24513

inv(c), (1 5)

inv(c), (1 5)5cs1 s3 = 42513

inv(c), (1 5), (2 4)(1 2), (1 5),(2 4), (3 4), (3 5)5cs3 s2 = 25413

inv(c), (1 5), (4 5)(1 2), (1 4),(3 4), (4 5)4cs3 s4 = 24531

inv(c), (1 3), (1 5)(1 2), (1 3),(3 4), (3 5)4cs1 s3 s2 = 45213

inv(c), (1 5),(2 4), (2 5)(1 2), (2 4), (2 5),(3 4), (3 5)5cs1 s3 s4 = 42531

inv(c), (1 3),(1 5), (2 4)(1 2), (1 3),(2 4), (3 4), (3 5)5cs3 s2 s4 = 25431

inv(c), (1 3),(1 5), (4 5)(1 2), (1 3),(3 4), (4 5)4c2 = cs1 s3 s2 s4= 45231

inv(c), (1 3), (1 5),(2 4), (2 5)(1 2), (1 3), (2 4),(2 5), (3 4), (3 5)6cs1 s3 s2 s1= 54213

inv(c), (1 5),(2 4), (2 5), (4 5)(1 2), (2 4),(3 4), (4 5)4c2 s1 = cs1 s3 s2 s4 s1= 54231

inv(c), (1 3), (1 5),(2 4), (2 5), (4 5)(1 2), (1 3), (2 4),(3 4), (4 5)5c2 s3 = cs1 s3 s2 s4 s3= 45321inv(c), (1 3), (1 5),(2 3), (2 4), (2 5)(1 2), (2 3),(3 4), (3 5)4c2 s1 s3 = cs1 s3 s2 s4 s1 s3= 54321inv(c), (1 3), (1 5),(2 3), (2 4), (2 5), (4 5)(1 2), (2 3),(3 4), (4 5)4j2 , since otherwise (i j2 ) would not be a Bruhat inversion. Thus w looks like · · · j1 · · · j2 · · · i · · · ,which contradicts to that w is 231-avoiding. Thus this map is an injection.
Next we will show that this map is a surjection. For i ∈ supp(w), we will show the following:(Claim): There exists some letter j such that i < j and j appears before i.
By Lemma 3.6.6, there exists some l with l > i such that l appears in the initial segment oflength i. Suppose that any letters left to i are less than i, that is, there is no j as in the above
claim. Then in particular i belongs to the initial segment of length i, and is left to l. Also by
Lemma 3.6.6 (2), there exists a letter i0 with i0 ≤ i such that i0 does not appear in the initial52

3. (JHP) AND GROTHENDIECK MONOIDS OF EXACT CATEGORIES

segment of length i. However we must have i0 6= i, thus i0 < i, since i appears in the initial segmentof length i. Now we have i0 < i < l and w looks like· · · i · · · l · · · i0 · · · , which contradicts to that w
is 231-avoiding. Thus the claim follows.
Take the rightmost letter j with the claimed property. Then (i j) is clearly a Bruhat inversion.
Thus the map Binv(w) → supp(w) is surjective.
Remark 3.6.19. In a paper [Eno5], we will show that the natural analogue for Theorem
3.6.14 holds for mod kQ with an arbitrary Dynkin quiver Q, or more generally, for mod Π for a
preprojective algebra Π of an arbitrary Dynkin type.
3.7. Computations of the Grothendieck monoids
So far, we do not know any method to compute the Grothendieck monoid M(E) for a given
length exact category E in general, except the following information:• M(E) is atomic, that is, generated by Atom M(E) = {[S] | S ∈ sim E} (Proposition 3.4.8).
• M(E) is just a free monoid if E turns out to satisfy (JHP) (Theorem 3.4.9).
• The cancellative quotient M(E)can , which is isomorphic to the positive part K+
0 (E) ofthe Grothendieck group (Proposition 3.A.15), is nothing but the monoid of dimension
vectors of modules belonging to E under mild assumption (Corollary 3.5.9).
Thus the problem is: we do not know how to check whether M(E) is cancellative or not (if M(E)is cancellative, then M(E) can be calculated in principle by the above), and the calculation seems
to be rather difficult if M(E) is not cancellative.
In this section, we will show (a bit artificial) examples of calculations of M(E) such that M(E)is not cancellative.
3.7.1. A category of modules with a designated set of dimension vectors. We willconvince the reader that lots of monoids can appear as the Grothendieck monoids of exact categories. First we consider the case of split exact categories.
Proposition 3.7.1. Let E be a split exact category, that is, every conflation splits. Then
M(E) is isomorphic to Iso E (with addition given by ⊕) as monoids.
Proof. This follows from the construction given in Proposition 3.3.3, or one can show thisby checking the universal property directly. The details are left to the reader.

Using this, we can realize any submonoids of Nn as Grothendieck monoids.
Proposition 3.7.2. Let M be a submonoid of Nn for some n ≥ 0. Then there exists a split
exact category E whose Grothendieck monoid is isomorphic to M .
Proof. Let k be a field and consider a semisimple k-algebra Λ := k n . Then Λ has n nonisomorphic simple modules, and we denote by dim X ∈ Nn for X ∈ mod Λ the dimension vectorof X, see Definition 3.5.3. Define the full subcategory E of mod Λ by the following:
E := {X ∈ mod Λ | dim X ∈ M }.
This is an extension-closed subcategory of mod Λ, and we claim that M(E) ∼= M holds. First
observe that every short exact sequence in mod Λ splits, so E is a split exact category. Thus it
suffices to show Iso E ∼= M by Proposition 3.7.1. The map X 7→ dim X induces an isomorphismnIso(mod Λ) ∼N,
andby construction, this map restricts an isomorphism Iso E ∼== M.
If E is a full subcategory of a module category, then by regarding E as an exact category witha split exact structure, Proposition 3.7.1 shows that our monoid M(E) only concerns with direct
sum decompositions of modules. In this case, M(E) is free if and only if the uniqueness of direct
sum decompositions holds (e.g. Krull-Schmidt categories), thus the combinatorial property of themonoid M(E) encodes information on the non-unique direct sum factorizations of modules. Thishas been studied by several authors (e.g. [Fac1, Fac2, BaGe]), and we refer the reader to therecent article [BaGe] and references therein for more information on this direction.
In a similar way to the construction of E above, we can attach extension-closed subcategories
of module categories to any submonoids of K0 (mod Λ) for an artin algebra Λ.

3.7. COMPUTATIONS OF THE GROTHENDIECK MONOIDS

53

Definition 3.7.3. Let Λ be an artin algebra with n non-isomorphic simple modules. For a
submonoid M of Nn , we define the subcategory EM of mod Λ by the following:
EM := {X ∈ mod Λ | dim X ∈ M }.
Then it is an extension-closed subcategory of mod Λ, thus an exact category.
By the universal property of M(EM ), we have a monoid homomorphism M(EM ) → M induced
by dim. Indeed M(EM ) is a thickening of M by some non-cancellative part:
Proposition 3.7.4. Let Λ be an artin algebra with n non-isomorphic simple modules and M a
submonoid of Nn . Then dim : M(EM ) → M induces an isomorphism M(EM )can ∼= M of monoids.
Proof. Let us introduce some notation. Write E := EM for simplicity. Let Si denote thesimple Λ-module with dim Si = ei for 1 ≤ i ≤ n, where ei is the standard basis of Nn . For annelement d ∈ NL
, we denote by P
Sd the unique semisimple Λ-module satisfying dim Sd = d, in otherwords, Sd := i Si⊕ai for d = i ai ei with ai ∈ N.
For d ∈ M , clearly we have Sd ∈ E. Since dim Sd = d holds, the map dim : M(E) → M is asurjection.
Thus, to prove M(E)can ∼= M , it suffices to show that for two objects X, Y ∈ E, we have[X] ∼can [Y ] in M(E) if and only if dim X = dim Y holds (see Proposition 3.A.15). The “only if”
part is clear since Nn (or M ) is cancellative, so we will prove the “if” part. To do this, it suffices
to show [X] ∼can [Sd ] for every X ∈ E with dim X = d.
Let lmax (X) be the maximum of lengths of indecomposable direct summands of X, and weshow [X] ∼can [Sd ] by induction on l := lmax (X). If l = 1, then X is semisimple and X ∼= Sd holds.
Suppose l > 1, and take a direct summand X1 of X with l(X1 ) = l. We have X = X1 ⊕X/X1 , andtake a simple submodule S of X1 . Note that S is a direct summand of Sd since S is a composition
factor of X. Thus there are two exact sequences in mod Λ:
0

S

X1

X1 /S

0

Sd /S

Sd /S ⊕ X/X1 ⊕ S

X/X1 ⊕ S

0,

and
0,

where the second one is just a split exact sequence. By taking direct sum of these two sequences,
we obtain the following exact sequence.
0Sd

X ⊕ SdS ⊕ X1 /S ⊕ X/X1

0

The dimension vectors of modules in this exact sequence is d, 2d and d respectively, so this is a
conflation of E. Therefore, we have [X] + [Sd ] = [X 0 ] + [Sd ] in M(E), hence [X] ∼can [X 0 ] holds,
where X 0 = S ⊕ X1 /S ⊕ X/X1 . Thus we can replace the direct summand X1 of X with S ⊕ X1 /S.
By iterating this process to all direct summands of X with length l, we obtain an object Z in E
such that lmax (Z) < l and [X] ∼can [Z]. Thus the claim follows from induction.
Thus our category EM has a monoid M as a cancellative quotient of its Grothendieck monoid
M(EM ), but M(EM ) is not cancellative in general. In the next subsection, we give an example ofcomputation of M(EM ).
3.7.2. The case of A2 quiver and monoids with one generator. Throughout this subsection, we denote by k a field and by Q a quiver 1 ← 2, and put Λ := kQ. Although the structureof the category mod Λ seems to be completely understood, this category has lots of extension-closed
subcategories, and the computation of their Grothendieck monoids is rather hard as we shall seein this subsection. A particular case of this computation was considered in [BeGr, Section 3.4].
First we recall basic properties of the category mod Λ. We have the complete list of indecomposable objects in mod Λ: two simple modules S1 and S2 , which are supported at vertices 1
and 2 respectively, and one non-simple projective injective module P . The Grothendieck group
K0 (mod Λ) is a free abelian group with basis {[S1 ], [S2 ]}, and we identify it with Z2 .
Now we apply the construction EM defined in the previous subsection to this case, where M isa submonoid of N2 . We will compute the monoid M(EM ) in the case that M is generated by one

54

3. (JHP) AND GROTHENDIECK MONOIDS OF EXACT CATEGORIES

element. In such a case, by Proposition 3.7.4, M(EM )can ∼= N holds, thus M(EM ) is a thickeningof N in some sense.
To present the monoid structure, we will use the Cayley quiver, which is a monoid version of
the Cayley graph of a group.
Definition 3.7.5. Let M be a monoid with a set of generators A ⊂ M . Then the Cayley
quiver of M with respect to A is a quiver defined as follows:• The vertex set is M .
a• For each a ∈ A and m ∈ M , we draw a (labelled) arrow m −→ m + a.
For an atomic monoid M , the natural choice of A above is the set Atom M of all atoms of M .
Now we can state our computation.
Proposition 3.7.6. Let M := N(m, n) be a submonoid of N2 generated by (m, n) ∈ N2 with(m, n) 6= (0, 0). Consider the exact category E := EM as in Definition 3.7.3. Then the following
hold, where we set l := min{m, n}.
(1) E has exactly l + 1 distinct simple objects A0 , · · · , Al , whereAi := P i ⊕ S1m−i ⊕ S2n−i
for 0 ≤ i ≤ l. We have dim Ai = (m, n) for every i.
(2) Put ai := [Ai ] ∈ M(E) for 0 ≤ i ≤ l, which are precisely the atoms of M(E). Then the
Cayley quiver of M(E) with respect to Atom M(E) is given as follows, where we draw
arrows  to represent l + 1 arrows a0 , · · · , al .
• (Case 1) The case m 6= n:
a0
a1

a0
a1

0
al

..
.

2a0

···

3a0

al

In particular, if m = 0 or n = 0, then M(E) ∼= N holds.
• (Case 2) The case m = n:
a0

a1
an

···

3a0

a1

a0

0

2a0

..
.
an

an

2an

an

3an

an

···

Here non-labelled arrows → represent n arrows a0 , · · · , an−1 .
Proof. For simplicity, put E := EM .
(1) It is clear that each Ai is a simple object in E since dim Ai = (m, n), and that Ai ∼= Ajholds if and only if i = j. On the other hand, we claim that every object in E is a direct sum of
Ai ’s, which clearly implies the assertion. In fact, take an object X ∈ E with dim X = (N m, N n)
for N > 0. Then X is isomorphic to P i ⊕ S1N m−i ⊕ S2N n−i for some 0 ≤ i ≤ N l. Take any integers
i1 , i2 , · · · , iN such that 0 ≤ ij ≤ l for each j and i = i1 + · · · + iN . Then it is straightforward to
see that X ∼= Ai1 ⊕ · · · ⊕ AiN holds.
(2) By Proposition 3.3.6, we have that Atom M(E) = {a0 , a1 , · · · , al } holds for ai := [Ai ], andall of them are distinct. On the other hand, we have a map ϕ : M(E) → N which sends an object
X with dim X = (N m, N n) to N for N ∈ N. This map is clearly a surjection (and actually this is
the universal cancellative quotient of M(E) by Proposition 3.7.4, but this fact is not needed here).
Denote by M(E)N the inverse image ϕ−1 (N ), and our strategy to compute M(E) is to compute
M(E)N . Clearly we have M(E)0 = {0 = [0]} and M(E)1 = Atom M(E) = {a0 , a1 , · · · , al }.

3.7. COMPUTATIONS OF THE GROTHENDIECK MONOIDS

55

Moreover, since every object of E is a direct sum of Ai ’s, we have x ∈ M(E)N if and only if there
exist integers 0 ≤ i1 , · · · , iN ≤ l satisfying x = ai1 + · · · + aiN . The key part of our computation
is to show the following equality in M(E):(Claim 1): For 1 ≤ i ≤ l and 0 ≤ j ≤ l, we have the equality ai + aj = ai−1 + aj in M(E),except the case j = m = n.
Proof of Claim 1. First, note that we have the following exact sequence in mod Λ.
0

S1

P

S2

0

Suppose that the equality j = m = n does not hold, then one can easily check that either j ≤ m−1
or j ≤ n − 1 (or both) holds. Suppose the former, and consider a split exact sequence below
0

P j ⊕ S1m−1−j ⊕ S2n−j

P i+j−1 ⊕ S12m−i−j ⊕ S22n−i−j

P i−1 ⊕ S1m−i+1 ⊕ S2n−i0.

By taking direct sum of the above two exact sequences, one obtains a conflation in E:0

Ai ⊕ Aj

Aj

Ai−1

0.

Thus ai + aj = ai−1 + aj holds in M(E).
Similarly, suppose that j ≤ n − 1 holds. Then by considering the following split sequence0

P i−1 ⊕ S1m−i ⊕ S2n−i+1

P i+j−1 ⊕ S12m−i−j ⊕ S22n−i−j

P j ⊕ S1m−j ⊕ S2n−1−j0,

as in the previous case, we obtain a conflation0

Ai ⊕ Aj

Ai−1

Aj

0,

which shows ai + aj = ai−1 + aj as well. This finishes the proof of Claim 1. 
In what follows, we consider two cases.
(Case 1): The case m 6= n. For every i, j with 0 ≤ i ≤ l and 0 ≤ j ≤ l, we have ai + aj =ai−1 + aj = · · · = a0 + aj = a0 + aj−1 = · · · = a0 + a0 = 2a0 by Claim 1. Thus M(E)2 = {2a0 }holds. We claim inductively that M(E)N = {N a0 } for N ≥ 2. This is because every element x in
M(E)N +1 can be written as x = ai + y for some 0 ≤ i ≤ l and y ∈ M(E)N , thus x = ai + N a0 =(ai + a0 ) + (N − 1)a0 = 2a0 + (N − 1)a0 = (N + 1)a0 . Now that M(E)N has been computed for
all N ≥ 0, the description of the Cayley quiver of M(E) easily follows.
(Case 2): The case m = n. Note that l = m = n in this case, so we only use the letter
n. For every i, j with 0 ≤ i ≤ n and 0 ≤ j ≤ n, if j 6= n, then ai + aj = ai−1 + aj = · · · =a0 + aj = a0 + aj−1 = · · · = 2a0 by Claim 1. Thus M(E)2 = {2a0 , 2an } holds (and later we willshow 2a0 6= 2an ). The same inductive argument as in (Case 1) shows that M(E)N = {N a0 , N an }holds for N ≥ 2, and it suffices to show that N a0 6= N an in M(E) for N ≥ 2.
N
Suppose that N a0 = N an holds in M(E), that is, [AN0 ] = [An ]. Denote by EP the fullnsubcategory of E consisting of direct sums of An = P , or equivalently, the subcategory of E
consisting of projective kQ-modules. Note that ANn ∈ EP holds. Then the following claim holds:(Claim 2): In (Case 2), for every conflation
0

X

Y

π

Z

0

in E, we have that Y ∈ EP holds if and only if both X ∈ EP and Z ∈ EP hold (in other words, EP
is a Serre subcategory of E).
Proof of Claim 2. If X ∈ EP and Z ∈ EP hold, then the above sequence splits since Z is
projective, thus Y ∼= X ⊕ Z ∈ EP holds. Conversely, suppose that Y ∈ EP holds. Since π induces
a surjection top Y  top Z, and top Y is a direct sum of top P = S2 , we have that so is top Z.
However, top Ai contains S1 as a direct summand if i < n. Therefore, Z, which is a direct sum of
appropriate Ai ’s, must be isomorphic to a direct sum of An . Thus Z ∈ EP holds. Moreover, theabove sequence splits since Z is projective, thus Y ∼= X ⊕ Z. Clearly EP is closed under direct
summands in E, so X belongs to EP . NLet us return to our situation, that is, [AN0 ] = [An ] holds. Since EP is a Serre subcategory ofN
E by Claim 2 and An belongs to EP , Proposition 3.3.7 implies that AN
0 belongs to EP . This is

56

3. (JHP) AND GROTHENDIECK MONOIDS OF EXACT CATEGORIESa contradiction since A0 is not projective. Therefore, we have N a0 6= N an , which completes theproof.

3.8. Counter-examples
In this section, we give examples of bad behavior of exact categories on several topics which
we have studied in the previous sections.
3.8.1. On the poset of admissible subobjects. In this subsection, we collect some counterexamples on the poset P(X) (see Section 3.2.1 for the detail)3.8.1.1. Subobject posets which are not lattices. The following example shows that P(X) is notnecessarily a lattice.
Example 3.8.1. Let k be a field and E a category of k-vector spaces whose dimensions are
not equal to 1 and 3. Then E is an extension-closed subcategory of mod k, thus an exact category.
Denote by X the 6-dimensional vector space with basis x1 , · · · , x6 , and put A := hx1 , x2 , x3 , x4 i andB := hx1 , x2 , x3 , x5 i. Then since X/A and X/B has dimension 2, both A and B are admissible
subobjects of X. Now the set-theoretic intersection of A and B is a 3-dimensional subspace
C := hx1 , x2 , x3 i, which does not belong to E, thus lower bounds of A and B in PE (X) are
precisely subspaces of C whose dimensions are exactly two or zero. Then A and B do not havethe greatest lower bound, since there are many two-dimensional subspaces of C.
The following is also an example in which P(X) is not a lattice, although the ambient category
is pre-abelian.
αβExample 3.8.2. Let k be a field and Λ an algebra Λ := k(1 ←−2←− 3)/(βα). Put E := proj Λ,then E has the natural exact structure induced from mod Λ, and every conflation splits in E. It
follows that a morphism ι : A → B in E is an inflation if and only if ι is a section. Moreover,since gl.dim Λ = 2 holds, kernels of morphisms between projective modules are projective, thus E
is pre-abelian.
We claim that PE (ΛΛ ) is not a lattice. By the previous argument, we can identify PE (Λ) withthe poset of submodules of Λ which are direct summands of Λ. We can write Λ as follows:ΛΛ = P1 ⊕ P2 ⊕ P3 = 1 ⊕ 21 ⊕ 32

(3.8.1)

Here Pi is an indecomposable projective modules corresponding to the vertex i, and the rightmost
notation indicates composition series of modules.
Consider the following two maps:


10
01
00



ι1 : P1 ⊕ P2 −−−−→ P1 ⊕ P2 ⊕ P3
1 0 
0 1
0β

ι2 : P1 ⊕ P2 −−−−→ P1 ⊕ P2 ⊕ P3
Here β : P2 → P3 denotes the multiplication map β · (−). We can check by matrix elimination thatthese maps are sections, thus Ni := Im ιi belongs to PE (Λ) for each i. In the notation of (3.8.1),
each Ni looks like the following:
N1 = {(a, cb , 00 ) ∈ P1 ⊕ P2 ⊕ P3 | a, b, c ∈ k}
N2 = {(a, cb , 0b ) ∈ P1 ⊕ P2 ⊕ P3 | a, b, c ∈ k}
We claim that N1 and N2 do not have the greatest lower bound in PE (Λ). Let N be a lower boundfor N1 and N2 in PE (Λ). Then N must be contained in the set-theoretic intersection N1 ∩ N2 :N1 ∩ N2 = 1 ⊕ 1 ⊕ 0 = {(a, 0c , 00 ) ∈ P1 ⊕ P2 ⊕ P3 | a, c ∈ k},which is a two-dimensional vector space spanned by e1 and α. Here (a, 0c , 00 ) in the above notation
corresponds to ae1 + cα. Consider the quotient module Λ/N1 ∩ N2 :
Λ/N1 ∩ N2 = S2 ⊕ P3 = 2 ⊕ 32

3.8. COUNTER-EXAMPLES

57

This is not projective, thus we have N1 ∩ N2 ∈/ PE (Λ). Hence we have N ( N1 ∩ N2 , so dim N = 0
or dim N = 1 holds. Therefore, to show that the meet of N1 and N2 does not exist in PE (Λ), it
suffices to show that there exist two distinct one-dimensional submodules of N1 ∩ N2 which belong
to PE (Λ). To show this, consider the following two maps:t[1,0,0]i1 : P1 −−−−→ P1 ⊕ P2 ⊕ P3
t[1,α,0]i2 : P1 −−−−→ P1 ⊕ P2 ⊕ P3
The images of these maps are one-dimensional submodules generated by e1 and e1 +α respectively.
By matrix elimination, we can show that these are direct summands of Λ, thus both belong toPE (Λ). Therefore PE (Λ) is not a lattice.
3.8.1.2. Subobject posets are lattices but not modular lattices. Recall from Proposition 3.2.15
that if E is a quasi-abelian category with the maximal exact structure, subobject posets arein fact lattices. However, there exist examples which are quasi-abelian but these lattices are notnecessarily modular. For such examples, we refer the reader to Section 3.8.2.1, since the modularity
implies the unique length property (Corollary 3.2.19).
3.8.2. (JHP) and the unique length property. Next let us see some counter-examples
on the unique factorization properties on exact categories.
3.8.2.1. Lengths are not unique. We collect some examples in which the unique length property
fails (see Section 3.2.2). Example D in the introduction is one of such example, but is rather
artificial and not idempotent complete. We give several idempotent complete examples (actually
torsion-free classes, so quasi-abelian) which do not satisfy the unique length property. Actually
we can obtain more examples in a quiver of type A by using the results in Section 3.6.
Example 3.8.3. Let Q be a quiver 1 ← 2 ← 3 → 4. Then the Auslander-Reiten quiver ofmod kQ is as follows. Here we use the notation introduced in Proposition 3.6.10.
M[4,5)

M[1,4)

M[1,5)
M[1,3)
M[1,2): E

M[2,4)

M[2,5)

M[2,3): simple objects in E

M[3,4)

M[3,5)

Define E as the additive subcategory of mod kQ corresponding to the gray region, then it isclosed under extensions (and actually a torsion-free class) in mod kQ. We can check that E has 5
simples indicated by circles: sim E = {M[1,2) , M[2,3) , M[3,5) , M[4,5) , M[1,4) }. Now consider an object
X := M[1,5) . Roughly speaking, the gray region below X looks like a module category of an A3
quiver, thus X seems to have length 3, and the gray region above X looks like a module categoryof an A2 quiver, thus X seems to have length 2. In fact, we have the following two composition
series of X in E:
0 < M[4,5) < X, and
0 < M[1,2) < M[1,3) < X.
Thus lengths of X are not unique.
Here is another example, which we have already encountered.
Example 3.8.4. Consider Example 3.6.17 and let w be the fourth one in Table 2, that is,
w = 42513. Then F(w) is the additive subcategory corresponding to {A, B, C, D, E, F } depicted
as follows:
D
B

F
C

A

E

58

3. (JHP) AND GROTHENDIECK MONOIDS OF EXACT CATEGORIES

We have that sim F(w) = {A, B, D, E, F }. Consider X := C ⊕ D. Then a conflation A  C  E
shows that C has length 2, hence X = C ⊕ D has length 3. On the other hand, a conflation
B  X  F implies that X has length 2. Thus lengths of X are not unique.
3.8.2.2. Length are unique but (JHP) fails. Here is an example of an exact category which
satisfies the unique length property, but does not satisfy (JHP).
Example 3.8.5. Let F be a hereditary torsion-free class of length abelian categories. ThenF satisfies the unique length property by Corollary 3.2.24. Typically, F arises in the following
way: Take any artin algebra Λ and chose any set of simple modules S = {S1 , · · · , Sl }. Define F
as follows:
F := {X ∈ mod Λ | HomΛ (X, Si ) = 0 for 1 ≤ i ≤ l}.
Then F is a torsion-free class, and the corresponding torsion class is the smallest Serre subcategorycontaining S. Thus F is a hereditary torsion-free class.
We have already encountered such examples which do not satisfy (JHP): E1 in Example E,
and F(c2 ) and F(c2 s1 ) in Example 3.6.17.
3.8.2.3. Examples which satisfy (JHP). Finally, we collect examples which satisfy (JHP) forthe convenience of the reader.
Example 3.8.6. The following examples satisfy (JHP).
(1) Krull-Schmidt categories together with the split exact structure. This follows by Proposition 3.7.1 and the fact that the uniqueness of direct sum decomposition holds.
(2) Length abelian categories. This is because the Jordan-Hölder theorem holds in any
abelian categories, see e.g. [Ste, p.92].
(3) Torsion(-free) classes of mod Λ for a Nakayama algebra Λ (Corollary 3.5.19).
(4) Torsion-free classes of mod kQ for a quiver Q of type A which satisfies the condition given
in Corollary 3.6.15. Explicit examples are given in Example 3.6.17 and E1 in Example
E.
(5) The category F(∆) of modules with ∆-filtrations over a quasi-hereditary algebra, or moregenerally, over a standardly stratified algebra (see e.g. [PR, Proposition 1.2]). Simple
objects in F(∆) are precisely standard modules.
3.8.3. Non-cancellative Grothendieck monoids. In this subsection, we will give someexamples in which the Grothendieck monoids are not finitely generated or not cancellative.
3.8.3.1. Functorially finite torsion-free class, but neither finitely generated nor cancellative.
In a length exact category, its Grothendieck monoid is atomic, thus it is not finitely generated asa monoid if and only if there exists infinitely many non-isomorphic simple objects (Proposition
3.4.8). We will give such an example.
Let k be an algebraically closed field and Q a Kronecker quiver, namely, Q : 1 ⇔ 2. We refer
the reader to [ARS, Section VIII.7] for the structure of mod kQ. We denote by E the subcategoryof mod kQ consisting of modules which does not contain S2 (simple modules supported at 2) asa direct summand. Then E is closed under extensions, thus an exact category. Actually, E is atorsion-free class associated with an APR cotilting module I1 ⊕ τ S2 , thus it is a functorially finite
torsion-free class (with infinitely many indecomposables).
Recall that for each element x = [x1 : x2 ] ∈ P1 (k) in the projective line P1 (k), we have aregular module Rx with dimension vector (1, 1), that is, Rx is the following representation of Q.
k

x1
x2

k

This assignment is injective, that is, we have Rx ∼= Ry if and only if x = y in P1 (k). Now we claim
the following:
sim E = {S1 } ∪ {Rx | x ∈ P1 (k)}.
Actually, for any indecomposable preprojective module X except S1 , it can be shown by explicit
calculation that there exists an exact sequence
0 → S1 → X → R → 0

3.8. COUNTER-EXAMPLES

59

such that R is a non-zero regular module. Thus such X is not simple in E. Moreover, any
indecomposable regular module is known to be written as a finite extension of Rx for some x ∈P1 (k). Therefore, indecomposable modules except S1 and Rx ’s are not simple. On the other hand,by considering dimension vectors, each Rx must be simple object, because dim Rx = (1, 1) is anatom of dim E. Thus Rx is simple in E for each x ∈ P1 (k).
Therefore, M(E) has infinitely many atoms (since k is an infinite field), so it is not finitely
generated.
Furthermore, we claim that M(E) is not cancellative. In fact, for each x = [x1 : x2 ] ∈ P1 (k),we have an exact sequence
h

0

S1

−x2
x1

i

P2

Rx

0

where P2 is an indecomposable projective module corresponding to 2 ∈ Q, and the left map is anembedding of S1 into the socle S1 ⊕ S1 of P2 . This shows that [S1 ] + [Rx ] = [P2 ] holds for everyx ∈ P1 (k). Since [Rx ] 6= [Ry ] in M(E) for x 6= y, we must conclude that M(E) is not cancellative.
Moreover, this gives an example such that non-isomorphic simples may represent the same∼element in the Grothendieck group. By Proposition 3.5.9, we have an isomorphism dim : K0 (E) −→2
1
Z . Thus for each x, y ∈ P (k), we have [Rx ] = [Ry ] in K0 (E).
3.8.3.2. Only finitely many indecomposables, but not cancellative. In the previous example,
the category we considered has infinitely many indecomposable objects. If an exact category E
has finitely many indecomposables, then M(E) is finitely generated by a trivial reason, but it may
not be cancellative as we shall see.
Let Λ be the following algebra, which is defined by the ideal quotient of the path algebra:βΛ := k α12  /(β 2 )Then the Auslander-Reiten quiver of mod Λ is as follows:
P2

···

P1
2
1

1

P1

I1

I1

1

1

I1
P2

P1
2
1

M
2
1

M

P2

2

2
1

M
P1

M
1

···

I1
P2

2

2
2

2

where we write composition factors of modules, and P1 = 11 , P2 = 1 , I1 = 1 2 and M := 1 1 2 .
1
1Let E denote the additive subcategory of mod Λ corresponding to these gray regions: ind E ={P1 , P2 , I1 , M }. Then E is shown to be closed under extensions in mod Λ 2 , and the AuslanderReiten quiver of the exact category E is as follows:
M
P1

I1
P2

2This does not seem to be so trivial. One categorical way to show this is to use results in [Eno1, Eno2]. By[Eno2, Corollary 3.10], one can endow E with the exact structure which corresponds to the Auslander-Reiten quiverof E drawn below. In this exact structure, E has a progenerator P1 ⊕ P2 = Λ, and one can consider the Morita typeembedding E(Λ, −) : E → mod Λ, which is nothing but the inclusion functor. Then by [Eno1, Proposition 2.8], its
image, E, is an extension-closed subcategory of mod Λ.

60

3. (JHP) AND GROTHENDIECK MONOIDS OF EXACT CATEGORIESBy checking subobjects, it can be shown that all 4 indecomposable objects in E are simple objects
in E. However, by the Auslander-Reiten quiver of mod Λ, it can be checked that we have thefollowing conflations in E:
0

P1

M ⊕ P2

I1

0,

0

M

P1 ⊕ I1

M

0,

0

P1

P2 ⊕ P2

I1

0.

This implies the equality [M ] + [M ] = [M ] + [P2 ] = [P2 ] + [P2 ] in M(E). Since [M ] 6= [P2 ] by
Proposition 3.3.6, we must have that M(E) is not cancellative.
3.9. Problems
In this section, we collect some open problems on several topics in this paper.
As we saw in Section 7, the computation of the Grothendieck monoid is rather difficult if it
fails to be cancellative.
Problem 3.9.1. Let E be an exact category. Is there any criterion to check whether M(E) iscancellative?
This leads to the following question.
Problem 3.9.2. Is there an example of an exact category E which satisfies the following
conditions:(1) E has finitely many indecomposable objects up to isomorphism.
(2) E = ⊥ U for a cotilting module U over an artin algebra Λ, or more strongly, E is atorsion-free class of mod Λ.
(3) M(E) is not cancellative.
If we drop (1) or (2), then we have such an example (Section 3.8.3.1 and 3.8.3.2 respectively).
In what follows, let F be a functorially finite torsion-free class of mod Λ for an artin algebra
Λ. The most general (and thus difficult) problem is the following:
Problem 3.9.3. Compute the Grothendieck monoid M(F), more precisely, draw the Cayley
quiver as we did in Proposition 3.7.6.
As a first approximation to M(F), the following problem naturally occurs, which is of interest
in its own right.
Problem 3.9.4. For a given torsion-free class F, classify (or count) simple objects in F.
Of course, we cannot expect the general classification of simples, but it may be done when Λand F are given explicitly.
The cancellative quotient M(F)can is easier to handle with by Corollary 3.5.9: it is nothing but
the monoid of dimension vectors dim F. Moreover, it is a finitely generated submonoid of Nn by
Proposition 3.5.16. Such a monoid is called a positive affine monoid, and this class is investigatedin the theory of combinatorial commutative algebra and toric geometry via its monoid algebra.
We refer the reader to [BrGu] for the details on affine monoids and affine monoid algebras.
Problem 3.9.5. For a given torsion-free class F, investigate the combinatorial property of
the affine monoid dim F, the monoid of dimension vectors of modules in F. In particular,• Is this monoid normal?
• Describe the minimal generating set of it (this is related to simple objects in F).
• When does this monoid homogeneous (this is related to the unique length property of F)?
• Compute invariants of this monoid, such as extreme rays, the class group, support hyperplanes, and so on.
Finally, we consider topics in Section 7. Let Q be a quiver of type A (or more generally,
any Dynkin type, see Remark 3.6.19). Theorem 3.6.12 gives an explicit description of torsion-freeclasses of mod kQ. Thus it is reasonable to expect that Problems 3.9.3 and 3.9.5 may be easier3.A. PRELIMINARIES ON MONOIDS

61

in this case. In addition, the author does not know when the uniqueness of lengths holds, except
Example 3.8.5.
Problem 3.9.6. In the situation of Theorem 3.6.12, is there a combinatorial criterion for w
to check whether F(w) has the unique length property?
Also the following (purely combinatorial) problem is of interest.
Problem 3.9.7. For a Coxeter element c, is there any closed formula which gives the numberof torsion-free classes satisfying (JHP), or equivalently, the number of c-sortable elements w suchthat # supp(w) = # Binv(w) holds?
3.A. Preliminaries on monoids
We collect some basic definitions and properties on monoids needed in this paper. Recall thatmonoids are always assumed to be a commutative monoid with a unit, and that we use an additive
notation with unit 0.
3.A.1. Basic definitions. First, we collect some basic definitions on monoids.
Definition 3.A.1. Let M be a monoid.
(1) M is reduced if a + b = 0 implies a = b = 0 for a, b ∈ M .
(2) M is cancellative if a + x = a + y implies x = y for a, x, y ∈ M .
There exists a natural pre-order ≤ on any monoid, defined as follows.
Definition 3.A.2. Let M be a monoid. We define x ≤ y if there exists some a ∈ M suchthat y = x + a.
It can be checked that x ≤ y if and only if y ∈ x + M if and only if y + M ⊂ x + M . This preorder is sometimes called Green’s pre-order in semigroup theory, e.g. [Gri1, Gri2], or divisibility
pre-order in the multiplicative theory of integral domains, e.g. [GH-K].
Definition 3.A.3. A monoid M is called naturally partially ordered if the pre-order ≤ on M
is a partial order, that is, x ≤ y and y ≤ x implies x = y.
The following properties can be easily checked.
Proposition 3.A.4. Let M be a monoid. Then M is reduced if and only if 0 ≤ x ≤ 0 implies
x = 0 for every x ∈ M . In particular, a naturally partially ordered monoid is reduced.
Although we cannot naively take quotients of monoids as in abelian groups since there may
not exist additive inverses, we can obtain some kind of quotient monoids by considering quotient
sets with respect to monoid congruences, defined as follows.
Definition 3.A.5. Let M be a monoid. The equivalence relation ∼ is called a (monoid)
congruence if x ∼ y implies a + x ∼ a + y for every a, x, y ∈ M . In this case, the quotient set
M/ ∼ of equivalence classes naturally has the structure of a monoid.
We often use the smallest monoid congruence generated by some binary relation. See e.g.
[Gri1, Propositions I.4.1, I.4.2] for the detail.
Proposition 3.A.6. Let M be a monoid and ∼ an arbitrary (not necessarily equivalence)
binary relation on M . Then there exists the smallest monoid congruence ≈ which contains ∼, thatis, x ∼ y implies x ≈ y for x, y ∈ M .
3.A.2. Factorization properties on monoids. Let us define several notions on the uniquefactorization property on monoids. We refer the reader to [GH-K] for the details in this subsection(be aware that monoids in [GH-K] are assumed to be cancellative, but contents in this subsection hold in non-cancellative monoids). The most typical one is the freeness of monoids, whichcorresponds to (JHP) for exact categories by our main result (Theorem 3.4.9).
Definition 3.A.7. Let M be a monoid.

62

3. (JHP) AND GROTHENDIECK MONOIDS OF EXACT CATEGORIES(1) We say that a monoid is finitely generated if there exists a finite subset of M which
generates M .
(2) For a subset A of M , we say that M is free on A if every element x ∈ M can be written
as a finite sum of elements in A in a unique way up to permutations. A monoid is called
free if it is free on some subset of M .
L
(A)
(A)
the submonoid:=a∈A Na of the free abelian group Z
L For a set A, we denote by N
Za
with
basis
A,
which
consistsoffinitesums
ofnon-negative
linearcombinationsofelea∈Aments in A. It is easy to see that N(A) is free on A and that a monoid is free if and only if it
is isomorphic to N(A) for some set A. If A is a finite set, then we often write NA = N(A) and
ZA = Z(A) . Moreover it is obvious that a free monoid is reduced.
Now let us consider elements of a monoid which cannot be decomposed into smaller ones.
Here, for simplicity, we only consider reduced monoids (this is a reasonable assumption sinceGrothendieck monoids are reduced by Proposition 3.3.5).
Definition 3.A.8. Let M be a reduced monoid.
(1) A non-zero element x of M is called an atom if x = y + z implies either y = 0 or z = 0
for y, z ∈ M . We denote by Atom M the set of all atoms in M .
(2) M is called atomic if Atom M generates M , that is, every element of M is a finite sumof atoms.
(3) M is called factorial if every element can be expressed as a finite sum of atoms, and thisexpression is unique up to permutations.
(4) M is called half-factorial if it is atomic, and for every element x and expressions
x = a1 + · · · + anwith ai ∈ Atom M , the number n depends only on x.
The following observations can be proved directly from the definitions.
Proposition 3.A.9. Let M be a reduced monoid. Then the following hold.
(1) If M is generated by a subset A, then Atom M ⊂ A holds. In particular, Atom M is a
finite set for a finitely generated monoid M .
(2) M is finitely generated and atomic if and only if M is atomic and Atom M is a finite
set.
(3) If M is free on a subset A of M , then A = Atom M holds.
(4) If M is free, then it is atomic, cancellative and factorial.
(5) M is free if and only if M is factorial.
3.A.3. Group completion. For a given monoid M , there is the universal construction whichtransforms M into a group. We call it a group completion in this paper.
Definition 3.A.10. Let M be a monoid. Then the group completion gp M is an abeliangroup gp M together with a map ι : M → gp M which satisfies the following universal property:(1) ι is a monoid homomorphism.
(2) Every monoid homomorphism f : M → G into a group G factors uniquely ι, that is,there exists a unique group homomorphism f : gp M → G which satisfies f = ιf .
Example 3.A.11. Let M be a monoid which is free on A. Then gp M is a free abelian groupwith basis A, that is, gp(N(A) ) = Z(A) . The original basis A can be reconstructed from M since
A = Atom M , but this information is lost when taking the group completion, since bases of thefree abelian group is far from unique. This is one reason why we consider monoids, not groups.
The explicit construction of the group completion is given as follows.
Proposition 3.A.12. Let M be a monoid. Define an equivalence relation ∼ on the set M ×M
by
(x1 , y1 ) ∼ (x2 , y2 ) :⇔ there exists an element a ∈ M such that x1 + y2 + a = x2 + y1 + a.

3.A. PRELIMINARIES ON MONOIDS

63

Then the quotient set (M × M )/ ∼, together with a map M → (M × M )/ ∼ given by x 7→ (x, 0),
is a group completion of M .
The cancellation property are related to the group completion as follows.
Proposition 3.A.13. Let M be a monoid. Then it is cancellative if and only if the naturalmap ι : M → gp M is an injection if and only if there is an injective monoid homomorphism intosome group. In this case, every injective monoid homomorphism ϕ : M → G such that ϕ(M )generates G is automatically a group completion of M .
It follows that the image of the group completion is always cancellative. One of the difficultyof dealing with a monoid M is that M may not be cancellative. Thus this image is much easier todeal with than M , and it has more information on its group completion gp M . This corresponds
to the positive part of the Grothendieck group, see Section 3.3.2.
Definition 3.A.14. Let M be a monoid. We denote by Mcan the image of the group completion ι : M → gp M , and call it the cancellative quotient of M .
We leave it the reader to check Mcan is actually the largest cancellative quotient of M :Proposition 3.A.15. Let M be a monoid and define an equivalence relation ∼can on M byx ∼can y :⇔ there exists an element a ∈ M such that x + a = y + a.
Then ∼can is a monoid congruence on M , and we have an isomorphism of monoids Mcan ∼=M/ ∼can . Consequently, for every monoid homomorphism ϕ : M → N to a cancellative monoid
N , there exists a unique monoid homomorphism ϕ : Mcan → N such that ϕ = ϕι.
3.A.4. Length-like functions on monoids. We will introduce a kind of length on monoids,
which corresponds to length-like functions on exact categories introduced in Definition 3.4.1.
Definition 3.A.16. Let M be a monoid. A length-like function on M is a monoid homomorphism ν : M → N such that ν(x) = 0 implies that x = 0.
The existence of a length-like function implies some nice properties.
Proposition 3.A.17. Let M be a monoid and suppose that there exists a length-like function
ν : M → N on M . Then the following hold.
(1) M is naturally partially ordered, hence reduced.
(2) M is atomic.
Proof. (1) Suppose that x ≤ y ≤ x holds for x, y ∈ M . Since ν is a monoid homomorphism,
it follows immediately that ν(x) ≤ ν(y) ≤ ν(x) holds, thus ν(x) = ν(y). On the other hand, we
have y = x + a for some a ∈ M by x ≤ y. Thus ν(y) = ν(x) + ν(a), which implies that ν(a) = 0.
Since ν is a length-like function, a = 0 holds, hence x = y. Thus M is naturally partially ordered,so it is reduced by Proposition 3.A.4.
(2) Suppose that M is not atomic. Then take an element x such that:(a) x 6= 0.
(b) x cannot be expressed as a finite sum of atoms.
(c) ν(x) is minimal among those x which satisfy (a) and (b).
Obviously x is not an atom by (b). Thus there is a decomposition x = y + z with y, z 6= 0. Clearly
either y or z satisfies (b), so let us assume that y satisfies (b). However, we have ν(x) = ν(y)+ν(z)with ν(y), ν(z) 6= 0 since ν is a length-like function. Therefore ν(y) < ν(x) and y satisfies (a) and(b). This contradicts the minimality of x.
For later use, we show the following characterization of half-factorial monoids.
Lemma 3.A.18. Let M be a monoid. Then the following are equivalent:(1) M is a half-factorial monoid.
(2) M has a length-like function ν satisfying ν(a) = 1 for every a ∈ Atom M .

64

3. (JHP) AND GROTHENDIECK MONOIDS OF EXACT CATEGORIESProof. (1) ⇒ (2) Suppose that M is half-factorial. We define a monoid homomorphismPll : M → N as follows: For every x in E, we can write x = i=1 ai in M(E) with ai ∈ Atom M
for each i since M is atomic. We set l(x) := l. Since M is half-factorial, l does not depend on
the choice of expressions, thus this map is well-defined. Furthermore, it is easy to see that l is alength-like function and that l(a) = 1 for every a ∈ Atom M .
(2) ⇒ (1): First observe that M is reduced and atomic by Proposition 3.A.17. Let x be anelement of M . Consider any expression
x = a1 + · · · + anwith ai ∈ Atom M . Then by (2), we have that ν(x) = ν(a1 ) + · · · + ν(an ) = n. Therefore, thenumber n depends only on x, so M is half-factorial.

3.A.5. Characterizations of free monoids. In what follows, we give a criterion for a given
monoid to be free. We use this results to check (JHP) in Section 4.
If a monoid M is free, then it has a length-like function (Lemma 3.A.18), its group completionis also free, and its rank coincides with the number of atoms. In general, we have the following
inequality.
Proposition 3.A.19. Let M be a reduced atomic monoid and suppose that gp M is a freeabelian group. Then the following inequality holds.
rank(gp M ) ≤ # Atom M
Proof. Let ι : M → gp M denote the group completion. As an abelian group, gp M isgenerated by ιM , so it is generated by ι(Atom M ). Thus rank(gp M ) ≤ # Atom M holds.

The following gives a kind of converse of this. This is an important characterization of freemonoids, which is very useful to our setting.
Theorem 3.A.20. Let M be a monoid, and denote by ι : M → gp M the group completion of
M . Then the following are equivalent:(1) M is a free monoid.
(2) M has a length-like function, ι is injective on Atom M , and gp M is a free abelian groupwith basis ι(Atom M ).
(3) M is reduced atomic, ι is injective on Atom M , and gp M is a free abelian group withbasis ι(Atom M ).
(4) M is reduced atomic, ι is injective on Atom M , and all elements in ι(Atom M ) are linearly
independent over Z.
Proof. (1) ⇒ (2): M has a length-like function by Proposition 3.A.18. Since M is free, it isfree on Atom M by Proposition 3.A.9 (3), and we have an isomorphism M ∼= N(Atom M ) . Thus we(Atom M )∼have gp M = Z
by Example 3.A.11. Thus (2) follows.
(2) ⇒ (3): This follows from Proposition 3.A.17.
(3) ⇒ (4): This is trivial.
(4) ⇒ (1): Define a monoid homomorphism ϕ : N(Atom M ) → M by ϕ(a) = a for each a ∈Atom M . We claim that this map is an isomorphism of monoids. It suffices to show that ϕ is abijection.
Since M is reduced and atomic, every element of M is a finite sum of atoms. Thus ϕ is asurjection. On the other hand, consider the following commutative diagram of monoids
N(Atom M )ϕi

Z(Atom M )

M
ι

ϕ

gp M,

where ι and i are group completions of M and N(Atom M ) respectively, and ϕ is a group homomorphism induced by ϕ. Here i is obviously an injection. Moreover, since ι is injective on Atom M andall elements in ι(Atom M ) are linearly independent, ϕ is an injection. Then the above commutative
diagram shows that so is ϕ. Therefore ϕ is a bijection.
3.A. PRELIMINARIES ON MONOIDS

65

Under the assumption of finite generation, we have a more convenient characterization, in
which we only have to count the number of atoms.
Corollary 3.A.21. Let M be a monoid, and denote by ι : M → gp M the group completionof M . Then the following are equivalent:(1) M is finitely generated and free as a monoid.
(2) M is a free monoid and # Atom M is finite.
(3) M is reduced and atomic, ι is injective on Atom M and gp M is a free abelian group offinite rank with basis ι(Atom M ).
(4) The following hold:(a) M is reduced and atomic.
(b) gp M is a free abelian group of finite rank.
(c) # Atom M = rank(gp M ) holds, where rank denotes a rank as an abelian group.
Proof. (1) ⇒ (2): This follows from Proposition 3.A.9 (1).
(2) ⇒ (3): This follows from (1) ⇒ (3) in Theorem 3.A.20.
(3) ⇒ (4): Trivial.
(4) ⇒ (1): We use the same notations and strategy as in the proof of (4) ⇒ (1) in Theorem
3.A.20. It suffices to show that the natural map ϕ : ZAtom M → gp M is an isomorphism. Since
ϕ is a surjection, so is ϕ. On the other hand, we have that gp M is free of rank # Atom M . By
counting ranks, it follows that ϕ is an isomorphism.
CHAPTER 4

Bruhat inversions in Weyl groups and torsion-free classes
over preprojective algebras
This chapter is based on [Eno5].
For an element w of the simply-laced Weyl group, Buan-Iyama-Reiten-Scott defined a subcategory F(w) of a module category over a preprojective algebra of Dynkin type. This paper aims at
studying categorical properties of F(w) via its connection with the root system. We show that bytaking dimension vectors, simple objects in F(w) bijectively correspond to Bruhat inversion rootsof w. As an application, we obtain a combinatorial criterion for F(w) to satisfy the Jordan-Hölder
property (JHP). To achieve this, we develop a method to find simple objects in a general torsionfree class by using a brick sequence associated to a maximal green sequence of it. For type A case,we give a diagrammatic construction of simple objects, and show that (JHP) can be characterized
via a forest-like permutation, introduced by Bousquet-Mélou and Butler in the study of Schubertvarieties.
4.1. Introduction
This paper focuses on the interplay between the preprojective algebras of Dynkin type andthe root system. More precisely, we study a certain subcategory F(w) of the module category of
the preprojective algebra via an inversion set in the root system.
4.1.1. Background. Let Φ be the simply-laced root system of type X and W the corresponding Weyl group. Let Q be a quiver of type X, that is, the underlying graph of Q is the
Dynkin diagram X. Then the celebrated Gabriel’s theorem gives a bijection between indecomposable representations of Q and positive roots in Φ, by taking dimension vectors.
A preprojective algebra Π of Φ is a finite-dimensional algebra which unifies the representation
theory of all quivers of type X, and has a lot of symmetry compared to path algebras. This algebra
has been one of the most important objects in the representation theory of algebras, for example,[AIRT, BIRS, GLS, IRRT, Miz], and also plays an important role in the theory of crystal basesof quantum groups, for example, [Lus, KaSa].
In this paper, we focus on a certain subcategory F(w) of mod Π associated to an element w
of W introduced by Buan-Iyama-Reiten-Scott [BIRS] (under the name Cw ). This category has
a nice structure related to cluster algebras, that is, a stably 2-Calabi-Yau Frobenius category
admitting a cluster-tilting object. Indeed, Geiss-Leclerc-Schröer later [GLS] proved that F(w)gives a categorification of the cluster algebra structure on the coordinate ring of the unipotent cellin the complex simple Lie group of Dynkin type X.
The category F(w) naturally arises also from the viewpoint of the representation theory ofalgebras, as well as the lattice theoretical study of the Weyl group. This category is a torsion-freeclass in mod Π, that is, closed under submodules and extensions. Mizuno [Miz] proved that themap w 7→ F(w) is actually a bijection from W to the set torf Π of all torsion-free classes in mod Π.
He also proved that this bijection is an isomorphism of lattices, where we endow W with the right
weak order and torf Π the inclusion order. Via this isomorphism, lattice theoretical properties of
W were investigated in [IRRT, DIRRT].
4.1.2. Main results. It is natural to expect that categorical properties of F(w) are related
to combinatorial properties of w. In this direction, we prove the two main results: we classify
664.1. INTRODUCTION

67

simple objects in F(w) (Theorem A), and give a criterion for the validity of the Jordan-Hölder
type theorem in F(w) (Theorem C).
A Π-module M in F(w) is called a simple object in F(w) if there is no non-trivial submodule
L of M satisfying L, M/L ∈ F(w). This notion was introduced in the context of Quillen’s exact
categories, and has been investigated by several papers such as [Eno4, BHLR]. In [Eno4], theauthor classified simple objects in a torsion-free class in mod kQ for type A case, and the original
motivation of this paper is to generalize this to other Dynkin types and to preprojective algebras.
Our strategy is to consider F(w) via the root system Φ. For M ∈ mod Π, we can regard its
dimension vector as a vector in the ambient space of Φ naturally. Let inv(w) be the set of inversionsof w, positive roots which are sent to negative by w−1 . Then any M ∈ F(w), its dimension vector
dim M is a non-negative integer linear combination of inversions of w (Corollary 4.4.5). In parallelwith simples in F(w), it is natural to consider an inversion of w which cannot be written as asum of inversions of w non-trivially. We call such a root a Bruhat inversion (Definition 4.2.11,
Theorem 4.2.17). Then the first main result of this paper is the following:
Theorem J (= Corollary 4.4.9). Let Π be the preprojective algebra of Dynkin type and w
an element of the corresponding Weyl group W . Then by taking dimension vectors, we have thebijection between the following two sets:(1) The set of isomorphism classes of simple objects in F(w).
(2) The set of Bruhat inversions of w.
This immediately deduces the similar result for the case of the path algebra kQ. A torsion-freeclass in mod kQ bijectively corresponds to a cQ -sortable element of W by [IT]. For such an element
w of W , we have a torsion-free class FQ (w) in mod kQ, which is actually equal to the restrictionof F(w) to mod kQ (Proposition 4.5.4). Then we have the same result for FQ (w):Theorem K (= Theorem 4.5.5). Let Q be the Dynkin quiver, w a cQ -sortable element of
the Weyl group W and FQ (w) the corresponding torsion-free class in mod kQ. Then by taking
dimension vectors, we have the bijection between the following two sets:(1) The set of isomorphism classes of simple objects in FQ (w).
(2) The set of Bruhat inversions of w.
As an application, we can characterize the validity of the Jordan-Hölder property (JHP) in
F(w) or FQ (w). We say that a torsion-free class F satisfies (JHP) if for any M in F, any relative
composition series of M in F are equivalent (see Definition 4.4.10 for the precise definition). By
using the characterization of (JHP) obtained in [Eno5], we prove the following second main result.
Theorem L (= Theorem 4.4.15, Proposition 4.A.6). Let Π be the preprojective algebra of
Dynkin type and w an element of the corresponding Weyl group W . Then the following are
equivalent:(1) F(w) satisfies (JHP).
(2) Bruhat inversions of w are linearly independent.
(3) The number of Bruhat inversions of w is equal to that of supports of w.
Moreover, for the type A case, the above statements are equivalent to the following:(4) w is forest-like in the sense of [BMB], that is, its Bruhat inversion graph is acyclic.
Here a support of w is a vertex i in the Dynkin diagram X such that the reduced expression of
w contains the simple reflection si . See Appendix 4.A.2 for the details on forest-like permutations.
The same result also holds for the case of path algebras (Corollary 4.5.6).
To show these results, we develop a method to find simple objects in a given torsion-free class
F by using a brick sequence associated to a maximal green sequence of F. A maximal green
sequence of F is just a saturated chain 0 = F0 l F1 l · · · l Fl = F of torsion-free classes. We
can associate to it a sequence of bricks (modules B such that EndΛ (B) is a division ring) by using
the brick labeling introduced in [DIRRT]. Once we can compute one brick sequence of F, thefollowing gives a way to determine all simple objects in F:

68

4. BRUHAT INVERSIONS AND TORSION-FREE CLASSES OVER PREPROJECTIVE ALGEBRAS

Proposition M (= Corollary 4.3.12). Let Λ be a finite-dimensional algebra and F a torsionfree class in mod Λ. Suppose that there is a maximal green sequence of F, and let B1 , . . . , Bl be
the associated brick sequence. Then the following hold:(1) Every simple object in F is isomorphic to Bi for some i.
(2) For 1 ≤ i ≤ l, the following are equivalent:(a) Bi is a simple object in F.
(b) Every morphism Bi → Bj with i < j is either zero or injective.
(c) There is no surjection Bi  Bj with i < j.
4.1.3. Organization. This paper is organized as follows. In Section 4.2, we give roottheoretical preliminaries and results. More precisely, we characterize Bruhat inversions (Theorem
4.2.17). Next we introduce a root sequence associated to a reduced expression of an element of W ,which plays an important role later. In Section 4.3, we develop a general theory of simple objects
in a torsion-free class, and prove Proposition M. In Section 4.4, we focus on torsion-free classes
over preprojective algebras of Dynkin type. We show that a brick sequence of F(w) categorifies
a root sequence of w, and prove Theorem J, K. In Section 4.5, we deduce several results on path
algebras using preprojective algebras, and prove Theorem L. In the appendix, we give two combinatorial interpretation of the results for type A case. In Section 4.A.1, we give a diagrammaticconstruction of simple objects in F(w) using arc diagrams, and in Section 4.A.2, we show that
F(w) satisfies (JHP) if and only if w is a forest-like permutation.
4.1.4. Conventions and notation. Throughout this paper, we assume that all categories
are skeletally small, that is, the isomorphism classes of objects form a set. In addition, all subcategories are assumed to be full and closed under isomorphisms. For a Krull-Schmidt category E,
we denote by ind E the set of isomorphism classes of indecomposable objects in E. We denote by|X| the number of non-isomorphic indecomposable direct summands of X.
For a poset P and two elements a, b ∈ P with a ≤ b, we denote by [a, b] the interval poset[a, b] := {x ∈ P | a ≤ x ≤ b} with the obvious partial order. For x, y ∈ P , we say that x covers y
if x > y holds and there exists no z ∈ P with x > z > y. In this case, we write x m y or y l x.
For a set A, we denote by #A the cardinality of A.
Acknowledgement. The author would like to thank his supervisor Osamu Iyama for many
helpful comments and discussions. He is also grateful to Yuya Mizuno for explaining to him arc
diagrams and bricks for type A case. This work is supported by JSPS KAKENHI Grant Number
JP18J21556.
4.2. Preliminaries on root system
In this section, we give some results on root systems which we need later. In particular, wegive a definition and characterization of Bruhat inversions, and introduce the notion of a root
sequence.
4.2.1. Basic definitions. First we recall some basic definitions and properties of root systems. We refer the reader to [Hum1, Hum2] for the details.
Let V be the Euclidean space, that is, a finite-dimensional R-vector space with the positive
definite symmetric bilinear form (−, −) : V × V → R.
For two vectors α, β ∈ V with α 6= 0, we put(β, α).
hβ, αi := 2(α, α)
Note that h−, αi is linear but hα, −i is not. For α ∈ V with α 6= 0, we denote by tα : V → V thereflection with respect to α, that is,tα (β) = β − hβ, αiα.
Definition 4.2.1. A subset Φ of the Euclidean space V is called a root system if it satisfies
the following axioms:(R0) Φ is a finite subset of V which spans V as an R-vector space, and does not contain 0.

4.2. PRELIMINARIES ON ROOT SYSTEM

69(R1) Φ ∩ Rα = {α, −α} for every α ∈ Φ.
(R2) tα (Φ) = Φ for every α ∈ Φ.
(R3) hβ, αi ∈ Z for every α, β ∈ Φ.
A root system is called simply-laced if it satisfies the following condition.
(R4) hα, βi = hβ, αi for every α, β ∈ Φ.
Obviously (R4) is equivalent to that for every α, β ∈ Φ, if (α, β) 6= 0, then α and β have thesame length, that is, (α, α) = (β, β).
Possible values of integers hα, βi in (R3) are very limited as follows.
Proposition 4.2.2 ([Hum1, 9.4]). Let Φ be a root system, and let α and β be two roots in Φwith (α, α) ≤ (β, β). Then the following hold:(1) hα, βi · hβ, αi ≥ 0 and (|hα, βi|, |hβ, αi|) ∈ {(0, 0), (1, 1), (1, 2), (1, 3), (2, 2)}.
(2) β = ±α if and only if (|hα, βi|, |hβ, αi|) = (2, 2).
(3) Φ is simply-laced if and only if hα, βi ∈ {−1, 0, 1} for every α, β ∈ Φ with α 6= β.
Throughout this section, we will use the following notation:• Φ is a root system in V .
• We fix a choice of simple roots ∆ of Φ.
• Φ+ (resp. Φ− ) is the set of positive roots (resp. negative roots) in Φ with respect to ∆.
• W is the Weyl group associated with Φ, that is, W is a subgroup of GL(V ) generated
by tα with α ∈ Φ.
• T ⊂ W is a set of reflections in W , that is, T = {tα ∈ W | α ∈ Φ}.
• We often write sα = tα if α is a simple root.
It is well-known that W is generated by simple reflections. For w ∈ W , let
w = s1 · · · sl
be such an expression. Then this expression is called a reduced expression if l is minimal among
all such expressions. In this case, l is called the length of w and we write `(w) := l.
On the reduced expression, we will need the following exchange property later.
Lemma 4.2.3 ([Hum2, 1.6, 1.7]). Let w = s1 · · · sl be a reduced expression of w ∈ W , and let
α be a simple roots. Then the following are equivalent:(1) s1 · · · sl sα is not reduced, that is, `(wsα ) < `(w).
(2) There exists i such that s1 · · · sl sα = s1 · · · sbi · · · sl (si omitted).
(3) w(α) ∈ Φ− .
4.2.2. Inversion sets and root sequences. For w ∈ W , we define inv(w) byinv(w) := Φ+ ∩ w(Φ− ),that is, inv(w) is the set of positive roots which are sent to negative roots by w−1 . We callan element of inv(w) an inversion of w. This set plays an important role in this paper, since
it corresponds to the set of dimension vectors of bricks in a torsion-free class F(w) over the
preprojective algebra (Corollary 4.4.5).
The following description of inversion sets is well-known.
Proposition 4.2.4 ([Hum2, 1.7]). Let w be an element of W . Take a reduced expression
w = sα1 · · · sαl of w with αi ∈ ∆. Then we haveinv(w) = {α1 , sα1 (α2 ), sα1 sα2 (α3 ), . . . , sα1 · · · sαl−1 (αl )},and all the elements above are distinct. In particular, we have # inv(w) = `(w).
By using this, we can easily show the following property.
Lemma 4.2.5. Let v and w elements of W satisfying `(vw) = `(v) + `(w). Then we haveinv(vw) = inv(v) t v(inv(w)).

70

4. BRUHAT INVERSIONS AND TORSION-FREE CLASSES OVER PREPROJECTIVE ALGEBRAS

If we choose a reduced expression of w, then Proposition 4.2.4 gives a sequence of positiveroots. It turns out that the order of appearance are important for our purpose. This leads to thenotion of root sequences.
Definition 4.2.6. Let W be the Weyl group of a root system Φ.
(1) Let w be an element of W and w = sα1 · · · sαl a reduced expression of w. Then a root
sequence of w associated to this expression is a (ordered) sequence of positive roots
α1 , sα1 (α2 ), sα1 sα2 (α3 ), . . . , sα1 · · · sαl−1 (αl ).
(2) A root sequence is a sequence of roots which arises as a root sequence of some reduced
expression of some element w in W . We call such a sequence a root sequence of w.
The notion of root sequences appeared in several papers: they are called compatible (convex)
orderings in [Pap], reflection orderings in [Dye]. We borrowed the terminology root sequences from
[GL, FS].
We will use the following characterization of inversion sets and root sequences due to Papi
later.
Theorem 4.2.7 ([Pap]). A sequence R of positive roots is a root sequence if and only if thefollowing two conditions are satisfied for any pair of positive roots α, β satisfying α + β ∈ Φ+ :(1) If α and β appear in R, then α + β appears between α and β in R.
(2) If α + β appears in R, then one of α and β appears and precedes α + β in R.
To give examples of root sequences (and its connection with brick sequence defined in the next
section), it is convenient to introduce the right weak order ≤R on W and its Hasse quiver. Define
a quiver Hasse(W, ≤R ) as follows:• A vertex set of Hasse(W, ≤R ) is W .
• We draw an arrow v ← vs if `(vs) = `(v) + 1 for v ∈ W and a simple reflection s ∈ W .
Then define a partial order ≤R on W by v ≤R w if and only if there is a path from w to v in
Hasse(W, ≤R ). It is known that (W, ≤R ) is actually a lattice, see e.g. [BB, 3.2], and Hasse(W, ≤R )is actually a Hasse quiver of (W, ≤R ) defined later.
By construction, each reduced expression of w = s1 s2 · · · sl gives a reverse path e ← s1 ←s1 s2 ← · · · ← w from e to w in Hasse(W, ≤R ), and this correspondence is a bijection. We define
the root labeling of arrows in Hasse(W, ≤R ) by attaching a positive root v(α) to an arrow v ← vsα .
Then a root sequence associated to a reduced expression w = s1 · · · sl is obtained by reading labelsof the corresponding reverse path e ← s1 ← · · · ← w.
Example 4.2.8. Let Φ be the root system of type A3 with its Dynkin graph 1—2—3. We
denote by si (resp. αi ) the simple reflection (resp. simple root) associated to the vertex i for1 ≤ i ≤ 3. Consider w = s1 s2 s3 s1 s2 ∈ W . Figure 1 shows all the reverse paths from e to w and
its root labeling. Here we write s1231 := s1 s2 s3 s1 and 110 := α1 + α2 for example. Then the left
011

s2312

111

s123

s121

111
100

010

s12

s231

011

110

s21

110

110

s1

s2
100

100

s1231

010

111

w = s12312

e

s23
011

010

Figure 1. Root sequences of w = s12312
most path corresponds to a reduced expression s12312 , and the right most corresponds to s23121 .
Their associated root sequences are 100, 110, 111, 010, 011 and 010, 011, 110, 111, 100 respectively.

4.2. PRELIMINARIES ON ROOT SYSTEM

71

Example 4.2.9. Let Φ be the root system of type D4 , whose Dynkin diagram is as follows:
2
1

0

3

Consider w = s012301230 (here we use the same notation as in Example 4.2.8). Figure 2 shows all the
reduced expressions of w and its root sequence. For example, the right most path gives a reduced
w = s012301230
1
111

s01230123
1
011

s0123023

1
110

0
111

1
1
110 011s0123013

1
110s0123012

1
011

0
111

0
111

s012303s012302 1

s012301

010

0
011

0
110

0
111

1
011

s01203s01230s01302s02301

1
121

1
121

1
121

1
121

s0130

s0230

1
110

s0120

s0123
0
011

1
110

s012

0
110

1
010
0
011

0
110

0
111

s013

0
110

0
011

1
010

1
011

s023
1
010

s01

s02
0
110

1
010

s03
0
011

s0
0
010

e
Figure 2. Root sequences for w = s012301230
0 , 0 , 1 , 1 , 1 , 0 , 0 , 1 , 1 (hereexpression w = s012030210 with its associated root sequence 010
110 010 110 121 011 111 011 111
1
as before, 011 denotes α0 + α2 + α3 for example).

4.2.3. Bruhat inversions. A simple root in a root system is indecomposable in Φ+ in thesense that we cannot write it as a positive linear combination of positive roots in a non-trivialway.
We are interested, not in the whole set of positive roots, but in the inversion set inv(w) fora fixed w ∈ W . It is natural to ask which elements are indecomposable in the above sense. This
leads to the notion of Bruhat inversions.
Let us begin with the property of reflections with respect to inversion roots of w.
Proposition 4.2.10 ([Hum2, Theorem 5.8], [BB, Theorem 1.4.3]). Let w be an element of W
and β a positive root. Then the following hold:(1) Then β ∈ inv(w) if and only if `(tβ w) < `(w) holds.
(2) Let s1 s2 · · · sl = w be a reduced expression with si = sαi , and let β1 , β2 , . . . , βl a root
sequence associated to it. Then we havetβi w = s1 s2 · · · sbi · · · sl .
Next we will define Bruhat inversions of elements in W . Recall that the Bruhat order on W is
the transitive closure of the following relation: for every t ∈ T and w ∈ W satisfying `(tw) < `(w),
we have tw ≤ w.

72

4. BRUHAT INVERSIONS AND TORSION-FREE CLASSES OVER PREPROJECTIVE ALGEBRASDefinition 4.2.11. Let w be an element of W and β a positive root. Then β is a Bruhat
inversion of w if it satisfies `(tβ w) = `(w) − 1. We denote by Binv(w) the set of Bruhat inversionsof w. We call an element of inv(w) \ Binv(w) a non-Bruhat inversion of w.
By the description in Proposition 4.2.10, the number of Bruhat inversions can be computed
as follows: fix a reduced expression w = s1 · · · sl of w, then # Binv(w) is the number of i’s suchthat deleting si from this expression still yields a reduced expression.
A Bruhat inversion is closely related to the covering relation in the Bruhat order as follows.
Proposition 4.2.12. For w ∈ W and β ∈ Φ+ , the following are equivalent:(1) β ∈ Binv(w), that is, β is a Bruhat inversion of w.
(2) w covers tβ w in the Bruhat order of w.
Proof. This easily follows from Proposition 4.2.10 and the chain property of the Bruhat
order, see [BB, Theorem 2.2.6] for example.

The following is an example of Bruhat inversions for type A case (we refer the reader to theappendix for the detail).
Example 4.2.13. Let Φ be the standard root system of type An , and α1 , . . . , αn the simple
roots. Then positive roots are of the form β(i j) := αi + αi+1 + · · · + αj−1 for 1 ≤ i < j ≤ n + 1. We
can identify W with the symmetric group Sn+1 . For w in W = Sn+1 , we have that β(i,j) ∈ inv(w)
if and only if (i, j) is the classical inversion of w, that is, i < j and w−1 (i) > w−1 (j) holds.
Moreover, we can easily see that β(i,j) ∈ Binv(w) if and only if β(i,j) ∈ inv(w) and there is noi < k < j with β(i,k) , β(k,j) ∈ inv(w). For example, if w = 42153 ∈ S5 , we have inv(w) ={β(1,2 , β(1,4) , β(2,4) , β(3,4) , β(3,5) } and Binv(w) = inv(w) \ {β(1,4) }.
For a fixed element w ∈ W , we will give a characterization of Bruhat inversions of w among
all inversions of w. To do this, we prepare some lemmas.
The following is known as the lifting property of the Bruhat order.
Lemma 4.2.14 ([BB, Proposition 2.2.7]). Let v and w be elements in W satisfying v < w, andlet s be a simple reflection. If `(sv) = `(v) + 1 and `(sw) = `(w) − 1 holds, then we have sv < w
and v < sw.
By using the lifting property, we can show the following technical lemma.
Lemma 4.2.15. Let w be an element of W and β ∈ Binv(w), and put t = tβ ∈ W . Then either
one of the following holds:(1) β is a simple root.
(2) There exists a simple reflection s ∈ W which satisfies the following two conditions:(a) `(sw) = `(w) + 1.
(b) `(stw) = `(tw) + 1.
Proof. Suppose that (2) does not hold. We will show that β must be simple.
Since (2) does not hold, for every simple reflection s ∈ W , we have that either (a) or (b) (orboth) fails to hold. We will show the following claim.
(Claim): There exists some simple reflection s ∈ W such that (a) does not hold and (b) holds.
Proof of (Claim). If this is not the case, then (b) does not hold for every simple reflections ∈ W . This means that tw is the longest element in W , which contradicts to `(w) = `(tw) + 1. 
Take such a simple reflection s. Then we have that `(sw) = `(w) − 1 and `(stw) = `(tw) + 1
hold, and that tw < w by the assumption. Then by Lemma 4.2.14, we have that tw ≤ sw < w
holds. Since w covers tw in the Bruhat order, we must have tw = sw, hence t = s. Therefore β isa simple root.
Now we can show that a Bruhat inversion can be transformed into a simple root:
Proposition 4.2.16. Let w be an element of W and β an inversion of w. Then the following
are equivalent:4.2. PRELIMINARIES ON ROOT SYSTEM

73(1) β is a Bruhat inversion of w.
(2) There exists some element v ∈ W which satisfies the following two conditions:(a) `(vw) = `(v) + `(w) holds.
(b) v(β) is a simple root.
Proof. (1) ⇒ (2): Suppose that β ∈ Binv(w) holds. If β is a simple root, then v = e satisfies
the conditions (2)(a) and (b).
From now on, we assume that β is not simple. Then by Lemma 4.2.15, there exists a simple
reflection s such that `(sw) = `(w)+1 and `(stβ w) = `(w) hold. Put w0 = sw and β 0 := s(β), thenβ 0 ∈ inv(w0 ) by Lemma 4.2.5. Moreover, `(stβ w) = `(w) implies that `(tβ 0 w0 ) = `(stβ s · sw) =`(w) = `(w0 ) − 1. Thus β 0 is a Bruhat inversion of w0 . If β 0 is a simple root, then v = s satisfies
the conditions (2)(a) and (b).
If β 0 is not simple, then we can iterate this process by considering β 0 and w0 instead of β and w.
Moreover, this process must stop at some point since otherwise we would have `(w) < `(w0 ) < · · · ,which contradicts to the existence of the longest element in W . Therefore, we have that the Bruhat
inversion at this point, which can be written as v(β) for some v, is a simple root.
(2) ⇒ (1): Since v(β) is a simple root and v(β) ∈ inv(vw) by Lemma 4.2.5, we have`(tv(β) vw) = `(vw) − 1 = `(v) + `(w) − 1. On the other hand, since tv(β) vw = vtβ v −1 vw = vtβ w
holds, we have `(tv(β) vw) ≤ `(v) + `(tβ w). Thus `(w) − 1 ≤ `(tβ w). Since β is an inversion of w,
we have `(tβ w) ≤ `(w) − 1 by Proposition 4.2.10. Therefore, we have `(tβ w) = `(w) − 1, that is,
β is a Bruhat inversion of w.
The following is a main result in this section, which gives a characterization of Bruhat inversions.
Theorem 4.2.17. Let w be an element of the Weyl group W of Φ, and β ∈ inv(w). Then thefollowing are equivalent:(1) β is a non-Bruhat inversion of w.
(2) There exists
P aγ ∈ R≥0 for each γ ∈ inv(w) which satisfies the following conditions:• β = γ∈inv(w) aγ γ holds.
• This expression is not of the form β = 1 · β, that is, there exists some γ ∈ inv(w)such that aγ 6= δβ,γ , where δ is the Kronecker delta.
(3) There exist γ1 , γ2 ∈ inv(w) with γ1 6= γ2 and a1 , a2 ∈ R>0 such that β = a1 γ1 + a2 γ2holds.
(4) There exist γ1 , γ2 ∈ inv(w) with γ1 6= γ2 and n ∈ {1, 2, 3} such that β = (γ1 + γ2 )/nholds.
Moreover, if Φ is simply-laced, then the above statements are equivalent to the following:(5) There exist γ1 , γ2 ∈ inv(w) satisfying β = γ1 + γ2 .
Proof. The implications (5) ⇒ (4) ⇒ (3) ⇒ (2) are clear.
(2) ⇒ (1): Suppose that β is a Bruhat inversion. Then by Proposition 4.2.16, we have anelement v ∈ W such that `(vw) = `(v) + `(w) and that v(β) is a simple root. Note that Lemma
4.2.5 implies v(inv(w)) ⊂ Φ+ .
P
Now take aγ ∈ R≥0 for γ ∈ inv(w) with β = γ aγ γ as claimed in (2). Then we haveXv(β) =aγ v(γ).
γ∈inv(w)Since v(β) is a simple root and all v(γ)’s are distinct positive roots, we must have that aγ = δβ,γ ,which contradicts to the condition in (2). Thus β is not a Bruhat inversion.
(1) ⇒ (4),(5): Take any reduced expression w = s1 s2 · · · sl of w, and let β1 , β2 , . . . , βl be theroot sequence associated to it. Then β = βm for some m. Since we have tβ w = s1 · · · sc
m · · · sl by
Proposition 4.2.10 and β is not a Bruhat inversion, the expression s1 · · · sc
m · · · sl is not reduced.
Take the minimal j such that s1 · · · sc
m · · · sj is not a reduced expression. Then by Lemma
4.2.3, there exists i such that s1 · · · sccm · · · sj−1 = s1 · · · sbi · · · s
m · · · sj . In this situation, we will
prove the following claim.

74

4. BRUHAT INVERSIONS AND TORSION-FREE CLASSES OVER PREPROJECTIVE ALGEBRAS(Claim): We have tβ (βj ) = −βi .
Proof of (Claim). First we have the following equation:tβ (βj ) = tβm (βj )= (s1 · · · sm−1 sm sm−1 · · · s1 )(βj )= (s1 · · · sm−1 sm sm−1 · · · s1 )(s1 · · · sj−1 )(αj )= (s1 · · · sc
m · · · sj−1 )(αj ),where αj is a simple root with sαj = sj . On the other hand, since s1 · · · sc
m · · · sj−1 sj is not
reduced, Lemma 4.2.3 implies that (s1 · · · sc
m · · · sj−1 )(αj ) is a negative root. Thus tβ (βj ) is anegative root.
Now consider the reflection with respect to tβ (βj ):ttβ (βj ) = tβ tβj tβ= (s1 · · · sm · · · s1 )(s1 · · · sm · · · sj · · · sm · · · s1 )(s1 · · · sm · · · s1 )= s1 · · · sccm · · · sj−1 sj sj−1 · · · s
m · · · s1
= (s1 · · · sbi · · · sccm · · · sj )sj sj−1 · · · s
m · · · s1
= s1 · · · si · · · s1
= tβi .
Thus the reflection with respect to tβ (βj ) coincides with that along βi . Since tβ (βj ) is a negative
root, we must have tβ (βj ) = −βi .
Now by (Claim), we have βj − hβj , βiβ = tβ (βj ) = −βi , thus hβj , βiβ = βi + βj . Since β, βi
and βj are positive roots, we must have hβj , βi > 0. We have hβj , βi ∈ {1, 2, 3} by Proposition
4.2.2, so (4) holds.
Assume that Φ is simply-laced. Since m < j, we have that βm = β and βj are distinct. Thus
hβj , βi = 1 holds by Proposition 4.2.2, so (5) holds.

By this theorem, a non-Bruhat inversion γ of w can be written as γ = α +β with α, β ∈ inv(w)if Φ is simply-laced. This kind of equation gives a restriction of the relative position of α, β andγ as follows.
Lemma 4.2.18. Suppose that Φ is simply-laced, and that α and β in Φ satisfies γ := α+β ∈ Φ.
Then hα, βi = −1, hα, γi = hβ, γi = 1 hold, thus they look as follows.
βγ =α+βπ3π3αProof. Clearly α, β, γ are distinct. Thus we have that hβ, αi, hγ, αi ∈ {−1, 0, 1} by Proposition 4.2.2. However, we have hγ, αi = hα + β, αi = hα, αi + hβ, αi = 2 + hβ, αi. Thus we musthave hβ, αi = −1 and hγ, αi = 1. The equation hγ, βi = 1 can be shown similarly.
Since (α, β), (α, γ) 6= 0 and Φ is simply-laced, α, β, γ has the same length. Then it easily
follows from hβ, αi = −1 that the angle between α and β is 32 π. Therefore the situation looks likethe figure.
Remark 4.2.19. If Φ is not simply-laced, then a non-Bruhat inversion γ of w may not be
written as a sum of other inversions. For example, let Φ be the root system of type B2 with α ashort simple root and β a long simple root:
βα+β2α + βα4.3. BRICK SEQUENCE OF A TORSION-FREE CLASS

75

Then we have four positive roots. Consider w = sβ sα sβ . Easy computation shows that its root
sequence is β, α + β, 2α + β, thus we have inv(w) = {β, α + β, 2α + β}. By using Theorem 4.2.17,
one can conclude that α + β is not a Bruhat inversion of w since α + β = β/2 + (2α + β)/2 holds (ofcourse this can be deduced by checking that sβ scα sβ is not reduced, which is trivial). Nevertheless,
we cannot write α + β as a sum of other inversions of w.
4.3. Brick sequence of a torsion-free class
In this section, we define a brick sequence of a torsion-free class F, which is associated toa maximal green sequence of F (a saturated chain of torsion-free classes between 0 and F). In
particular, we focus on the relation between brick sequences and simple objects in F.
Throughout this section, let Λ be a finite-dimensional k-algebra over a field k. We denoteby mod Λ the category of finitely generated right Λ-modules. A module always means a finitely
generated right modules.
4.3.1. Brick labeling. We briefly recall the lattice structure of torsion-free classes in mod Λ,
following [DIRRT]. Note that in [DIRRT], torsion classes, the dual notion of torsion-free classes,
were mainly studied, but the same theory works also for torsion-free classes by the standard
duality.
A subcategory F of mod Λ is a torsion-free class if it is closed under extensions and submodules
in mod Λ. We denote by torf Λ the set of all torsion-free classes in mod Λ. Then torf Λ is a posetwith respect to inclusion. Moreover, since intersection of any torsion-free classes is also a torsionfree class, torf Λ is a complete lattice.
For a poset P , its Hasse quiver Hasse P is a quiver defined as follows:• A vertex set of Hasse P is P .
• We draw a unique arrow x → y in Hasse P if x m y, that is, x covers y in a poset P .
Now Hasse(torf Λ) has the additional structure called the brick labeling, established in [DIRRT].
Let us introduce some terminologies to state it. A Λ-module M in mod Λ is called a brick if
EndΛ (M ) is a division ring. For a collection C of Λ-modules, we will use the following notations:• add C denotes the category of direct summands of finite direct sums of modules in C.
• Sub C denotes the category of modules X such that there exists an injection from X toa module in add C.
• Filt C denotes the category of modules X such that there exists a filtration 0 = X0 ≤X1 ≤ · · · ≤ Xn = X of submodules of X satisfying Xi /Xi−1 ∈ C for each i.
• F(C) denotes the smallest torsion-free class containing C, or equivalently, F(C) = Filt(Sub C).
• ⊥ C denotes the category of modules X satisfying HomΛ (X, C) = 0.
• C ⊥ denotes the category of modules X satisfying HomΛ (C, X) = 0.
• ind C denotes the set of isomorphism classes of indecomposable modules in C.
• brick C denotes the set of isomorphism classes of bricks in C.
For two collections C and D of Λ-modules, C ∗ D denotes the category of modules X such that
there exist an exact sequence
0

C

X

D

0

with C ∈ C and D ∈ D. From the classical torsion theory, we have mod Λ = ⊥ F ∗ F for a
torsion-free class F in mod Λ, see e.g. [ASS, VI.1]. It can be easily checked that the operation ∗ is
associative, that is, we have (C ∗ D) ∗ E = C ∗ (D ∗ E) holds for collections C, D and E of Λ-modules.
Thus we omit parentheses and just write as C ∗ D ∗ E.
In [DIRRT], the following basic observation was established.
Proposition 4.3.1 ([DIRRT, Theorems 3.3, 3.4]). Let G ⊂ F be two torsion-free classes inmod Λ. Then there exists an arrow q : F → G in Hasse(torf Λ) if and only if brick(⊥ G ∩ F) contains
exactly one element Bq . In this case, we have ⊥ G ∩F = Filt(Bq ), F = Filt(Bq )∗G and G = F ∩Bq⊥ .
By this, to each arrow q in Hasse(torf Λ) we can associate a brick Bq , which we call the labelof q. This labeling is called the brick labeling of torf Λ.

76

4. BRUHAT INVERSIONS AND TORSION-FREE CLASSES OVER PREPROJECTIVE ALGEBRAS

4.3.2. Brick sequence associated to a maximal green sequence. To study the structure of a fixed torsion-free class F, a brick sequence associated to a maximal green sequence of F
plays an important role later. Let us introduce these notions.
Definition 4.3.2. Let F be a torsion-free class in mod Λ. Then a maximal green sequenceof F is a finite path in Hasse(torf Λ) which starts at F and ends at 0. Or equivalently, a mximal
green sequence of F is a saturated chain in torf Λ of the form 0 = F0 l F1 l · · · l Fl = F.
Maximal green sequences were introduced by Keller in the context of quiver mutations incluster algebras, and have been investigated from various viewpoints. We refer the reader to therecent article [DK] and the reference therein for the details on this notion. The above definition
is a straightforward generalization of maximal green sequences of abelian categories, which wasintroduced in [BST].
To a maximal green sequence of F, we can associate a sequence of bricks in F as follows.
This is analogous to the root sequence associated to a reduced expression in the Weyl group (andactually gives a categorification as we shall see in Proposition 4.4.3).
Definition 4.3.3. Let 0 = F0 ← F1 ← . . . ← Fl = F be a maximal green sequence of a
torsion-free class F in mod Λ. Then a brick sequence associated to it is a sequence B1 , B2 , . . . , Blof bricks where each Bi is the label of the arrow Fi ← Fi−1 in Hasse(torf Λ). We simply call abrick sequence associated to some maximal green sequence of F a brick sequence of F.
As in the case of root sequences, we take the appearance order of bricks into account. Fora fixed F, brick sequences of F heavily depend on the choice of maximal green sequences. In
general, the lengths of brick sequences may differ, as the following example shows.
Example 4.3.4. Let Q be a quiver 1 ← 2. Then Hasse(torf kQ) and its brick labeling are as
follows:
mod kQ = add{1, 21, 2}2

1

add{1, 21}21

add{2}

add{1}

210Here we write composition series to indicate kQ-modules. Thus there are exactly two brick sequences of mod kQ, namely, 1, 21, 2 and 2, 1.
The fundamental property of a brick sequence is the following.
Theorem 4.3.5. Let B1 , . . . , Bl be a brick sequence of a torsion-free class F in mod Λ. Thenthe following hold.
(1) HomΛ (Bj , Bi ) = 0 for j > i.
(2) B1 , . . . , Bl are pairwise non-isomorphic.
(3) F = Filt(Bl ) ∗ Filt(Bl−1 ) ∗ · · · ∗ Filt(B2 ) ∗ Filt(B1 ) holds. In particular, we have F =Filt(B1 , . . . , Bl ).
Proof. Let 0 = F0 ← F1 ← . . . ← Fl = F be a maximal green sequence of F which gives abrick sequence B1 , . . . , Bl .
(1) By Proposition 4.3.1 and the definition of the brick labeling, each Bi is contained in⊥
Fi−1 ∩Fi . Thus Bj ∈ ⊥ Fj−1 ⊂ ⊥ Fi holds since Fj−1 ⊃ Fi for j > i. Therefore HomΛ (Bj , Bi ) = 0
by Bi ∈ Fi .
(2) This is clear from (1).
(3) We will prove by backward induction on i that ⊥ Fi−1 ∩F = Filt(Bl )∗· · ·∗Filt(Bi+1 )∗Filt(Bi )holds for 1 ≤ i ≤ l. For i = l, this holds by Proposition 4.3.1.

4.3. BRICK SEQUENCE OF A TORSION-FREE CLASS

77

Suppose this holds for i = j + 1 with 1 ≤ j < l, and take M ∈ ⊥ Fj−1 ∩ F. Since ⊥ Fj ∗ Fj =mod Λ holds, we have an exact sequence in mod Λ
0

T

M

F

0

⊥

with T ∈ Fj and F ∈ Fj . Since F is closed under submodules and M ∈ F, we have T ∈⊥Fj ∩ F = Filt(Bl ) ∗ · · · ∗ Filt(Bj+1 ) by the induction hypothesis. On the other hand, since⊥
Fj−1 is closed under quotients, we have F ∈ ⊥ Fj−1 ∩ Fj = Filt(Bj ). Thus we have M ∈Filt(Bl ) ∗ · · · ∗ Filt(Bj+1 ) ∗ Filt(Bj ).

We remark that this statement appeared in [Tre, Corollary 6.5] for the case F = mod Λ,and the filtration given in (3) above is called the Harder-Narasimhan filtration associated to a
maximal green sequence. Also this filtration was considered in [Tat, Theorem 6.8] in the settingof quasi-abelian categories. Since torsion-free classes are quasi-abelian, we can apply his result toour setting to obtain this theorem.
4.3.3. Simple objects in a torsion-free class. We introduce the notion of simple objects
in a torsion-free class.
Definition 4.3.6. Let F be a torsion-free class in mod Λ.
(1) M ∈ F is a simple object in F if M 6= 0 and for any short exact sequence
0

L

M

N

0

of Λ-modules with L, N ∈ F, we have L = 0 or M = 0.
(2) We denote by sim F the set of isomorphism classes of simple objects in F.
Note that whether a given module M is simple object or not depends on the torsion-free classwhich contains M . Originally, the notion of simple objects was introduced and studied in thecontext of exact categories by several papers such as [Eno4, BHLR]. Since we are only interested
in torsion-free classes, we will not work in full generality.
For a torsion-free class, a simple object can be described by the following property.
Lemma 4.3.7. Let F be a torsion-free class in mod Λ and M a non-zero object in F. Thenthe following are equivalent:(1) M is simple in F.
(2) Every morphism M → F with F ∈ F is either zero or injective.
Proof. (1) ⇒ (2): Take a non-zero morphism ϕ : M → F with F ∈ F. Then we have anexact sequence
0
Ker ϕ
M
Im ϕ
0
in mod Λ. Since Im ϕ is a submodule of F ∈ F, we have Im ϕ ∈ F. Similarly Ker ϕ ∈ F holds
since so is M . On the other hand, Im ϕ 6= 0 by assumption. Therefore, Ker ϕ = 0 holds by thesimplicity of M .
(2) ⇒ (1): Take a short exact sequence
0

L

M

π

N

0

with L, N ∈ F. Then by assumption, π should be zero or injection. We have N = 0 in the former
case, and L = 0 in the latter. Thus M is a simple object in F.
We will investigate the relation between simple objects in F and brick sequences of F. Oneof the remarkable property is that simples always appear in every brick sequence of F:
Proposition 4.3.8. Let F be a torsion-free class in mod Λ and S a simple object in F. Then
S appears exactly once in every brick sequence of it (if exists).
Proof. Let B1 , . . . , Bl be a brick sequence of F. Then we have F = Filt(B1 , . . . , Bl ) holds
by Theorem 4.3.5. This means that S has a filtration consisting of B1 , . . . , Bl , but since S is a
simple object in F, clearly S ∼= Bi holds for some i. By Theorem 4.3.5, all the bricks in this bricksequence are pairwise non-isomorphic, thus S must appear exactly once.
78

4. BRUHAT INVERSIONS AND TORSION-FREE CLASSES OVER PREPROJECTIVE ALGEBRAS

By this, to find a simple object in F, we only have to work in a fixed brick sequence of F.
In order to give a criterion for a brick in a brick sequence to be simple, we will introduce sometechnical condition and lemmas.
Definition 4.3.9. Let M be a Λ-module and C a collection of Λ-modules. Then we say that
C has the zero-mono property for M if every morphism M → C with C ∈ C is either zero or
injective in mod Λ.
Then Lemma 4.3.7 amounts to that M is simple in a torsion-free class F if and only if F has
the zero-mono property for M .
The important advantage of the zero-mono property is that this property is closed under
extensions in the following sense:
Lemma 4.3.10. Let M be a Λ-module and C, D collections of Λ-modules. If C and D have thezero-mono property for M , then so does C ∗ D.
Proof. Consider any short exact sequence
0

ι

C

X

π

D

0

with C ∈ C and D ∈ D. Take any morphism ϕ : M → X, and it suffices to show that ϕ is either
zero or injective.
Consider the following diagram.
M
ϕ

0

C

ϕ
ι

X

π

D

0

Since D has the zero-mono property for M , either πϕ is an injection or πϕ = 0. In the former
case, ϕ is an injection, so suppose the latter. Then there exists a morphism ϕ : M → C with
ϕ = ϕι. Since C has the zero-mono property for M , we have ϕ is either zero or injective. Thus ϕis either zero or injective respectively.

As a first application of this corollary, we can show the following.
Corollary 4.3.11. Let F be a torsion-free class in mod Λ. Then taking labels gives aninjection{Arrows in Hasse(torf Λ) starting at F} ,→ sim F.
Proof. Let F → F 0 be an arrow in Hasse(torf Λ) with B its label. We will show that B issimple in F, or equivalently, F has the zero-mono property for B by Lemma 4.3.7.
By Proposition 4.3.1, we have F = Filt(B) ∗ F 0 and F 0 = F ∩ B ⊥ . Therefore, according toLemma 4.3.10, it suffies to show that B and F 0 have the zero-mono property for B. Clearly B
has the zero-mono property for B since B is a brick. On the other hand, since F 0 ⊂ B ⊥ , every
morphism from B to F ∈ F 0 should be zero. Thus F 0 has the zero-mono property.
Another application of Lemma 4.3.10 is the following complete description of a brick in a given
brick sequence to be simple.
Corollary 4.3.12. Let B1 , . . . , Bl be a brick sequence of a torsion-free class F in mod Λ.
Then any simple objects in F are contained in {B1 , . . . , Bl }, and the following are equivalent for1 ≤ i ≤ l.
(1) Bi is simple in F.
(2) Every morphism Bi → Bj is either zero or injective for each j 6= i, or equivalently, j > i.
(3) {B1 , . . . , Bl } has the zero-mono property for Bi .
(4) There is no surjection Bi  Bj with i < j.
Proof. Since we have HomΛ (Bi , Bj ) = 0 for j < i by Theorem 4.3.5, the two conditions in(2) are equivalent.
(1) ⇒ (2): This is clear by Lemma 4.3.7 since Bj ∈ F for every j.

4.3. BRICK SEQUENCE OF A TORSION-FREE CLASS

79(2) ⇒ (3): It suffices to recall that Bi has the zero-mono property for Bi since Bi is a brick.
(3) ⇒ (1): Lemma 4.3.10 shows that Filt(B1 , . . . , Bl ) has the zero-mono property for Bi . Since
F = Filt(B1 , . . . , Bl ) holds by Theorem 4.3.5, we have that Bi is simple in F by Lemma 4.3.7.
(2) ⇒ (4): This is clear since Bi and Bj with j > i are non-isomorphic by Theorem 4.3.5.
(4) ⇒ (2): Suppose that there are some j > i and a map ϕ : Bi → Bj which is neither zeronor injective. Take minimal j with this property. We claim that ϕ is actually a surjection.
Let 0 = F0 ← F1 ← · · · ← Fl = F be a maximal green sequence of F corresponding to thebrick sequence B1 , . . . , Bl . Then we have Fj = Filt(Bj ) ∗ Fj−1 by Proposition 4.3.1. Moreover,since (1)-(3) are equivalent and B1 , . . . , Bj−1 is a brick sequence of Fj−1 , the minimality of j
implies that Bi is a simple object in Fj−1 , that is, Fj−1 has the zero-mono property for Bi .
We have Im ϕ ∈ Fj since it is a submodule of Bj ∈ Fj . Thus by Fj = Filt(Bj ) ∗ Fj−1 , wehave the following exact sequence
Bi

ϕ

Bj

π

0

X

Im ϕ

F

0

with X ∈ Filt(Bj ) and F ∈ Fj−1 . Suppose that X = 0, then Im ϕ ∼= F ∈ Fj−1 holds. Thus
π is either zero or injective. Since ϕ 6= 0, we have that π is injective, hence so is ϕ, which
is a contradiction. Thus X 6= 0. It follows from X ∈ Filt(Bi ) and X 6= 0 that dim Bi ≤dim X holds. On the other hand, since we have injections X ,→ Im ϕ ,→ Bi , we have dim X ≤dim(Im ϕ) ≤ dim Bi . It follows that dim(Im ϕ) = dim Bi , hence the inclusion map Im ϕ ,→ Bi is
an isomorphism. Therefore, ϕ is a surjection.
Thus, the problem to determine simple objects in a torsion-free class F can be solved by thefollowing process in principle.
(1) Find and fix a maximal green sequence of F (if exists).
(2) Compute a brick sequence B1 , . . . , Bl associated to it.
(3) For each i, check whether every morphism Bi → Bj with i < j is either zero or injective(or equivalently, check whether there is no surjection Bi  Bj with i < j).
(4) sim F consists of Bi with such a property.
Probably the most non-trivial part of the above computation is (1) and (2). If F is functoriallyfinite, then (1) can be computed by using mutations of support τ -tilting modules in [AIR], and (2)can be computed by using the description of brick labels associated to mutations due to [Asa1].
Example 4.3.13. We will give two examples of computation of simple objects by considering
algebras borrowed from [DIRRT, Example 1.14]. Let Λ1 and Λ2 be k-algebras defined byv /(v 2 ).
Λ1 = k u
1
2 /(u2 ) and Λ2 = k 1
2Then torf Λi and their brick labeling for i = 1, 2 is as follows:

2

F1

1
2

F2

1
1
2

F3

0

1

2

mod Λ1
1

F4

2

and

G1

1
2
2

G2

1
2

G3

0

1

mod Λ2
1

G4

2

We omit the description of each torsion-free class. By using Corollary 4.3.12, one can obtain thefollowing table of simple objects. Note that torf Λ1 and torf Λ2 are isomorphic as posets, and F2
corresponds to G2 . However, the number of simple objects in F2 differs from that in G2 . This showsthat to determine (the number of) simple objects, the poset structure of torf Λ is not enough, andwe indeed have to compute brick labels and homomorphisms between labels in general.
Fortunately, in the case of preprojective algebras (or path algebras) of Dynkin type, we canmake use of root-theoretical properties developed in the previous section to determine simples, as
we shall see in the next section.

80

4. BRUHAT INVERSIONS AND TORSION-FREE CLASSES OVER PREPROJECTIVE ALGEBRAS

torsion-free class in mod Λ1 simple objects torsion-free class in mod Λ1 simple objects
0
∅
0
∅
F1
2
G1
2
1
F2
2, 12
G2
2, 2
2
1
F3
2, 12, 1
G3
2, 12
2
F4
1
G4
1
mod Λ1
1, 2
mod Λ1
1, 2
Table 1. Simple objects in torsion-free classes in mod Λ1 and mod Λ2

We end this section by the following small lemma, which we need later.
Lemma 4.3.14. Let F be a torsion-free class in mod Λ such that the interval [0, F] in torf Λis finite. Then every brick B in F appears at least one brick sequence of F.
Proof. Consider F(B), the smallest torsion-free class containing F. Then F(B) ⊂ F holds,
that is, F(B) ∈ [0, F].
Since [0, F] is a finite lattice, clearly there exists a maximal green sequence of F which is of
the form 0 ← . . . ← F(B) ← . . . ← F. In [DIRRT, Theorem 3.4], it was shown that there is only
one arrow which starts at F(B) in Hasse(torf Λ), and that its label is B. Therefore, this maximal
sequence gives the desired brick sequence.

4.4. Torsion-free classes over preprojective algebras of Dynkin typeIn this section, we will classify simples in torsion-free classes in preprojective algebras of
Dynkin type, by using root systems and brick sequences.
4.4.1. Notation and preliminary results. Let us briefly recall the definition of preprojective algebras of Dynkin type, with emphasis on their relation to root systems.
Let Φ be the simply-laced root system of the Dynkin type X, namely, X ∈ {An , Dn , E6 , E7 , E8 }.
We denote by W its Weyl group, and we fix a choice of simple roots of Φ.
Let Q = (Q0 , Q1 ) be a Dynkin quiver of the same type X, that is, Q is a quiver whose
underlying graph is the Dynkin diagram of type X. Then we may identify Q0 with the index setof simple roots of Φ. For i ∈ Q0 , we denote by αi the simple root of Φ corresponding to the vertex
i in the Dynkin diagram, and by si = sαi ∈ W the simple reflection with respect to αi .
Let Q be a double quiver of Q, which is obtained from Q by adding an arrow a∗ : j → i foreach arrow a : i → j in Q. The preprojective algebra of Q is defined by,XΠ = ΠΦ := kQ aa∗ − a∗ a .
a∈Q1

It is known that Π is a finite-dimensional k-algebra, and it only depends on Φ and does not dependon the choice of Q (the choice of orientations of the Dynkin diagram) up to isomorphism. Thus
we call it a preprojective algebra of Φ.
It is convenient to consider the dimension vector of Π-modules inside the ambient space ofΦ. Let V be the ambient space of Φ, then V has a basis {αi | i ∈ Q0 } as a R-vector space.
Define a group homomorphism dim : K0 (mod Π) → V by [Si ] 7→ αi , where K0 (mod Π) denotes theGrothendieck group of mod Π and Si denotes the simple Π-module corresponding to the vertexi ∈ Q0 . We simply write dim M := dim[M ] for M ∈ mod Π.
To sum up, throughout this section, we keep the following notation:• Φ is a simply-laced root system of Dynkin type X in the ambient space V .
• Q is a Dynkin quiver of the same type X.
• W is the Weyl group of Φ.
• Π is the preprojective algebra of Φ (or Q).
• Si is the simple Π-module corresponding to the vertex i ∈ Q0 .

4.4. TORSION-FREE CLASSES OVER PREPROJECTIVE ALGEBRAS OF DYNKIN TYPE

81

• dim : K0 (mod Π) → V is a map defined by [Si ] 7→ αi for i ∈ Q0 .
Torsion-free classes in mod Π was completely classified by Mizuno in [Miz], and they are in
bijection with elements in W . Let us briefly explain his result in our context.
For a vertex i ∈ Q0 , we denote by ei the corresponding idempotent of Π. We denote by Ii
the two-sided ideal of Π generated by 1 − ei . For an element w ∈ W , we can define a two-sided
ideal Iw of Π as follows: Take any reduced expression w = su1 su2 · · · sul of w in W . Then I(w) isdefined by
I(w) = Iul ...u2 u1 := Iul Iul−1 · · · Iu2 Iu1 .
This construction does not depend on the choice of reduced expressions of w by [BIRS, Theorem
III.1.9].
The following result of Mizuno gives the key connection between the representation theory ofΠ and its root system Φ.
Theorem 4.4.1 ([Miz, Theorem 2.30]). Let Φ be a simply-laced root system of Dynkin type,
W its Weyl group and Π its preprojective algebra.
(1) F(w) := Sub(Π/I(w)) is a torsion-free class in mod Π.
(2) The map w 7→ F(w) gives a bijection∼

W −→ torf Π.
Moreover, this bijection is actually an isomorphism of finite lattices, where we endow W
with the right weak order.
4.4.2. Brick sequences in preprojective algebras. We begin with studying the relation
between brick sequences of F(w) and root sequences of w.
By Theorem 4.4.1, we can identify a maximal green sequence of F(w) with a saturated chain
in the interval [e, w] in (W, ≤R ), which in turn is identified with a particular choice of reduced
expression of w (see e.g. [BB, Proposition 3.1.2]). In this way, we can talk about a brick sequenceof F(w) associated to a reduced expression of w. More precisely, let w = su1 · · · sul be a reduced
expression of w ∈ W . Then we have a maximal green sequence 0 = F(e) ← F(su1 ) ← F(su1 su2 ) ←· · · ← F(su1 · · · sul ) = F(w) of F(w), thus we obtain the corresponding brick sequence of F(w).
We remark that this sequence coincides with a sequence of layer modules considered in [AIRT].
By a lattice isomorphism in Theorem 4.4.1, an arrow in Hasse(torf Π) is of the form F(w) ←F(wsi ) for some w ∈ W and i ∈ Q0 satisfying `(w) < `(wsi ). In this case, we have I(wsi ) ⊂ I(w),and the following holds.
Proposition 4.4.2 ([IRRT, Theorem 4.1]). The brick label of F(w) ← F(wsi ) is given by
I(w)/I(wsi ).
Next we will compute dimension vectors of brick labels following [AIRT].
Proposition 4.4.3. Let w = su1 . . . sul be a reduced expression of w ∈ W and B1 , B2 , . . . , Bl
its associated brick sequence of F(w). Then the following equality holds in V for 1 ≤ m ≤ l:dim Bm = su1 · · · sum−1 (αum ).
In particular, {dim B1 , . . . , dim Bl } = inv(w) ⊂ Φ+ holds, and dim B1 , dim B2 , . . . , dim Bl is a root
sequence associated to the above reduced expression of w, defined in Definition 4.2.6.
Proof. In [AIRT, Theorem 2.7], it was shown that
[Bm ] = [I(sum−1 · · · su1 )/I(sum · · · su1 )] = Ru1 · · · Rum−1 [Suj ]holds in K0 (mod Π), where Ri : K0 (mod Π) → K0 (mod Π) for i ∈ Q0 is a group homomorphism
defined by
Ri ([Sj ]) := [Sj ] − (2δij − mij )[Si ]on the free basis {[Sj ] | j ∈ Q0 } of K0 (mod Π). Here δ is the Kronecker delta, and mij is thenumber of edges in the Dynkin diagram of Φ which connect i and j.

82

4. BRUHAT INVERSIONS AND TORSION-FREE CLASSES OVER PREPROJECTIVE ALGEBRASTherefore, it suffices to check that the following diagram commutes:
Ri

K0 (mod Π)

K0 (mod Π)

dim

dim
si

V

V

We only have to check on the basis [Sj ] for j ∈ Q0 . This follows from the following equality:
if i = j,22δij − mij = hαj , αi i = 0(4.4.1)
if i and j are not connected by an arrow in Q,−1 if i and j are connected by an arrow in Q,
which can be checked directly.



Therefore, a brick sequence of F(w) serves as a categorification of a root sequence of w. Notethat different bricks may have the same dimension vector.
Example 4.4.4. Consider an element w in Example 4.2.8. Then Figure 3 is the Hasse quiverof the interval [0, F(w)] with its brick labels. For example, two bricks 12 and 12 are non-isomorphic
3
2

•
3

•

•
13
2

2
1

•

2
1

1
2

•

•

1

0

•

3
2

1

2

•

1

• 3

2

2
1

F(w)

•

1
2

3
2

2

Figure 3. Brick sequences of w = s12312
but have the same dimension vector α1 + α2 .
The same computation can be done for Example 4.2.9, and actually Figure 3 can be seen asa Hasse quiver of [0, F(w)] with dimension vectors of its brick labels.
Corollary 4.4.5. Let w be an element of W , then the following hold.
(1) For any M ∈ F(w), we havedim M =

Xnβ β,β∈inv(w)where nβ is some non-negative integer for each β ∈ inv(w).
(2) For any B ∈ brick F(w), we have dim B ∈ inv(w). Thus dim induces a surjection
dim : brick F(w) → inv(w).
Proof. (1) Let B1 , . . . , Bl be a brick sequence of F(w). Then by Proposition 4.4.3, we haveinv(w) = {dim B1 , . . . , dim Bl }.
Let M ∈ F(w). Then Theorem 4.3.5 implies that M ∈ Filt(B1 , . . . , Bl ). Thus by taking thedimension vector, the assertion immediately follows.
(2) Let B be a brick in F(w). Then Lemma 4.3.14 implies that there is a brick sequence ofF(w) which contains B, since [0, F(w)] ⊂ torf Π ∼= W is a finite lattice. Now Proposition 4.4.3
implies that dim B ∈ inv(w). Thus the map dim : brick F(w) → inv(w) is well-defined. Moreover,
it is clearly surjective by Proposition 4.4.3.
4.4. TORSION-FREE CLASSES OVER PREPROJECTIVE ALGEBRAS OF DYNKIN TYPE

83

4.4.3. Simple objects versus Bruhat inversions. By Proposition 4.3.8, to find simple
objects in F(w), it suffices to check whether Bi is simple or not for a fixed brick sequence B1 , . . . , Blof F(w). For this, the homological lemma due to Crawley-Boevey is useful. To state this, let us
introduce the symmetric bilinear form h−, −iΠ on K0 (mod Π) defined by
X
X
X
X
h
ai [Si ],bj [Sj ]iΠ = 2
ai bi −
(ai bj + aj bi ).
i

j

i∈Q0

i→j∈Q1This coincides with the standard homological symmetric bilinear form associated to the quiver Q(or twice of it, see e.g. [ASS, VII.4]). It can also be interpreted as the restriction of the Euler formb of the extended Dynkin type corresponding to Π, see e.g. [IRRT,of the preprojective algebra Π
Section 3]. Then we have the following formula due to Crawley-Boevey.
Lemma 4.4.6 ([C-B, Lemma 1]). Let M, N ∈ mod Π. Then we have the following equation:h[M ], [N ]iΠ = dim HomΠ (M, N ) + dim HomΠ (N, M ) − dim Ext1Π (M, N ).
We remark that this formula can be shown easily by using the 2-Calabi-Yau property of theb of the extended Dynkin type.
preprojective algebra ΠWe can check the compatibility of the bilinear form on K0 (mod Π) and the value hα, βi in
α, β ∈ V defined in Section 2 as follows.
Lemma 4.4.7. Suppose that M, N ∈ mod Π satisfy dim M, dim N ∈ Φ+ . Then we haveh[M ], [N ]iΠ = hdim M, dim N i.
Proof.
P Since dim M and dim N are positive roots, we can write as dim M =dim N = j nj αj for non-negative integers mi , nj . Then we haveXX
mi αi ,
nj αj ihdim M, dim N i = hi=i mi αi and

j


X

P



mi hαi ,iXnj α j i

j




X
mi h
nj αj , αi i

=

X

=

XX

i

i

j

mi nj hαj , αi i.

j

The first equality follows from definition, the second and the last follow since h−, −i is linear withrespect to the first variable, and the third follows since Φ is simply-laced and both vectors insideh−, −i are roots. By using the equation (4.4.1), we can compute this as follows:XXX
X
mi nj hαj , αi i = 2
mi mi −
(mi nj + mi nj )
i

j

i∈Q0

i→j∈Q

Thus we have the assertion.



By using this, we can show the following main result of this paper.
Theorem 4.4.8. Let w be an element of W and B1 , . . . , Bl a brick sequence of F(w). Thenthe following are equivalent for 1 ≤ m ≤ l(1) dim Bm ∈ Binv(w) holds.
(2) Bm is a simple object in F(w).
Proof. (1) ⇒ (2) Suppose that Bm is not a simple object in F(w). Then there exists anexact sequence
0
L
Bm
N
0

84

4. BRUHAT INVERSIONS AND TORSION-FREE CLASSES OVER PREPROJECTIVE ALGEBRAS

with L, N 6= 0. By applying dim, we obtain
dim Bm = dim L + dim Nwith dim L, dim N 6= 0. By Corollary 4.4.5, both dim L and dim N are non-negative integer linear
combinations of inversions of w such that at least one of the coefficients should be strictly positive.
Thus Theorem 4.2.17 implies that dim Bm is a non-Bruhat inversion.
(2) ⇒ (1): Suppose that dim Bm is a non-Bruhat inversion of w. We will use Lemma 4.3.7 toshow that Bm is not a simple object in F(w).
By Theorem 4.2.17, there exists some α, β ∈ inv(w) such that α + β = dim Bm since Φ issimply-laced. Moreover, these satisfy hdim Bm , αi = hdim Bm , βi = 1 by Lemma 4.2.18.
On the other hand, since {dim B1 , . . . , dim Bl } = inv(w) by Proposition 4.4.3, there are i andj such that dim Bi = α and dim Bj = β. By exchanging α and β if necessary, we may assume
that i < j. Moreover, since dim B1 , . . . , dim Bl is a root sequence and dim Bm = dim Bi + dim Bj
holds, Theorem 4.2.7 implies i < m < j.
To summarize, we have found i and j with i < m < j such that dim Bm = dim Bi + dim Bj
and hdim Bm , dim Bj i = 1 hold. Then Lemma 4.4.7 implies
h[Bm ], [Bj ]iΠ = hdim Bm , dim Bj i = 1.
By combining this with Lemma 4.4.6, we have
dim HomΠ (Bm , Bj ) + dim HomΠ (Bj , Bm ) = 1 + dim Ext1Π (Bj , Bm ) ≥ 1.
On the other hand, since m < j, we must have HomΠ (Bj , Bm ) = 0 by Theorem 4.3.5. Therefore,
we have dim HomΠ (Bm , Bj ) ≥ 1, that is, HomΠ (Bm , Bj ) 6= 0.
Take any non-zero morphism ϕ : Bm → Bj . Since dim Bm = dim Bk + dim Bm , we havedim Bm > dim Bk , hence ϕ cannot be an injection. Therefore Lemma 4.3.7 implies that Bm is nota simple object in F(w).

In conclusion, we have the following classification of simple objects.
Corollary 4.4.9. Let w be an element of W . Then the following hold.
(1) A brick B in F(w) is a simple object in F(w) if and only if dim B ∈ Binv(w) holds.
(2) The map dim : brick F(w) → inv(w) in Corollary 4.4.5 restricts to a bijection∼

dim : sim F(w) −→ Binv(w).
In other words, simple objects in F(w) bijectively correspond to Bruhat inversions of w
by taking dimension vectors.
Proof. (1) Lemma 4.3.14 implies that B appears in some brick sequence of F(w). Then (1)is obvious by Theorem 4.4.8.
(2) Fix a brick sequence B1 , . . . , Bl of F(w). If S is a simple object in F(w), then S ∼= Bj forsome j by Proposition 4.3.8. Thus dim S ∈ Binv(w) holds by Theorem 4.4.8. Thus we obtain amap dim : sim F → Binv(w).
We claim that this map is a bijection. Let γ ∈ Binv(w). Then since {dim B1 , . . . , dim Bl } =inv(w) ⊃ Binv(w) by Proposition 4.4.3, there exists j with dim Bj = γ. This Bj is simple in F(w)by Theorem 4.4.8. Thus dim : sim F → Binv(w) is surjective.
On the other hand, let S and S 0 be simple objects in F(w) satisfying dim S = dim S 0 . Thenby the above argument, we have S ∼= Bi and S 0 ∼= Bj for some i and j, hence dim Bi = dim Bj
holds. Since elements in {dim B1 , . . . , dim Bl } = inv(w) are pairwise distinct by Proposition 4.2.4,
we have i = j, which shows S ∼
= S 0 . Thus dim : sim F → Binv(w) is injective.
4.4.4. Characterization of the Jordan-Hölder property. Next we will characterize
when F(w) satisfies the Jordan-Hölder property in the sense of [Eno4] in terms of the combinatorics of w. Let us recall some related definitions and results from [Eno4].
Definition 4.4.10. Let F be a torsion-free class in mod Λ for a finite-dimensional k-algebraΛ.

4.4. TORSION-FREE CLASSES OVER PREPROJECTIVE ALGEBRAS OF DYNKIN TYPE

85(1) For M in F, a composition series of M in F is a series of submodules of M
0 = M0 ⊂ M1 ⊂ · · · ⊂ Mm
such that Mi /Mi−1 is a simple object in F for each i.
(2) For M in F, let 0 = M0 ⊂ · · · ⊂ Mm = M and 0 = M10 ⊂ · · · ⊂ Mn0 = M be twocomposition series of M in F. We say that these are equivalent if m = n holds and there
00exists a permutation σ of the set {1, 2, . . . , n} such that Mi /Mi−1 ∼/Mσ(i)−1
holds= Mσ(i)for each i.
(3) We say that F satisfies the Jordan-Hölder property, abbreviated by (JHP), if any composition series of M are equivalent for every object M in F.
In [Eno4, Theorem 5.10], the author gives a numerical criterion for (JHP). To rephrase his
result in our context, we introduce the support of modules or torsion-free classes.
Definition 4.4.11. Let Λ be a finite-dimensional k-algebra and sim(mod Λ) the set of isomorphism classes of simple Λ-modules.
(1) For a module M , the support of M is a set of simple Λ-modules defined bysupp M := {S ∈ sim(mod Λ) | S is a composition factor of M }.
(2) For a collection C of modules, the support of C is a set of simple Λ-modules defined by[supp C :=supp M.
M ∈C

Then the following gives a numerical criterion for (JHP). Here for a set A, we denote by Z(A)the free abelian group with basis A. We simply write ZA := Z(A) if A is a finite set.
Theorem 4.4.12. Let F be a torsion-free class in mod Λ for a finite-dimensional k-algebra Λ.
Suppose that F = Sub M holds for some M ∈ F. Then the following are equivalent:(1) F satisfies (JHP).
(2) The natural map Z(sim F ) → K0 (mod Λ) which sends M ∈ sim F to [M ] is an injection.
(20 ) The natural map Z(sim F ) → Z(supp F ) is an isomorphism, where we identify Z(supp F ) witha subgroup of K0 (mod Λ) generated by [S] with S ∈ supp F.
(3) # sim F = # supp F holds.
Moreover, the map in (20 ) is always surjective.
Proof. We give a proof using τ -tilting theory and the Grothendieck group K0 (F) of the exact
category F, for which we refer to [AIR] and [Eno4] respectively. It is shown in [Eno4, Theorem
4.12, Corollary 5.14] that the following are equivalent:(i) F satisfies (JHP).
(ii) The natural map Z(sim F ) → K0 (F), which is always surjective, is an isomorphism.
(iii) # sim F = |U | holds, where U is a support τ − -tilting module with F = Sub U and |U | isa number of non-isomorphic indecomposable direct summands of U .
On the other hand, [AIR, Proposition 2.2] implies |U | = # supp U . Since F = Sub U , clearlysupp F = supp U holds, hence we have |U | = # supp F. Therefore, (1) and (3) are equivalent.
To see that (2) and (20 ) are also equivalent, let us consider K0 (F). By using [Eno4, Lemma5.7], one can show that the natural map K0 (F) → K0 (mod Λ) is an injection, and that its image
is precisely Z(supp F ) . Thus all the conditions are equivalent.

To describe a characterization of (JHP) for F(w), we introduce the support of w ∈ W . Recall
that simple roots and simple reflections are parametrized by Q0 in our setting.
Definition 4.4.13. Let w be an element of W . Then its support is a subset supp(w) of Q0
defined as follows:supp(w) = {i ∈ Q0 | there is a reduced expression of w which contains si }Then the support of w coincides with the support of F(w) in the following sense:

86

4. BRUHAT INVERSIONS AND TORSION-FREE CLASSES OVER PREPROJECTIVE ALGEBRAS∼

Proposition 4.4.14. Let w be an element in W . Then a natural bijection Q0 −→ sim(mod Π),which sends i to the simple Π-module corresponding to i, restricts to a bijection∼

supp(w) −→ supp F(w).
P
Proof. For a positive root β ∈ Φ+ , we can write β = i∈Q0 ni αi with ni ≥ 0 in a uniqueway. Denote by supp(β) the set of i ∈ Q0 with ni > 0. Then Corollary 4.4.5 clearly implies thatthe bijection Q0 → sim(mod Π) restricts to a bijection[∼
supp(β) −→ supp F(w).
β∈inv(w)S

Thus it suffices to show β∈inv(w) supp(β) = supp(w).
Let w = su1 · · · sul be a reduced expression of w, and β1 , . . . , βl its associatedroot sequence.
S
Then we have inv(w) = {β1 , . . . , βl }. First suppose that i belongs to β∈inv(w) supp(β), theni ∈ supp(βm ) for some 1 ≤ m ≤ l. Recall that βm = su1 · · · sum−1 (αum ), and that {αu |u ∈ Q0 } isa basis of V . Since each su : V → V changes only the αu -component of roots with respect to thisbasis, i should appear in {u1 , u2 , . . . , um }. Thus i ∈ supp(w) holds.
Conversely, suppose i ∈ supp(w). Take the minimal m such that um = i holds. We claimi ∈ supp(βm ). Indeed, we have βm = su1 su2 · · · sum−1 (αi ), and i does not appear in u1 , . . . , um−1
by the minimality of m. Since su changes only the αu -component, the αi -component of βm is 1,
hence i ∈ supp(βm ) holds.

Now the following immediately follows from these observations.
Theorem 4.4.15. Let w be an element of W . Then the following are equivalent:(1) F(w) satisfies the Jordan-Hölder property. P
PP(2) A map ϕw : ZBinv(w) → Zsupp(w) defined by ϕw ( i ni αi ) = i ni ei for i ni αi ∈ Binv(w)is a bijection, where ei denotes the basis of Zsupp(w) corresponding to i ∈ supp(w).
(3) Elements in Binv(w) are linearly independent in V .
(4) # Binv(w) = # supp(w) holds.
Moreover, the map in (2) is always surjective.
Proof. Since F(w) = Sub(Π/Iw ) by definition, we can apply Theorem 4.4.12 to F(w). Hencethe following are equivalent:(i) F(w) satisfies (JHP).
(ii) The map Z(sim F (w)) → Zsupp F (w) , which is always a surjection, is an isomorphism.
(iii) # sim F(w) = # supp F(w) holds.
By identifying Q0 with simple Π-modules, supp F(w) bijectively corresponds to supp(w) by Proposition 4.4.14. Moreover, sim F(w) bijectively corresponds to Binv(w) by taking dimension vectors
by Corollary 4.4.9. Thus the map in (ii) are exactly
P same as ϕw in the assertion under theidentification Z(sim F (w)) ∼= ZBinv(w) and Zsupp F (w) ∼= i∈supp(w) Zαi .
The left hand side in (iii) is equal to # Binv(w) by Corollary 4.4.9, and the right hand sideis equal to # supp(w) by Proposition 4.4.14, thus (1), (2) and (4) are equivalent. Moreover, it isclear that (2) is equivalent to (3) since ϕw is always surjective.

We will use the map ϕ above later in the appendix to relate our characterization of (JHP) toforest-like permutations defined in [BMB] and the Schubert variety Xw for type A case.
Remark 4.4.16. The equality # Binv(w) = # supp(w) naturally arises when one consider the
Bruhat interval in W and its Poincaré polynomial. Let w be an element of W and [e, w] theinterval with respect to the Bruhat order. A Pincaré polynomial Pw (q) of w is defined byX
Pw (q) :=q `(v) .
v∈[e,w]

P`(w)

Let us write Pw (q) = i=0 ai q i . Then we have # supp(w) = a1 and # Binv(w) = al−1 , since
supports of w are precisely simple reflections which are below w in the Bruhat order, and Bruhat

4.5. TORSION-FREE CLASSES OVER PATH ALGEBRAS OF DYNKIN TYPE

87

inversions of w are in bijection with elements which are covered by w in the Bruhat order by
Proposition 4.2.12. Thus our criterion is equivalent to a1 = al−1 .
Example 4.4.17. Consider an element w = s12312 in Example 4.2.8. Then there are three
Bruhat inversions of w, namely, 100, 010 and 011. This number is equal to the number of supp(w) ={1, 2, 3}, thus F(w) satisfies (JHP).
On the other hand, consider an element w = s012301230 in Example 4.2.9. Then we have1 } (for example, this follows
supp(w) = {0, 1, 2, 3}, but a computation shows Binv(w) = inv(w) \ {121from the fact that deleting any letter from s012301230 yields a reduced expression except for the
middle 0). so there are eight Bruhat inversions of w. Thus F(w) does not satisfy (JHP).
4.4.5. Conjectures. In this subsection, we give some natural conjectures on the existenceof the particular kind of short exact sequences related to Theorem 4.4.8.
The most non-trivial part of the proof of Theorem 4.4.8 is to show that Bm is non-simple if
dim Bm is non-Bruhat. If dim Bm is non-Bruhat, then as in the proof, there is a brick sequence
B1 , . . . , Bi , . . . , B, . . . , Bj , . . . , Bl of F(w) such that dim Bi + dim Bj = dim B. Then the following
conjecture naturally occurs.
Conjecture 4.4.18. Let w be an element of W . Take a brick sequence B1 , . . . , Bl of F(w),and suppose dim Bm = dim Bi + dim Bj for 1 ≤ i < m < j ≤ l. Then there is an exact sequence
0

Bi

Bm

Bj

0.

In fact, in the proof of Theorem 4.4.8, we only construct a non-zero non-injection ϕ : Bm → Bj ,which is enough for our purpose. This conjecture can be seen as a natural generalization of theresult of Proposition 4.5.7, where the path algebra case was shown over an algebraically closed
field.
We have another conjecture on non-simple objects. A semibrick S in mod Π is a set of bricks
in mod Π such that HomΠ (S, T ) = 0 holds for every two distinct elements S, T ∈ S.
Conjecture 4.4.19. Let w be an element of W and B a non-simple object in F(w). Then
there is a semibrick {S, T } in F(w) and an exact sequence
0

S

B

T

0.

This conjecture is closely related to the lattice property (forcing order) of the interval [e, w]
or [0, F(w)], and the root-theoretical combinatorial property of the inversion set (contractibilityof inversion triples defined in [GL]). The author has obtained the proof of Conjecture 4.4.19 fortype An , Dn using combinatorics of (signed) permutations and E6 using computer.
Conjecture 4.4.19 can be shown to be equivalent to the following conjecture. Recall that asimple object in a torsion-free class F appears in every brick sequence of F by Proposition 4.3.8.
Then it is natural to ask whether the converse holds:
Conjecture 4.4.20. Let w be an element of W . If a brick B appears in every brick sequenceof F(w), then B is a simple object in F(w).
This conjecture makes sense for any torsion-free classes over any finite-dimensional algebras,but this fails in general. For example, consider G3 in Example 4.3.13. Then there is only one brick1
1
sequence of G3 , namely, 2, 2, 12. However, 2 is non-simple in G3 .
2

2

4.5. Torsion-free classes over path algebras of Dynkin typeIn this section, we use the results in the previous section to study torsion-free classes overpath algebras of Dynkin type. Throughout this section, let Q be a Dynkin quiver, and we use thesame notation as in Section 4.4.1. We have a natural surjection of algebras Π  kQ, definedby annihilating all arrows in Q which do not appear in Q. Thereby we have an embedding
mod kQ ,→ mod Π, and we often identify mod kQ with a subcategory of mod Π.
Let us recall the celebrated theorem of Gabriel:

88

4. BRUHAT INVERSIONS AND TORSION-FREE CLASSES OVER PREPROJECTIVE ALGEBRASTheorem 4.5.1. The assignment M 7→ dim M for M ∈ mod kQ induces a bijection∼

dim : ind(mod kQ) −→ Φ+ .
In other words, indecomposable kQ-modules bijectively correspond to positive roots by taking dimension vectors.
4.5.1. Coxeter-sortable elements and torsion-free classes. We begin with introducing
some terminology which we need to give a description of torf kQ. Put n := #Q0 . Then a Coxeter
element cQ of Q is an element cQ = su1 · · · sun of W with Q0 = {u1 , . . . , un } which satisfies thefollowing condition: if there is an arrow i ← j in Q, then si appears before sj in this expressionof cQ .
Let c = cQ be a Coxeter element of Q, and w an element of W . We say that w is c-sortable if
there exists a reduced expression of the form w = c(0) c(1) · · · c(m) such that each c(i) is a subwordof c satisfying supp(c(0) ) ⊃ supp(c(1) ) ⊃ · · · ⊃ supp(c(m) ). We call such an expression a c-sorting
word of w.
Now we can state the classification of torsion-free classes in mod kQ, which was first establishedby [IT], and then generalized to any acyclic quiver by [AIRT] and [Tho].
Theorem 4.5.2 ([IT, Theorem 4.3]). Let Q be a Dynkin quiver and W its Weyl group. Forw ∈ W , define a subcategory FQ (w) of mod kQ by
FQ (w) := add{M ∈ ind(mod kQ) | dim M ∈ inv(w)}.
Then the assignment w 7→ FQ (w) gives a bijection∼{w | w is cQ -sortable} −→ torf kQ.
4.5.2. Simple objects versus Bruhat inversions. Let w be a cQ -sortable element, thenwe have a torsion-free class FQ (w) in mod kQ and a torsion-free class F(w) in mod Π. The relation
between these two was stated implicitly in [AIRT] and the proof was involved, thus we present a
brief explanation of it.
We begin with the following observation.
Proposition 4.5.3 ([AIRT, Theorem 3.3]). Let w be a cQ -sortable element in W and B1 , . . . , Bla brick sequence of F(w) associated with a cQ -sorting word of w. Then we have Bi ∈ mod kQ foreach i.
Using this, we obtain the following description of FQ (w) via a brick sequence (c.f. [AIRT,
Theorem 3.11]).
Proposition 4.5.4. Let w be a cQ -sortable element in W , and let B1 , . . . , Bl be a bricksequence of F(w) associated with a cQ -sorting word of w. Then we have FQ (w) = add{B1 , . . . , Bl }.
Moreover, FQ (w) = F(w) ∩ mod kQ holds.
Proof. By Proposition 4.5.3, we have B1 , . . . , Bl ∈ mod kQ, and {dim B1 , . . . , dim Bl }=inv(w) holds by Proposition 4.4.3. Since Theorem 4.5.1 implies that there exists exactly one
indecomposable kQ-module which has a fixed dimension vector, every indecomposable kQ-module
M with dim M ∈ inv(w) should appear in {B1 , . . . , Bl }. Therefore, by the definition of FQ (w),
we have FQ (w) = add{B1 , . . . , Bl }.
We will prove FQ (w) = F(w) ∩ mod kQ. Since each Bi belongs to F(w) ∩ mod kQ, we have
FQ (w) ⊂ F(w)∩mod kQ. Conversely, let M ∈ F(w)∩mod kQ. Then we have M ∈ Filt(B1 , . . . , Bl )by Theorem 4.3.5, where Filt is considered inside mod Π. On the other hand, since mod kQ ⊂ mod Π
is closed under subquotients, clearly we have M ∈ FiltkQ (B1 , . . . , Bl ), where FiltkQ means we
consider it inside mod kQ. However, add{B1 , . . . , Bl } = FQ (w) is known to be closed under
extensions in mod kQ since it is a torsion-free class by Theorem 4.5.2. Thus M ∈ FQ (w) holds. 
Now we can state our classification of simple objects in FQ (w).
Theorem 4.5.5. Let w be a cQ -sortable element of W . Then a bijection dim : ind FQ (w) →inv(w) restricts to a bijection∼
dim : sim FQ (w) −→ Binv(w).

4.A. DESCRIPTION AND ENUMERATION FOR TYPE A

89

In other words, simple objects in FQ (w) bijectively correspond to Bruhat inversions of w by taking
dimension vectors.
Proof. Let B1 , . . . , Bl be a brick sequence of F(w) (not FQ (w)!) associated to a cQ -sorting
word of w. Then Proposition 4.5.4 says that FQ (w) = {B1 , . . . , Bl }. Thus it suffices to show thefollowing:(Claim): The following are equivalent for 1 ≤ m ≤ l:(1) dim Bm ∈ Binv(w) holds.
(2) Bm is a simple object in F(w)(3) Bm is a simple object in FQ (w).
The equivalence of (1) and (2) is nothing but Theorem 4.4.8, thus it suffices to show that (2)and (3) are equivalent. This immediately follows from the fact that FQ (w) = F(w) ∩ mod kQ
holds by Proposition 4.5.4 and that mod kQ is closed under subquotients in mod Π.
As in the case of F(w), we can characterize the validity of the Jordan-Hölder property as
follows.
Corollary 4.5.6. Let w be an element of W . Then FQ (w) satisfies (JHP) if and only if# Binv(w) = # supp(w) holds.
Proof. Immediate from Theorems 4.4.12 and 4.5.5, once we observe that supp F(w) are in
bijection with supp(w), which can be proved similarly to Proposition 4.4.14.
These results generalize the results in [Eno4], in which the type A case was proved by directcomputation of representations of Q.
If we assume that the base field k is algebraically closed, then we can use the following resultof [DR] to give a more quick proof of Theorem 4.5.5 without using preprojective algebras.
Proposition 4.5.7 ([DR]). Let k be an algebraically closed field and Q a Dynkin quiver. Take
indecomposable kQ-modules L, M, N such that dim L + dim N = dim M holds in Φ+ . Then byinterchanging L and N if necessary, there is an exact sequence in mod kQ of the following form:
0

L

M

N

0

Note that the proof of this given in [DR] is algebro-geometric. The author do not knowwhether the same method can be used to study preprojective algebras, and whether this kind ofexact sequence always exists in the preprojective algebra case (see Conjecture 4.4.18).
Now we can give another proof of Theorem 4.5.5 provided that k is algebraically closed.
Proof of Theorem 4.5.5. Recall that we have a bijection dim : ind FQ (w) → inv(w). We
will show that M ∈ ind FQ (w) is simple in FQ (w) if and only if dim M ∈ Binv(w). If M is not
simple, then Theorem 4.2.17 clearly implies that dim M is a non-Bruhat inversion of w by taking
dimension vectors. Thus it suffices to show that if dim M is a non-Bruhat inversion of w, then M
is not a simple object in FQ (w).
By Theorem 4.2.17, there are α, β ∈ inv(w) such that dim M = α + β holds since Φ is simplylaced. Take indecomposable kQ-modules Mα and Mβ with dim Mα = α and dim Mβ = β, which
exist by Theorem 4.5.1. By definition we have Mα , Mβ ∈ FQ (w) holds. Then Proposition 4.5.7
implies that there exists a short exact sequence
0

Mα

M

Mβ

0

by interchanging α and β if necessary. Clearly this implies that M is not a simple object in
FQ (w).

4.A. Description and enumeration for type A
In this appendix, we focus on type A case and give more explicit and combinatorial descriptionof results in this paper. First, we give an explicit diagrammatic construction of simple objects inF(w) by using a Bruhat inversion graph Gw . Next, we characterize elements w such that F(w)satisfies (JHP) in terms of Gw , and deduce some numerical consequences.

90

4. BRUHAT INVERSIONS AND TORSION-FREE CLASSES OVER PREPROJECTIVE ALGEBRAS

Throughout this appendix, we will use the following notation:• Q is a quiver whose underlying graph is the following Dynkin diagram of type An :
1

2

···

n.

• Φ is a standard root system of type An in V := Rn+1 , that is, Φ = {εi − εj | 1 ≤ i, j ≤n + 1}, where εi denotes the standard basis of V .
• We fix simple roots by αi := εi − εi+1 for 1 ≤ i ≤ n.
• W is the Weyl group of Φ, and we often identify W = Sn+1 with the symmetric group
Sn+1 acting on the set [n + 1] := {1, 2, . . . , n, n + 1} so that w(εi ) = εw(i) holds.
• For i, j ∈ [n + 1], we denote by (i j) ∈ Sn+1 the transposition of the letter i and j, andput β(i,j) := εi − εj . Then (i j) is identified with the reflection with respect to β(i,j) .
• For w ∈ Sn+1 , we often use the one-line notation for w, that is, we write as w =w(1)w(2) · · · w(n + 1).
• Π is the preprojective algebra of Φ.
• F(w) ∈ torf Π is the torsion-free class in mod Π defined in Theorem 4.4.1.
First, let us introduce the combinatorial variants of (Bruhat) inversion sets.
Definition 4.A.1. Let w be an element of Sn+1 .
(1) Inv(w) consists of a pair (i, j) with 1 ≤ i < j ≤ n + 1 such that the letter j appears left
to i in the one-line notation for w.
(2) BInv(w) consists of a pair (i, j) with 1 ≤ i < j ≤ n + 1 such that the letter j appears left
to i and there is no k with i < k < j such that the letter k appears between j and i.
This notation is justified by the following, which can be proved by direct calculation.
Proposition 4.A.2. Let w be an element of W = Sn+1 and i, j ∈ [n + 1]. Then the followinghold.
(1) β(i,j) ∈ inv(w) if and only if (i, j) ∈ Inv(w).
(2) β(i,j) ∈ Binv(w) if and only if (i, j) ∈ BInv(w).
Let us introduce a way to visualize Bruhat inversions, a Bruhat inversion graph. Let w be anelement of W = Sn+1 . Consider a square array of boxes with (n + 1) rows and (n + 1) columns.
We name (i, j) to the box in the i-th row and j-th column, and put a dot in (i, w(i)) for each1 ≤ i ≤ n + 1. We call it a diagram of w. A Bruhat inversion graph Gw is obtained by connecting
every two dots in the diagram of w which correspond to the Bruhat inversion of w, that is, weconnect (w−1 (i), i) and (w−1 (j), j) if (i, j) ∈ BInv(w) holds. Figure 4 is examples for Gw for1 2 3 4 5
1
2
3
4
5

1 2 3 4 5
1
2
3
4
5

w = 42153

1 2 3 4 5
1
2
3
4
5

w = 42513

1 2 3 4 5
1
2
3
4
5

w = 42351

w = 12543

Figure 4. Bruhat inversion graphs Gw
elements w in S5 . It is clear from definition that Gw is obtained by connecting every two dots inthe diagram of w which look as follows,

such that there are no dots in the gray region. It is also clear from definition that an edge in Gw
bijectively corresponds to a Bruhat inversion of w.

4.A. DESCRIPTION AND ENUMERATION FOR TYPE A

91

4.A.1. Diagrammatic description of simples in F(w). In this subsection, we will give a
combinatorial description of simple objects in F(w) using arc diagrams introduced in [Rea2] andthe description of bricks given in [Asa2]. The author would like to thank Y. Mizuno for explaining
to him the interpretation of the description in [Asa2] in terms of arc diagrams.
Let w be an element of Sn+1 = W . We will construct a Π-module Be for each edge e in Gπ
in the following way:
(1) Remove all the edges in Gw except e.
(2) Move down all the dots into a single horizontal line, allowing e to curve, but not to passthrough any dots. We call this diagram an arc diagram of e.
(3) Draw n vertical dashed lines between adjacent dots in the arc diagram, and name theselines as 1, 2, . . . , n from left to right.
(4) Define a (not necessarily full) subquiver Q(e) of Q by the following rule:• The vertex set of Q(e) consists of i ∈ Q0 such that e and the line i intersect in thearc diagram of e.
• Suppose that we have i, i + 1 ∈ Q(e)0 . If the segment of e cut by the lines i and
i + 1 is above the unique dot between these lines, then we put an arrow i → i + 1,
and put i ← i + 1 if the segment is below the dot.
We call Q(e) the defining quiver of Be .
(5) Define a Π-module Be as follows, where we construct Be as a representation of Q.
• To each i ∈ Q0 , we assign k if i ∈ Q(e)0 , and 0 otherwise.
• To each arrow i → j ∈ Q, we assign the identity map if i → j ∈ Q(e)1 , and 0
otherwise.
Since two cycles in kQ annihilates Be by construction, we can regard Be as a Π-module.
Example 4.A.3. Figure 5 is an example of this construction for w = 42513 and all the edges
in Gw . The middle part is an arc diagram of three edges, and the right part shows defining quiversof Be corresponding to magenta, green, red, gray and blue edges from top to bottom.
Figure 5. Example of Be for w = 42513
1

2

3

Q: 1
1

4

Q(e): 1
Gw

2

3

2
2

3
3
3
3

Arc diagrams

4

4
4

Figure 6 is an example for w = 42351 and two particular edges in Gw . Note that the orientations of edges between 2 and 3 in Q(e) may differ as in this example.
Figure 6. Example of Be for w = 42351
1

2

3

4

Q: 1

2

3

4

1

2
2

3
3

4

The above construction of arc diagrams is due to [Rea2]. More precisely, in [Rea2], arcs
were assigned only to descents of w, which are inversions (i, j) ∈ Inv(w) such that w is of theform · · · ji · · · . Similarly, our construction of Π-modules Be is a generalization of the one given
in [Asa2, Theorem 4.6], where Be was given (without using arc diagrams) for elements w withunique descent.

92

4. BRUHAT INVERSIONS AND TORSION-FREE CLASSES OVER PREPROJECTIVE ALGEBRAS

We will confirm that Be is the simple object in F(w) associated with the Bruhat inversion
corresponding to e.
Proposition 4.A.4. Let w be an element of W = Sn+1 . Take (i, j) ∈ BInv(w), and let e
denotes the edge in Gw corresponding to it. Then Be is the unique simple object in F(w) with
dim Be = β(i,j) .
Proof. First, we will construct another element we of W with a unique descent. The following
picture illustrates the construction, where all the dots lie in the gray regions.
(A)

(D)
(G)

(B)

(E)

(C)

(F)

⇒

⇒ Sort (G) and (H)

⇒

we

(H)

The leftmost diagram is the diagram of w, and the red edge indicates e. Then perform the following
procedure, requiring that all the diagrams in each step are diagrams of some elements in W :(1) Move all the dots in (B) and (C) to (A), and those in (D) and (E) to (F).
(2) Sort all the dots in (G) and (H) so that the column number increases from top to bottom.
Denote by we the resulting element. Then it is clear from the construction that we has the uniquedescent. See Figure 7 for the example of this process, where w = 56723814.
Alternatively, in terms of the one-line notation, we can describe we as follow. Let w =· · · (a) · · · j · · · (b) · · · i · · · (c) · · · be the one-line notation for w. First, move all the numbers in (b)and (c) which are smaller than i to (a), and all the numbers in (a) and (b) which are larger thanj to (c). Then we obtain an element of the form · · · (a) · · · ji · · · (c) · · · , since (i, j) is a Bruhat
inversion of w. Next, sort the part (a) and (c) in ascending order, and denote by we the resulting
element. Then we = · · · ji · · · has the unique descent at the underlined part.
It is straightforward to see that the module Be is the same as the module Bwe := Be0 , wheree0 is the (unique) edge in the Bruhat inversion graph of we . Moreover, it is easily checked that
Bwe is nothing but the brick constructed in [Asa2, Theorem 4.6] associated to we , which has theunique descent.
It is shown in [Asa2, Theorem 3.1] that Bwe is the label of the unique arrow starting at F(we ).
In particular, we have Bwe ∈ brick F(we ). On the other hand, by construction, it is straightforward
to check that Inv(we ) ⊂ Inv(w) holds, which implies we ≤ w in the right weak order in W , seee.g. [BB, Proposition 3.1.3]. Thus we have F(we ) ⊂ F(w) in torf Π by Theorem 4.4.1. Hence wehave Bwe ∈ brick F(w). Therefore, Corollary 4.4.9 implies that Bwe is the unique simple object inF(w) with its dimension vector β(i,j) , since β(i,j) ∈ Binv(w).

By this, we can obtain all the simple objects in F(w) by computing Be for each edge e in Gw ,as we have done in Figure 5.
Figure 7. Example for we in Proposition 4.A.4
1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8⇒

w = 56723814

1 2 3 4 5 6 7 8⇒52163784

we = 12563478

4.A. DESCRIPTION AND ENUMERATION FOR TYPE A

93

4.A.2. Forest-like permutations and the Jordan-Hölder property. Next, we will investigate elements w ∈ W = Sn+1 such that F(w) satisfy the Jordan-Hölder property. By Theorem
4.4.15, this is equivalent to that # supp(w) = # Binv(w). By using [Sage], we calculated the number of such elements in Sn+1 and obtained a sequence 2, 6, 22, 89, 379, 1661, . . . . This coincides
with [OEIS, A111053], a sequence of the number of forest-like permutations defined in [BMB].
These conditions are equivalent (up to a multiplication by the longest element), as we shall see
later.
The following is the result of [BMB] in our context.
Theorem 4.A.5 ([BMB, Theorem 1.1]). Let w be an element of W = Sn+1 . Then thefollowing are equivalent:(1) The Bruhat inversion graph Gw is a forest, that is, it does not contain any cycles as an
undirected graph.
P(2) Define a map Lw : Zn → ZBinv(w) by Lw (ei ) = {β | β ∈ Binv(w), i ∈ supp(β)}, where ei
denotes the i-th standard basis of Zn . Then this map is surjective.
(3) w avoids the patterns 4231 and 3412 with Bruhat restriction 4 ↔ 1, that is, there exist
no 1 ≤ a < b < c < d ≤ n + 1 such that the one-line notation for w is of the formw = · · · d · · · b · · · c · · · a · · · or w = · · · c · · · da · · · b · · · .
We call w forest-like if Gw is a forest. We remark that w is forest-like in our sense if and only
if w0 w is forest-like in the sense of [BMB], where w0 = (n + 1)n · · · 21 is the longest element.
This turns out to be equivalent to our characterization of w such that F(w) satisfies (JHP):
Proposition 4.A.6. Let w be an element of W = Sn+1 . Then the following are equivalent:(1) F(w) satisfies (JHP), that is, # supp(w) = # BInv(w) holds.
(2) w is forest-like.
Proof. We will show that the surjectivity of Lw : Zn → ZBinv(w) is equivalent to the bijectivityof the map ϕw : ZBinv(w) → Zsupp(w) in Theorem 4.4.15.
Since supp(w) is a subset of Q0 = {1, 2, . . . , n}, we have the natural inclusion ι : Zsupp(w) ,→ Zn .
Then it is clear from the definition of Lw that Lw is surjective if and only if so is L0w := Lw ◦
ι : Zsupp(w) → ZBinv(w) .
Now it is straightforward to check that L0w is nothing but the Z-dual of ϕw by calculating
matrix representations of L0w and ϕw : the transpose of the matrix representing L0w coincides withthe matrix representing ϕw . In particular, since ϕw is always surjective, L0w should be injective.
Therefore, the surjectivity of Lw is equivalent to the bijectivity of L0w , which is in turn equivalent
to the bijectivity of ϕw , since Z-dual HomZ (−, Z) is a duality of the category of finitely generated
free Z-modules.

The motivation of forest-like permutations in [BMB] comes from the study of Schubert varieties
in the flag variety. Consider the flag variety SLn+1 (C)/B, where B is the subgroup of upper
triangular matrices. For w ∈ Sn+1 , let ew denote the permutation matrix for w. Then the
Schubert variety Xw is the Zariski closure of the B-orbit of ew in the flag variety.
It is known that Xw is a projective variety, which is not necessarily smooth. There are various
studies on the relation between algebro-geometric properties of Xw and combinatorial properties
of w. For example, Xw is smooth if and only if w avoids the patterns 4231 and 3412. Since there
are many excellent papers and books on Schubert varieties, we only refer the reader to the recent
survey article [AbBi] for the details.
We say that a variety is factorial if the local ring at every point is a unique factorization
domain. In [WY, Proposition 2], it was proved that Xw is factorial if and only if the map Lw
in Theorem 4.A.5 is surjective. Thus Theorem 4.A.5 characterizes the factoriality of Xw in a
combinatorial way.
By combining this result to Theorems 4.4.15 and 4.A.5, Proposition 4.A.6 and [BMB, Theorem3.1], we immediately obtain the following summary:Corollary 4.A.7. Let w be an element of W = Sn+1 . Then the following are equivalent:

94

4. BRUHAT INVERSIONS AND TORSION-FREE CLASSES OVER PREPROJECTIVE ALGEBRAS(1) Gw is forest.
(2) # supp(w) = # BInv(w) holds.
(3) Elements in Binv(w) are linearly independent.
(4) F(w) satisfies (JHP).
(5) Xw is factorial.
(6) w avoids the pattern 4231 and 3412 with Bruhat restriction 4 ↔ 1.
We remark that there is an explicit formula for the generation function of the number offorest-like permutations in [BMB, (2)], and the number grows exponentially, which says that
although there are lots of forest-like permutations, the number is relatively small compared to#Sn+1 = (n + 1)!.
Theorem 4.4.15 shows that (2)-(4) above are equivalent for all Dynkin types. It seems thatthey are also equivalent to (5) for other Dynkin types, by the same argument as in Proposition
4.A.6 using the maps ϕw in Theorem 4.4.15 and Lw in Theorem 4.A.5 and the Monk-Chevalley
formula. Since the author does not have appropriate knowledge on algebraic geometries andSchubert varieties, we state it in the following conjecture:
Conjecture 4.A.8. Let G be a simple algebraic group over C, B a Borel subgroup, Φ theassociated root system and W the Weyl group. Fix w ∈ W , and denote by Xw the Schubert variety
in G/B. Then the following are equivalent:(1) Xw is factorial.
(2) # supp(w) = # Binv(w) holds, or equivalently, Bruhat inversions of w are linearly independent, or equivalently, the torsion-free class F(w) in mod ΠΦ satisfies (JHP).
For type D case, we can use the realization of the Weyl group as a group of signed permutations
with even number of negatives. In particular, it seems to be an interesting problem to find ananalogue of Corollary 4.A.7 for type D case.

CHAPTER 5

Monobrick, a uniform approach to torsion-free classes andwide subcategories
This chapter is based on [Eno6]
For a length abelian category, we show that all torsion-free classes can be classified by using
only the information on bricks, including non functorially-finite ones. The idea is to consider
the set of simple objects in a torsion-free class, which has the following property: it is a set ofbricks where every non-zero map between them is an injection. We call such a set a monobrick.
In this paper, we provide a uniform method to study torsion-free classes and wide subcategories
via monobricks. We show that monobricks are in bijection with left Schur subcategories, whichcontains all subcategories closed under extensions, kernels and images, thus unifies torsion-freeclasses and wide subcategories. Then we show that torsion-free classes bijectively correspond tocofinally closed monobricks. Using monobricks, we deduce several known results on torsion(-free)classes and wide subcategories (e.g. finiteness result and bijections) in length abelian categories,
without using τ -tilting theory. For Nakayama algebras, left Schur subcategories are the same as
subcategories closed under extensions, kernels and images, and we show that its number is related
to the large Schröder number.
5.1. Introduction
For a finite-dimensional algebra Λ, several classes of subcategories of mod Λ have been investigated in the representation theory of algebras. Among them, torsion classes and torsion-freeclasses have been central, together with their connection to the tilting theory and various triangulated categories.
Recently, Adachi-Iyama-Reiten’s paper [AIR] made a major breakthrough in a classification ofthese subcategories. They show that functorially finite torsion-free classes can be classified using
support τ − -tilting modules. Their method is to consider Ext-injective objects.
In this paper, we take a different approach, which enables us to classify all torsion-free classes
in any length abelian categories. Our method is to consider simple objects (Definition 5.2.2).
We observe that every simple object in a torsion-free class is a brick (a module with a division
endomorphism ring), and we classify torsion-free classes using only the information on bricks:
Theorem N (= Theorem 5.3.15). Let A be a length abelian category. Then we have a bijection
between the following sets.
(1) The set of all torsion-free classes F in A.
(2) {M | M is a set of bricks in A satisfying the following two conditions: }(MB) Every non-zero map between objects in M is injective.
(CC) If there is an injection N ,→ M for a brick N 6∈ M and M ∈ M, then there is a
non-zero non-injection N → M 0 for some M 0 ∈ M.
The map from (1) to (2) is given by the set sim F of simple objects in F, and from (2) to (1) isgiven by taking the extension closure Filt M of M.
Example 5.1.1 (= Example 5.7.4). Let Λ be any Nakayama algebra whose quiver is given
by 1  2. Then there are 4 bricks in mod Λ, namely, 1, 2, 12, 21. Only from the information on(non-)injections between them, we can combinatorially list up a set M satisfying (MB) and (CC)above, namely, ∅, {1}, {2}, {1, 21}, {2, 12}, {1, 2}. Hence there are 6 torsion-free classes in mod Λ,
namely, 0 = Filt ∅, Filt{1}, Filt{2}, Filt{1, 21}, Filt{2, 12}, Filt{1, 2} = mod Λ.
95

96

5. MONOBRICKSIn this paper, we call a set M of bricks satisfying (MB) a monobrick. A well-known example
is a semibrick, a pairwise Hom-orthogonal set of bricks. It is classical that semibricks in A are
in bijection with wide subcategories in A by the same maps as in Theorem N (c.f. [Rin1, 1.2]).
The aim of this paper is to provide a uniform theory to study monobricks and several kinds ofsubcategories including torsion-free classes and wide subcategories, thereby giving a systematic
framework for studying these subcategories.
Our starting point is the bijection between mbrick A, the set of monobricks in A, and SchurL A,
the set of left Schur subcategories. A left Schur subcategory is a category whose simple objects
satisfy the one-sided Schur’s lemma (see Definition 5.2.5). The class of left Schur subcategories
contains any subcategories of A which are closed under extensions, kernels and images, thus unifies
torsion-free classes and wide subcategories.
Theorem O (= Theorems 5.2.11, 5.3.15, 5.4.5). Let A be a length abelian category. Then we
have bijections between the set of left Schur subcategories and monobricks in A:
SchurL A

simmbrick A

Filt

Moreover, this bijection restricts to the following bijections:• wide A  sbrick A between the set of wide subcategories of A and semibricks in A, and• torf A  mbrickc.c. A between the set of torsion-free classes in A and cofinally closed
monobricks in A (monobricks satisfying (CC) in Theorem N).
In the case of Nakayama algebras, we show that left Schur subcategories are precisely subcategories closed under extensions, kernels and images (Theorem 5.6.1), and the number of left Schursubcategories is related to the large Schröder number (Theorem 5.6.12).
We establish Theorem O by using two natural maps W : SchurL A  wide A and F : SchurL A 
torf A, where W(E) is the same as in [IT, MS] (Definition 5.4.3) and F(E) is the smallest torsion-freeclass containing E. We describe these maps in terms of a natural poset structure of monobricks
M, namely, L ≤ M in M if there is an injection L ,→ M . Then semibricks and cofinally closed
monobricks can be characterized by this poset structure (see Proposition 5.4.2 and Definition
5.3.2). Now the maps W and F are easily described in terms of the poset structure as follows.
Theorem P (= Theorems 5.3.15, 5.4.5). Let A be a length abelian category.
(1) The following diagram commutes, and the horizontal maps are bijections.
SchurL A

sim

max

W

wide A

mbrick A

Filtsimsbrick A

Filt

Here max M for a monobrick M denotes the set of maximal elements of M.
(2) The following diagram commutes, and the horizontal maps are bijections.
SchurL A

sim

(−)

F

torf A

mbrick A

Filt
sim
Filt

mbrickc.c. A

Here M for a monobrick M denotes the cofinal closure of M (Definition 5.3.5).
As an application, we can quickly prove the finiteness result in [DIJ]: torf A is a finite set if
and only if there are only finitely many bricks (Theorem 5.5.5). In addition, we can easily deduce
the following bijections between torf A and wide A using only some combinatorial observation on
posets. This was proved in [MS] in the case of finite-dimensional algebras by using τ -tilting theory.
Corollary Q (= Corollary 5.5.4). Let A be a length abelian category. Suppose that torf A
is a finite set. Then the maps W : torf A  wide A : F are mutually inverse bijections.

5.2. BIJECTION BETWEEN MONOBRICKS AND LEFT SCHUR SUBCATEGORIES

97

Comparison to τ -tilting theory. For the convenience of the reader, we summarize advantages and disadvantages of monobricks compared to τ -tilting theory.
(Advantages)• τ -tilting theory uses Ext-projectives, while we use simple objects. This enables us towork with any length abelian categories, where there may not be any projective objects.
• τ -tilting theory cannot classify non functorially finite cases, while monobricks can. Thisis because non-functorially finite subcategories may not have Ext-projectives.
• Using monobricks, we can study both wide subcategories and torsion-free classes in thesame framework, and the relation between them become more transparent.
• Left Schur subcategories, or subcategories closed under extensions, kernels and images,
seem to be new objects to study. Our enumerative result on Nakayama algebras via theSchröder number suggests that there are more hidden combinatorics in other algebras.
(Disadvantages)• In general, left Schur subcategories are complicated to deal with. Actually, there are left
Schur subcategories which are not even closed under direct summands (Example 5.7.2).
• One of the benefits of τ -tilting theory is a mutation, which provides a way to create
various torsion-free classes starting from mod Λ. So far, we have no such theory formonobricks.
• We cannot investigate functorial finiteness by monobricks. More precisely, two isomorphic monobricks (as posets) can correspond to functorially finite and non-functoriallyfinite torsion-free classes (Example 5.7.5).
Finally, we should mention the relation of this paper to [Asa1], where a bijection between
functorially finite torsion-free classes and semibricks satisfying some conditions was established.
Although we cannot reprove his result (due to the last diasadvantage), his map can be easily
described via monobricks: F 7→ max(sim F). See Remark 5.4.6 for the more detail.
Organization. This paper is organized as follows. In Section 5.2, we study basic propertiesof left Schur subcategories and monobricks, and establish a bijection between them. In Section
5.3, we study the cofinal closure M and show its relation to torsion-free classes. In Section 5.4,
we study the map W and its relation to max M. In Section 5.5, we apply previous results to show
results on torsion-free classes and wide subcategories. In Section 5.6, we give a combinatorialclassification of monobricks over Nakayama algebras, and enumerate its number. In Section 5.7,
we show some examples of the classification of monobricks and the computation of M and max M.
Conventions and notation. Throughout this paper, we assume that all categories are
skeletally small, that is, the isomorphism classes of objects form a set. In addition, all subcategories
are assumed to be full and closed under isomorphisms. We often identify an isomorphism classin a category with its representative. We always denote by A a skeletally small length abeliancategory, that is, an abelian category in which every object has finite length. For a collection Cof objects in A, we denote by add C the subcategory of A consisting of direct summands of finite
direct sums of objects in C. For a finite-dimensional algebra Λ, we denote by mod Λ the categoryof finitely generated right Λ-modules. For a set A, we denote by #A its cardinality.
5.2. Bijection between monobricks and left Schur subcategories
First we introduce a monobrick in a length abelian category A. Recall that a brick in A is anobject M such that EndA (M ) is a division ring.
Definition 5.2.1. Let M be a set of isomorphism classes of bricks in A.
(1) M is called a monobrick if every morphism between elements of M is either zero or an
injection in A. We denote by mbrick A the set of monobricks in A.
(2) M is called a semibrick if every morphism between elements of M is either zero or an
isomorphism. We denote by sbrick A the set of semibricks in A.

98

5. MONOBRICKS

Note the assumption that a monobrick M consists of bricks is automatically satisfied if we
require the above property, since every non-zero endomorphism of M in M should be an injection,
thus an isomorphism since M has finite length.
Clearly every semibrick is a monobrick, thus sbrick A ⊂ mbrick A holds. Next we introduce
left Schur subcategories of A. Roughly speaking, it is an extension-closed subcategory of A suchthat the “one-sided Schur’s lemma” holds. Let us define some notations.
Definition 5.2.2. Let E be a subcategory of A.
(1) E is closed under extensions or extension-closed in A if it satisfies the following condition:for every short exact sequence
0

X

Y

Z

0

in A, if X and Z belong to E, then so does Y .
(2) Suppose that E is extension-closed in A. Then a non-zero object M in E is a simple
object in E if there is no exact sequence of the form
0

L

M

N

0

in A satisfying L, M, N ∈ E and L, N 6= 0. We denote by sim E the set of isomorphismclasses of simple objects in E.
Clearly sim A is nothing but the set of the usual simple objects in an abelian category A.
Thus sim E is an analogue of simple objects inside E. Simple objects can be considered as one of
the invariants of extension-closed subcategories, and has an application such as a characterizationof the Jordan-Hölder type property in E [Eno4].
To define and study left Schur subcategories, the following terminology is useful.
Definition 5.2.3. Let C be a collection of objects in A. Then a non-zero object M ∈ A isleft Schurian for C if every morphism M → C with C ∈ C is either zero or an injection in A.
Note that we do not require that M belongs to C. It is clear that a collection M of non-zero
objects in A is a monobrick if and only if every object in M is left Schurian for M. Simple objects
in A (in the usual sense) are left Schurian for any collection C.
The fundamental relation between left Schurian objects and simple objects is as follows.
Proposition 5.2.4. Let E be an extension-closed subcategory of A. Then the following hold.
(1) {M ∈ E | M is left Schurian for E} is a monobrick.
(2) If M in E is left Schurian for E, then M is simple in E.
Proof. (1) Suppose that M and N in E are left Schurian for E. Then since M is left Schurian
for E, every morphism M → N is either zero or an injection. Thus the assertion holds.
(2) Suppose that M in E is left Schurian for E, and take an exact sequence
0

L

M

π

N

0

in A with L, N ∈ E. Then π should be either zero or an injection. In the former case, we have
N = 0, and in the latter, we have L = 0. Thus M is simple in E.
Then we can define a left Schur subcategory as follows.
Definition 5.2.5. A subcategory E of A is left Schur if it satisfies the following conditions:(1) E is closed under extensions in A.
(2) Every simple object in E is left Schurian for E, that is, for a simple object M in E, every
morphism M → X with X ∈ E is either zero or an injection in A.
We denote by SchurL A the set of left Schur subcategories of A.
The following immediately follows from Proposition 5.2.4:
Corollary 5.2.6. Let E be a left Schur subcategory of A and M an object in E. Then M issimple in E if and only if M is left Schurian for E. Moreover, sim E is a monobrick.

5.2. BIJECTION BETWEEN MONOBRICKS AND LEFT SCHUR SUBCATEGORIES

99

We are going to show that all wide subcategories and torsion-free classes are left Schur. Let
us recall the definitions of these subcategories.
Definition 5.2.7. Let E be a subcategory of A.
(1) E is closed under kernels (resp. cokernels, images) if for every morphism X → Y in E,
we have Ker f (resp. Coker f , Im f ) belongs to E.
(2) E is closed under subobjects (resp. quotients) if every subobject (resp. quotient object)of X belongs to E for every X in E.
(3) E is a torsion-free class (resp. torsion class) in A if it is closed under extensions andsubobjects (resp. extensions and quotients) in A. We denote by torf A the set of torsionfree classes in A.
(4) E is a wide subcategory of A if it is closed under extensions, kernels and cokernels. We
denote by wide A the set of wide subcategories of A.
It can be easily shown that every wide subcategory or every torsion-free class in A is closed
under extensions, kernels and images. We prove that this condition implies left Schur, thus
wide A ⊂ SchurL A and torf A ⊂ SchurL A hold.
Proposition 5.2.8. Let E be a subcategory of A which is closed under extensions, kernels andimages in A. Then E is a left Schur subcategory of A.
Proof. Let M be a simple object in E and f : M → X be a morphism with X ∈ E. Then we
have the following exact sequence in A:
0

Ker f

M

Im f

0

Since E is closed under kernels and images, we have Ker f, Im f ∈ E. Thus either Ker f = 0 or
Im f = 0 since M is simple in E. In the former case, we have that f is an injection in A, and in
the latter, we have f = 0. Thus M is left Schurian for E, hence E is left Schur.

Example 5.2.9. Let k be a field and Q be a quiver 1 ← 2 ← 3. Then the Auslander-Reiten
quiver of mod kQ is as follows:
3
2
1
2
1
1

3
2
2

3

3
3
Now E1 = add{1, 21, 2, 2} is a torsion-free class, and sim E1 = {1, 2, 2}. It can be checked that every
11

simple object in E1 is left Schurian for E1 , thus is a left Schur subcategory (this follows also from
3
Proposition 5.2.8). On the other hand, consider E2 = add{21, 2, 2}. This subcategory is closed
1
under extensions, and all the three indecomposables are simple objects in E2 . However, we have a
non-zero non-injection 21  2. Thus E2 is not a left Schur subcategory.
For a left Schur subcategory E of A, we have a monobrick sim E by Corollary 5.2.6. Conversely,
for a given monobrick M, we will construct a left Schur subcategory whose simples are M. To dothis, we will use the following operation.
Definition 5.2.10. Let C be a collection of objects in A. Then Filt C denotes the subcategoryof A consisting of objects X such that there is a chain
0 = X0 < X1 < · · · < Xn = X
of subobjects of X such that Xi /Xi−1 is in C for each i. We call such a chain a C-filtration of X,
and n a length of this C-filtration.
It follows from the Noether isomorphism theorem that Filt C is extension-closed, and it isobvious from the construction that it is the smallest extension-closed subcategory of A containingC.
Now we can state our first main result of this paper.

100

5. MONOBRICKS

Theorem 5.2.11. Let A be a length abelian category. Then sim and Filt give mutually inverse
bijections between left Schur subcategories of A and monobricks in A, which extends the bijection
between wide subcategories and semibricks:
sim : SchurL A∼

mbrick A : Filtwide A∼sbrick A

We need some preparation to prove it. For two collections C and D of objects in A, we denote
by C ∗ D the subcategory of A consisting of objects X such that there is an exact sequence
0

C

X

D

0

in A with C ∈ C and D ∈ D. As for this, the following lemma in [Eno5, Lemma 3.10] is useful.
We give a proof for the convenience of the reader.
Lemma 5.2.12. Let M be an object in A. If M is left Schurian for two collections C and Dof objects in A, then M is left Schurian also for C ∗ D.
Proof. Take a short exact sequence in A
0

ι

C

X

π

D

0

with C ∈ C and D ∈ D. Let ϕ : M → X be any morphism, and we will prove that ϕ is either zeroor an injection in A. Consider the following commutative diagram.
M
ϕ

0

C

ϕ
ι

X

π

D

0

Since M is left Schurian for D, either πϕ is an injection or πϕ = 0. In the former case, ϕ is aninjection, so suppose the latter. Then there exists a morphism ϕ : M → C with ϕ = ϕι. Since M
is left Schurian for C, we have that ϕ is either zero or injective. Thus ϕ is either zero or injective
respectively.
Now we are ready to prove Theorem 5.2.11.
Proof of Theorem 5.2.11. For a left Schur subcategory E of A, we have sim E ∈ mbrick A
by Corollary 5.2.6, thus we have a map sim : SchurL A → mbrick A.
For the converse direction, let M be a monobrick in A. Then Filt M is closed under extensions
in A. We will prove that Filt M is a left Schur subcategory of A. We show the following claim:(Claim): Let M be a monobrick. Then the following are equivalent for M ∈ Filt M.
(1) M is simple in Filt M.
(2) M is in M.
(3) M is left Schurian for Filt M.
Proof of (Claim).
(1) ⇒ (2): By the construction of Filt M and the definition of a simple object, every simple
object in Filt M should belong to M.
(2) ⇒ (3): Let M ∈ M. Since M is a monobrick, M is left Schurian for M. Then by using
Lemma 5.2.12 repeatedly, M is left Schurian for Filt M.
(3) ⇒ (1): This follows from Proposition 5.2.4.

In particular, the implication (1) ⇒ (3) implies that Filt M is left Schur. Therefore we obtaina map Filt : mbrick A → SchurL A. We will prove that these maps are mutually inverse to eachother. Since A is length, it can be easily shown by induction on lengths that E = Filt(sim E) holds
for any extension-closed subcategory E of A. Thus Filt ◦ sim : SchurL A → SchurL A is the identity.
Conversely, (Claim) implies sim(Filt M) = M for a monobrick M. Therefore, we have mutually
inverse bijections sim : SchurL A  mbrick A : Filt.
∼
Finally, we claim that this bijections sim : SchurL A −→ mbrick A : Filt restricts to bijections
wide A ↔ sbrick A. Ringel’s result [Rin1, 1.2] implies that Filt M is a wide subcategory of A if M

5.3. MAPS TO TORSION-FREE CLASSES AND COFINALLY CLOSED MONOBRICKS

101

is a semibrick. Conversely, let E be a wide subcategory of A. Then E is an abelian category, and
it is easy to see that simple objects in E coincides with usual simple objects in an abelian category
E. Thus the Schur’s lemma in E implies that sim E is a semibrick.

5.3. Maps to torsion-free classes and cofinally closed monobricks
In this section, we will show that a left Schur subcategory E of A is a torsion-free class if andonly if sim E is a cofinally closed monobrick. Then we construct a map from mbrick A to the set ofcofinally closed monobricks, taking the cofinal closure, which corresponds to the map F(E) which
sends E to the smallest torsion-free class containing E.
5.3.1. Cofinal extension and cofinal closure of monobricks. First we observe that eachmonobrick has a natural poset structure, which will play a central role in this paper.
Definition 5.3.1. Let M be a monobrick in A. For M, N in M, we define M ≤ N if there
is an injection M → N in A. Since A is length, it is easily checked that ≤ is actually a partial
order on M. We call this order the submodule order on M.
We introduce a notion of cofinal extension between monobricks, and cofinally closed monobricks.
Definition 5.3.2. Let M and N be two monobricks in A. Then we say that N is a cofinal
extension of M, or M is cofinal in N , if it satisfies the following conditions:(1) M ⊂ N holds.
(2) For every N ∈ N , there exists M ∈ M satisfying N ≤ M in N .
We say that a monobrick M is cofinally closed if there is no proper cofinal extension of M. We
denote by mbrickc.c. A the set of cofinally closed monobricks in A.
Note that this is a purely poset theoretical notion, and has nothing to do with the actual
module structure of each brick.
Cofinal extensions of M is closed under unions in the following sense:
Proposition 5.3.3. Let M be Sa monobrick in A. Suppose that {Ni | i ∈ I} is a familyof cofinal extensions of M. Then i∈I Ni is a cofinal extension of M (in particular, it is amonobrick).
S
Proof. Clearly, we only have to see that N := i∈I Ni is actually a monobrick. Take N1and N2 in N with N1 ∈ Ni1 and N2 ∈ Ni2 , and let f : N1 → N2 be any map. Since Ni2 is acofinal extension of N , there is an injection ι : N2 ,→ M with M ∈ M. Then the compositionιf : N1 → M is a map between elements in Ni1 , thus it should be either zero or an injection sinceNi1 is a monobrick. Then the injectivity of ι implies that f is either zero or an injection.
This immediately implies the existence of the largest cofinal extension of a given monobrick,which is cofinally closed.
Corollary 5.3.4. Let M be a monobrick in A. Then the union M of all cofinal extensionsof M satisfies the following properties:(1) M is a cofinal extension of M.
(2) For every cofinal extension N of M, we have M ⊂ N ⊂ M.
(3) M is cofinally closed. Moreover, if N is a cofinal extension of M which is cofinallyclosed, then N = M holds.
Proof. (1), (2) Clear from Proposition 5.3.6 and the definition of M.
(3) Let M0 be a cofinal extension of M. Then it is easy to see that M0 is also a cofinal
extension of M. Thus (2) implies M0 ⊂ M, thus M0 = M. Therefore, M is cofinally closed.
On the other hand, let N be a cofinal extension of M which is cofinally closed. Then we haveM ⊂ N ⊂ M holds by (2). It is obvous that M is a cofinal extension of N , thus we have N = M
by the definition of the cofinal closedness.
102

5. MONOBRICKS

Definition 5.3.5. Let M be a monobrick. We denote by M the union of all cofinal extensionsof M, and call it the cofinal closure of M. Then M is the unique cofinal extension of M whichis cofinally closed by Corollary 5.3.4.
Taking the cofinal closure defines a map (−) : mbrick A  mbrickc.c. A, which is the identity
on mbrickc.c. A by the definition of cofinal closedness. Similarly, we can check that a monobrickM is cofinally closed if and only if M = M holds.
Next, we will characterize the cofinal closure as in the theory of integral extensions of commutative rings: an integral closure of a ring is given by the set of all elements which is integral
over the base ring. To this purpose, we will introduce the following notion.
Definition 5.3.6. Let M be a monobrick in A. We say that a brick N in A is cofinal over
M if it satisfies the following conditions:(1) There exist M ∈ M and an injection N ,→ M in A.
(2) Every map N → M 0 with M 0 ∈ M is either zero or an injection.
Proposition 5.3.7. Let M be a monobrick in A and N a brick in A. Then N is cofinal over
M if and only if M ∪ {N } is a cofinal extension of M.
Proof. The “if” part is clear. Conversely, suppose that N is cofinal over M, and we claimthat M ∪ {N } is a cofinal extension of M. Obviously it suffices to show that M ∪ {N } is amonobrick.
Clearly we only have to show that every map f : M → N with M ∈ M is either zero or
an injection. By the assumption, there is an injection ι : N ,→ M 0 with M 0 ∈ M. Then thecomposition ιf : M → M 0 is a map between elements in M, thus is either zero or an injection.
Since ι is injective, this implies that f is either zero or an injection.

Now we can describe a cofinal extension of a monobrick in terms of elements:
Corollary 5.3.8. Let M be a monobrick in A and N a set of bricks in A satisfying M ⊂ N(we do not require that N is a monobrick). Then the following are equivalent:(1) N is a cofinal extension of M (in particular, N is a monobrick).
(2) Every element in N is cofinal over M.
Proof. (1) ⇒ (2): It follows immediately from definition.
(2) ⇒ (1): By Proposition 5.3.7,
we have that M ∪ {N } is a cofinal extension of M for eachS
N ∈ N . Since we have N = N ∈N (M ∪ {N }), Proposition 5.3.6 implies that N is a cofinal
extension of M.

Similarly, we have the following description of the cofinal closure.
Corollary 5.3.9. Let M be a monobrick in A. Then we have
M = {N ∈ brick A | N is cofinal over M}.
In particular, M is cofinally closed if and only if the following condition is satisfied:(CC) If a brick N has an injection N ,→ M of some M ∈ M and N 6∈ M, then there is some
non-zero non-injection N → M 0 to M 0 ∈ M.
Proof. Since M is a cofinal extension of M, every element in M is cofinal over M. Conversely, suppose that a brick N is cofinal over M. Then M ∪ {N } is a cofinal extension of M by
Proposition 5.3.7. Thus we have M ∪ {N } ⊂ M by Corollary 5.3.4, hence N ∈ M.
5.3.2. Torsion-free classes and cofinally closed monobricks. In this subsection, we willshow that the map F : SchurL A  torf A corresponds to the map (−) : mbrick A  mbrickc.c. A
defined in the previous subsection.
First of all, we can construct a torsion-free class from any collection of objects in A as follows.
Definition 5.3.10. Let C be a collection of objects in A.
• We denote by sub C the collection of all subobjects of objects in C (where subobjects
mean the usual subobjects in an abelian category A).

5.3. MAPS TO TORSION-FREE CLASSES AND COFINALLY CLOSED MONOBRICKS

103• We denote by F(C) := Filt(sub C).
Lemma 5.3.11. Let C be a collection of objects in A. Then F(C) is the smallest torsion-freeclass containing C.
Proof. Although this is well-known (e.g. [MS, Lemma 3.1]), we give a proof here for theconvenience. Clearly it suffices to show that F(C) is a torsion-free class in A. Since F(C) =Filt(sub C) is extension-closed in A, it is enough to show that F(C) is closed under subobjects.
Take any M ∈ F(C) and its subobject X ,→ M . We will show X ∈ F(C) by the induction ofa (sub C)-filtration length n of M . If n = 1, then M ∈ sub C holds, thus M is a subobject of someC ∈ C. Then it follows that X is also a subobject of C, which proves X ∈ sub C ⊂ F(C).
ιπ
Now suppose n > 1. Then there is a short exact sequence 0 → L →− M −→ N → 0 with
L, N ∈ F(C) such that L and N has (sub C)-filtrations of length smaller than n. We can obtainthe following exact commutative diagram,
0

L∩X

0

L

X
ι

M

π(X)

0

N

0

π

where all the vertical maps are injections. By the induction hypothesis, we have L∩X, π(X) ∈ F(C).
Thus X ∈ F(C) holds since F(C) is extension-closed.

The following basic observation is used later.
Lemma 5.3.12. Let C be a collection of objects in A. Then F(C) = F(Filt C) holds.
Proof. Since C ⊂ Filt C ⊂ F(Filt C), we have F(C) ⊂ F(Filt C) by the minimality of F(C). On
the other hand, F(C) = Filt(sub C) ⊃ Filt C holds, thus F(C) ⊃ F(Filt C).

Thus we have the following commutative diagram.
mbrick A∼
Filt
F

SchurL A
F

torf A
mbrick A∼
sim

SchurL A

The following claims that the dotted map is nothing but taking the cofinal closure.
Proposition 5.3.13. Let M be a monobrick in A. Then we have M = sim F(M).
Proof. First, we will prove M ⊂ sim F(M), which is equivalent to that every object in M
is left Schurian for F(M) by Corollary 5.2.6. Take any M ∈ M. Then M is left Schurian for M
since M is a monobrick. Since every object in sub M admits an injection into some object in M,
it is easily checked that M is left Schurian also for sub M. Then Lemma 5.2.12 implies that M isleft Schurian for Filt(sub M) = F(M).
Next, we will prove that M ⊂ sim F(M) is a cofinal extension. Let X be a simple object inF(M) = Filt(sub M). Then clearly we must have X ∈ sub M. It follows that there is an injection
X ,→ M with M ∈ M. This shows that sim F(M) is a cofinal extension of M.
Therefore, we have sim F(M) ⊂ M by Corollary 5.3.4. On the other hand, since M is amonobrick, every object in M is left Schurian for M, thus so is for M. Then the same argument
as the first part implies M ⊂ sim F(M). Hence sim F(M) = M holds.
As a corollary, we have the following description of simple objects in F(E) for a left Schurcategory E.
Corollary 5.3.14. Let E be a left Schur subcategory of A. Then sim F(E) consists of bricks
N in A which satisfy the following conditions:104

5. MONOBRICKS(1) There is an injection N ,→ M with M ∈ sim E.
(2) Every map N → M 0 with M 0 ∈ sim E is either zero or an injection.
Proof. This immediately follows from Corollary 5.3.9, since we have sim F(E) = sim E by
Proposition 5.3.13 and Theorem 5.2.11.

Now we can state our characterization of torsion-free classes via monobricks.
Theorem 5.3.15. Let A be a length abelian category. Then we have the following commutative
diagram, and all the horizontal maps are bijective.
torf A

sim
Filt

1

SchurL A

mbrickc.c. A

simmbrick A(−)

F

torf A

1

Filt
sim
Filt

mbrickc.c. A

Proof. Clearly it suffices to show that the following are equivalent for a monobrick M in A:(1) M is cofinally closed.
(2) Filt M is a torsion-free class.
(1) ⇒ (2): If M is cofinally closed, then M = M holds. Therefore, we have Filt M = Filt M =Filt(sim F(M)) = F(M) by Proposition 5.3.13, thus Filt M is a torsion-free class in A.
(2) ⇒ (1): Since Filt M is a torsion-free class, we have F(M) = F(Filt M) = Filt M holds
by Lemma 5.3.12. Thus we have M = sim F(M) = sim(Filt M) = M by Proposition 5.3.13.
Therefore M is cofinally closed.

We can obtain the following characterization of left Schur subcategories:
Corollary 5.3.16. Let E be a subcategory of A. Then E is left Schur if and only if there
exist a torsion-free class F of A and a subset M of sim F such that E = Filt M holds.
Proof. Since sim F (and its subset) is a monobrick for a torsion-free class F, the “if” part isclear. Conversely, let E be a left Schur subcategory of A. Then we have sim E ⊂ sim E = sim F(E)by Proposition 5.3.13. Thus F := F(E) satisfies the desired condition.

As a similar result, we can prove the following.
Corollary 5.3.17. Let M be a set of isomorphism classes of bricks in A. Then M is amonobrick if and only if there exist a cofinally closed monobrick N of A such that M is a subset
of N .
Proof. The “if” part is clear, and take N := M for the “only if” part.



5.4. Maps to semibricks and wide subcategories
We construct maps max : mbrick A → sbrick A and W : SchurL A → wide A, which are theidentities if restricted to sbrick A and wide A respectively. These maps correspond to each other
under Theorem 5.2.11.
First we consider the following operation on monobricks.
Proposition 5.4.1. Let M be a monobrick in A. Then the set max M of maximal elements
for the submodule order on M is a semibrick.
Proof. It suffices to show that every map f : M → N with M, N ∈ max M is zero if M 6= N .
Suppose that f is non-zero. Then f should be an injection in A since M is a monobrick. It follows
that M ≤ N , which contradicts to the maximality of M .
Thus we obtain the map max : mbrick A → sbrick A. Actually we have the following characterization of a semibrick in terms of the poset structure.

5.4. MAPS TO SEMIBRICKS AND WIDE SUBCATEGORIES

105

Proposition 5.4.2. Let S be a monobrick in A. Then the following are equivalent:(1) S is a semibrick.
(2) S is a discrete poset, that is, M ≤ N in S implies M = N .
(3) max S = S holds.
Proof. This is immediate from definitions and Proposition 5.4.1.



Next we introduce a map W : SchurL A → wide A. This extends the map W : torf A → wide A
defined by Marks-Št’ovı́ček [MS].
Definition 5.4.3. Let E be a left Schur subcategory of A. Then W(E) is a subcategory of E
consisting of objects W ∈ E satisfying the following condition: For every map f : W → X withX ∈ E, we have Coker f ∈ E, where Coker f denotes the cokernel of f in A.
The following is a key lemma to show that W(E) is actually a wide subcategory.
Lemma 5.4.4. Let E be a left Schur subcategory of A. Then the following holds.
(1) For M ∈ sim E, the following are equivalent:(a) M belongs to W(E).
(b) M is maximal in the submodule order on sim E.
(c) Every non-zero morphism f : M → X with X ∈ E is an injection in A and satisfies
Coker f ∈ E.
(2) If we have a short exact sequence
0

i

L

M

p

N

0

in A with L, M, N ∈ E, then M is in W(E) if and only if both L and N are in W(E).
Proof. (1) Let M be a simple object in E.
(a) ⇒ (b): Suppose that M is not maximal in sim E. Then we have a proper injection
ι : M ,→ M 0 with M 0 ∈ sim E. Consider the following exact sequence in A:
0

ι

M

M0

Coker ι0

Since ι is not an isomorphism, Coker ι is non-zero. Then Coker ι does not belong to E, sinceotherwise M 0 would not be simple in E. This implies that M does not belong to W(E).
(b) ⇒ (c) Let f : M → X be a non-zero map with X ∈ E. Then f is an injection in A since
E is left Schur. We will prove Coker f ∈ E by induction on a (sim E)-filtration length of X.
If X belongs to sim E, then the maximality of M clearly implies that f should be an isomorphism. Suppose that X has (sim E)-filtration of length n > 1. Then we have a short exactιπsequence 0 → L →− X −→ N → 0 with L ∈ sim E and N has (sim E)-filtration of length n − 1 (inparticular, L, N ∈ E). Consider the following diagram:
M
f

0

L

ι

X

π

N

0

We consider two cases.
(Case 1): πf = 0. In this case, there is a map f : M → L satisfying f = ιf . By the induction
hypothesis, f is either zero or an injection with Coker f ∈ E. In the former case, we have f = 0,
which is a contradiction. Thus the latter holds. Then we obtain the following exact commutative106

5. MONOBRICKS

diagram.
0

0

M

M
f

f
ι

π

0

L

X

0

Coker fCoker f

0

0

N

0

N

0

Since Coker f and N belong to E, so does Coker f since E is extension-closed.
(Case 2): πf 6= 0. In this case, πf is an injection with Coker(πf ) ∈ E by the induction
hypothesis. Then we obtain the following exact commutative diagram.
0

0

M

M

f

0

L

0

L

ι

πf
π

X

N

0

Coker fCoker(πf )

0

0

0

Since L and Coker(πf ) belong to E, so does Coker f since E is extension-closed.
(c) ⇒ (a): Clear from the definition of W(E).
(2) Suppose that L and N belong to W(E), and we will prove M ∈ W(E). Take any map
f : M → X with X ∈ E. Then we obtain the following exact commutative diagram, where f is a
map induced from the universality of the cokernel N of i.
0

L
L

i

fi

p

M

N

f

f

X

Coker(f i)00

Since L is in W(E) and X is in E, we have Coker(f i) ∈ E. Therefore, we have Coker f ∈ E since
N is in W(E). On the other hand, it can be shown that the right square is a pushout diagram.
Thus Coker f ∼= Coker f holds, which proves Coker f ∈ E. Therefore M ∈ W(E) holds.
Conversely, suppose that M belongs to W(E), and we will show that L and N belong to W(E).
First we will prove L ∈ W(E). Take any map f : L → X with X ∈ E. Then by taking the pushout,
we obtain the following exact commutative diagram.
0
0

L

i

M

f

f

X

E

p

N

0

N

0

Since the left square is pushout, we have Coker f ∼= Coker f . On the other hand, we have E ∈ E
since E is extension-closed and X, N ∈ E. Thus Coker f ∈ E holds by M ∈ W(E). Therefore
Coker f ∈ E, which proves L ∈ W(E).

5.5. APPLICATIONS

107

Next we will prove N ∈ W(E). Take any map f : N → X with X ∈ E. Then since p is asurjection, Coker f ∼= Coker(f p) holds. On the other hand, Coker(f p) ∈ E holds by M ∈ W(E)and X ∈ E. Therefore Coker f ∈ E, which proves N ∈ W(E).

Now we are ready to prove the main result in this section.
Theorem 5.4.5. Let A be a length abelian category. Then the following hold.
(1) W(W) = W holds for a wide subcategory W of A.
(2) W(E) is a wide subcategory of A for a left Schur subcategory E of A.
(3) The following diagram commutes, and all the horizontal maps are bijective.
wide Asim

sbrick A

Filt

1

SchurL A

sim

1

max

W

wide A

mbrick A

Filtsimsbrick A

Filt

Proof. (1) Clear from the definition of W(W) since W is closed under cokernels.
(2), (3) Let E be a left Schur subcategory of A and put M := sim E. By Theorem 5.2.11,
it clearly suffices to show that W(E) = Filt(max M), since max M is a semibrick by Proposition
5.4.1. By Lemma 5.4.4 (1), we have max M ⊂ W(E). Since E is extension-closed, it can be easily
checked that Lemma 5.4.4 (2) implies Filt(max M) ⊂ W(E).
Conversely, take any M in W(E). We will prove M ∈ Filt(max M) by the induction on a
M-filtration length n of M .
If n = 1, we have M ∈ M = sim E. Thus Lemma 5.4.4 (1) implies M ∈ max M, hence inparticular M ∈ Filt(max M). Suppose n > 1, then there is a short exact sequence in A
0

L

M

N

0.

such that L and N has M-filtrations of length smaller than n. In particular, we have L, N ∈ E.
Then Lemma 5.4.4 (2) implies that L, N ∈ W(E) by M ∈ W(E). By the induction hypothesis, wehave L, N ∈ Filt(max M), which shows M ∈ Filt(max M).
Remark 5.4.6. Let F be a torsion-free class in A. Then max(sim F) = sim W(F) holds by
Theorem 5.4.5. In [AP, Proposition 6.5], it is shown that this coincides with the set of brick labelsof arrows starting at F, which is introduced in [DIRRT]. Therefore, in our context, considering
the brick labels of F is nothing but taking the maximal element of the simple objects in F. In[Asa1], the bijection between functorially finite torsion-free classes and semibricks satisfying somecondition was established, and its bijection is given by taking brick labels of arrows starting at F,
thus coincides with max(sim F).
5.5. Applications
In this section, we give an application of the theory of monobricks to torsion-free classes andwide subcategories. We give new proofs of several results on these subcategories, such as DemonetIyama-Jasso’s finiteness results [DIJ] and Marks-Št’ovı́ček’s bijection [MS], and make the relation
between torsion-free classes and wide subcategories more transparent, which can be applied to any
length abelian categories, without using any τ -tilting theory.
5.5.1. Maps between torsion-free classes and wide subcategories via monobricks.
In this section, we consider the restrictions of our maps W and F to W : torf A → wide A andF : wide A → torf A. By using monobricks, we can reprove and generalize a Marks-Št’ovı́ček’s
result by using only the easy poset theoretical argument.
Proposition 5.5.1. Let A be a length abelian category. Then the following hold.
(1) Let M be a monobrick in A and N a cofinal extension of M. Then max M = max N
holds. In particular, we have max M = max M holds.

108

5. MONOBRICKS(2) Let S be a semibrick. Then we have max S = S. Thus the composition max ◦(−) : sbrick A →sbrick A is the identity.
(3) [MS, Proposition 3.3] The composition W ◦ F : wide A → wide A is the identity.
Proof. (1) Let M be a maximal element of M. If M is not maximal in N , then there is someN ∈ N with M < N . However, since N is cofinal in M, there is some M 0 ∈ M with N ≤ M 0 ,which implies M < M 0 . This is a contradiction, thus we have max M ⊂ max N . Conversely, let N
be a maximal element of N . Then since N is cofinal in M, there is some M ∈ M with N ≤ M .
Then the maximality implies N = M ∈ M, thus N ∈ max M holds.
(2) Obvious from (1).
(3) This follows from (2) and Theorems 5.3.15 and 5.4.5.

In general, the map F : wide A → torf A is not a surjection, and its image is studied in [AP].
We give a description of its image in terms of monobricks.
Proposition 5.5.2. Let F be a torsion-free class in A. Then the following are equivalent.
(1) F = F(W(F)) holds.
(2) There is a wide subcategory W satisfying F = F(W).
(3) There is a semibrick S satisfying sim F = S.
(4) sim F is a cofinal extension of some semibrick.
(5) max(sim F) is cofinal in sim F, that is, for every element M in sim F, there is an element
S ∈ sim F such that M ≤ S holds and S is maximal in sim F.
Proof. (1) ⇒ (2): Obvious.
(2) ⇒ (3): Clear from Theorem 5.4.5 and Theorem 5.3.15.
(3) ⇒ (4): This is clear since sim F = S is a cofinal extension of S.
(4) ⇒ (5): Let S be a semibrick such that sim F is a cofinal extension of S. Then we have
S = max S = max(sim F) holds by Propositions 5.4.2 and 5.5.1. Thus max(sim F) is cofinal in
sim F.
(5) ⇒ (1): (5) implies that sim F is a cofinal extension of max(sim F), and sim F is cofinallyclosed by Theorem 5.3.15. Thus we have max(sim F) = sim F holds by Corollary 5.3.4. This isnothing but (1) under the bijections in Theorems 5.3.15 and Theorem 5.4.5.

Example 5.5.3. Let us consider the 2-Kronecker case, see Example 5.7.5 for the detail andnotations. By checking which cofinally closed monobrick M satisfies that max M is cofinal in M,
any monobricks except case (M2) satisfy this. Thus any torsion-free classes except (M2) (torsionfree classes consisting of all preprojective modules) belong to the image of F : wide A → torf A.
As a corollary, we can quickly prove a Marks-Št’ovı́ček’s bijection (c.f. [MS, Corollary 3.11]).
Corollary 5.5.4. Let A be a length abelian category. Then the maps F : wide A  torf A : W
induces a bijection between wide A and F(wide A). If A has only finitely many torsion-free classes,
then F(wide A) = torf A holds, thus F and W are mutually inverse bijections between wide A andtorf A.
F

W

Proof. Since the composition wide A −→ torf A −→ wide A is the identity by Proposition
5.5.1, it suffices to show the last assertion. Suppose that A has finitely many torsion-free classes,
and it suffices to prove that max M is cofinal in M for every (cofinally closed) monobricks by
Proposition 5.5.2. We will see in Theorem 5.5.5 that there are only finitely many bricks in A up
to isomorphism. Therefore, every monobrick M is a finite poset, thus clearly max M is cofinal inM.

5.5.2. Finiteness conditions. In this subsection, we study several finiteness conditions onmonobricks. First we consider when torf A or wide A or mbrick A is finite. We denote by brick A
the set of isomorphism classes of bricks in A.
Theorem 5.5.5. Let A be a length abelian category. Then the following are equivalent:(1) brick A is finite, that is, there are only finitely many bricks in A up to isomorphism.

5.5. APPLICATIONS

109(2) mbrick A is finite.
(2)0 SchurL A is finite.
(3) mbrickc.c. A is finite.
(3)0 torf A is finite.
(4) sbrick A is finite.
(4)0 wide A is finite.
(5) There are only finitely many subcategories of A which are closed under extensions, kernels
and images.
Proof. First, note that (i) and (i)0 are equivalent for i = 2, 3, 4 by Theorems 5.2.11 and5.3.15.
(1) ⇒ (2): This is clear since mbrick A is a subset of 2brick A , the power set of brick A.
(2) ⇒ (3): This is clear by mbrickc.c. A ⊂ mbrick A.
(3) ⇒ (4): This is clear by an injection (−) : sbrick A ,→ mbrickc.c. A shown in Proposition
5.5.1.
(4) ⇒ (1): A map brick A → sbrick A defined by S 7→ {S} is clearly injective.
(2)0 ⇒ (5): This is clear since every subcategory of A closed under extensions, kernels andimages is left Schur by Proposition 5.2.8.
(5) ⇒ (3)0 : This is clear since every torsion-free class in A is closed under extensions, kernels
and images.
Definition 5.5.6. We call a length abelian category A brick-finite if it satisfies the equivalentconditions of Theorem 5.5.5.
Remark 5.5.7. In the case A = mod Λ for a finite-dimensional algebra Λ, the equivalence of(1) and (3)0 is a particular case of [DIJ, Theorems 3.8, 4.2], and such an algebra is called τ -tilting
finite. Actually it was shown in [DIJ] that mod Λ is brick-finite if and only if there are only finitely
many functorially finite torsion-free classes, a little stronger result than ours.
Next we consider when each monobrick consists of finitely many bricks. We begin with thefollowing general observation on posets. A subset X of a poset P is called a chain if X is totallyordered, and an antichain if every two distinct elements in X are incomparable. For an element
m of a poset P , we put ↓ m := {x ∈ P | x ≤ m}.
Lemma 5.5.8. Let P be a poset such that every chain in ↓ m is finite for every m in P . Then
P is finite if and only if it satisfies the following two conditions.
(1) max P is cofinal in P , that is, every element is below some maximal element.
(2) Every antichain of P is a finite set.
Proof. If P is finite, then it clearly satisfies (1) and (2).
Conversely, suppose that P isSan infinite set. Since max P is an antichain of P , it is a finite
set by (2). By (1), we have P = m∈max P (↓ m). Since P is infinite and max P is finite, we may
assume that ↓ m1 is an infinite set for m1 ∈ max P .
Put P1 := (↓ m1 ) \ {m1 } = {x ∈ P | x < m1 }. Clearly P1 also satisfies (2), since every
antichain of P1 is also an antichain of P . Suppose that there is an element x ∈ P1 which is notbelow any maximal elements in P1 . Since x is not maximal in P1 , there is some x < x1 withx1 ∈ P1 , and x1 is not below any maximal elements in P1 . By iterating this, we obtain an infinite
chain inside P1 ⊂ (↓ m1 ), which is a contradiction. Thus P1 satisfies (1).
Now we can repeat the same process as P to find an element m2 ∈ P1 such that ↓ m2 is
infinite. Then apply the same process to P2 := {x ∈ P | x < m2 }. We can iterate this procedure,and obtain an infinite chain m1 > m2 > m3 > · · · in ↓ m1 . This is a contradiction.

By using this, we can prove the following criterion of the finiteness of a monobrick.
Proposition 5.5.9. Let M be a monobrick in A. Then M is a finite set if and only if itsatisfies the following two conditions:(1) Every element in M is below some maximal element in M.
(2) Every semibrick S with S ⊂ M is a finite set.

110

5. MONOBRICKS

Proof. First we will check that a poset M satisfies the assumption and the conditions inLemma 5.5.8. Take any M ∈ M and consider ↓ M . Clearly 0 ≤ l(X) < l(M ) holds for every
element X in ↓ M , where l(−) denotes the lengths of objects in A. If X < X 0 in M, thenl(X) < l(X 0 ) holds. Thus clearly ↓ M cannot contain any infinite chains.
The condition (1) and (2) in Lemma 5.5.8 is nothing but (1) and (2) in this proposition
respectively. Moreover, we can check that a subset S of M is an antichain if and only if S is asemibrick. Thus the assertion holds.
As an application, we have the following criterion of the finiteness of the number of simple
objects in a given torsion-free class.
Corollary 5.5.10. Let F be a torsion-free class in A. Then sim F is a finite set if and only
if F satisfies the following conditions.
(1) F = F(W) holds for some wide subcategory W of A.
(2) Every semibrick S satisfying S ⊂ sim F is finite.
The following is the fundamental relation between brick-finiteness and the finiteness of eachmonobrick.
Theorem 5.5.11. Let A be a length abelian category. Then the following are equivalent.
(1) Every monobrick in A is a finite set.
(2) Every cofinally closed monobrick in A is a finite set.
(2)0 # sim F is finite for every torsion-free class F in A.
(3) Every semibrick in A is a finite set, and the map F : wide A ,→ torf A is surjective.
Moreover, if A = mod Λ for a finite-dimensional algebra Λ, then the following are also equivalent.
(4) mod Λ is brick-finite, that is, there are only finitely many bricks in mod Λ.
Proof. (1) ⇒ (2): Obvious.
(2) ⇒ (1): Let M be a monobrick in A. Then M ⊂ M holds for the cofinal closure of M.
Since M is cofinally closed, it is finite by (2), thus so is M.
(2) ⇔ (2)0 : Clear from Theorem 5.3.15.
(1) + (2)0 ⇒ (3): The surjectivity of the map F : wide A ,→ torf A follows from Corollary
5.5.10. Since every semibrick is a monobrick, it is finite by (1).
(3) ⇒ (2)0 : Clear from Corollary 5.5.10.
Now we have shown the equivalence of (1), (2), (2)0 and (3). From now on, suppose that Λ isa finite-dimensional algebra and A = mod Λ.
(2)0 ⇒ (4): Suppose that mod Λ is not brick-finite. Then by [DIJ, Theorem 3.8], there isa torsion-free class F in mod Λ which is not functorially finite. Put F0 := 0 ∈ torf(mod Λ).
Then [DIJ, Theorem 3.1] implies that there is a functorially finite torsion-free class F1 satisfying
F0 ( F1 ⊂ F. Since F is not functorially finite, we have F1 ( F. By repeating this process,weSobtain a strictly ascending chain 0 = F0 ( F1 ( F2 ( · · · of torsion-free classes. Put G := i≥0 Fi .
Then it is clearly a torsion-free class, and sim G is finite by (2)0 . Therefore, there is some i suchthat sim G ⊂ Fi holds. Since Fi is extension-closed, this would imply G = Filt(sim G) ⊂ Fi ⊂ G,
thus Fi = Fi+1 = · · · = G, which is a contradiction.
(4) ⇒ (1): Clear.

We propose the following conjecture related to this, which is of interest in its own.
Conjecture 5.5.12. Let Λ be a finite-dimensional algebra. If every semibrick in mod Λ is a
finite set, then mod Λ is brick-finite, that is, Λ is τ -tilting finite.
Roughly speaking, Proposition 5.5.9 and Theorem 5.5.11 says that in order to show brickfiniteness, we have to show the finiteness of antichains (semibricks) and chains of bricks. Thus
this conjecture is roughly equivalent to the following question: if every monobrick has finite width(finite antichains), then does every monobrick have a finite height?
Regarding this, it was recently shown in [ST, Theorem 1.1] that the finiteness of height implies
brick-finiteness. More precisely, it was shown that if there is an upper bound on the lengths ofbricks, then mod Λ is brick-finite.

5.6. MONOBRICKS OVER NAKAYAMA ALGEBRAS

111

5.6. Monobricks over Nakayama algebras
For a finite-dimensional algebra Λ, we put mbrick Λ := mbrick(mod Λ) and so on. In thissection, we investigate monobricks and left Schur subcategories of mod Λ for a Nakayama algebra
Λ. For the details on Nakayama algebras, we refer the reader to standard texts such as [ASS, V.3].
First of all, we show that left Schur subcategories are precisely categories closed under extensions, kernels and images.
Theorem 5.6.1. Let Λ be a Nakayama algebra and E a subcategory of mod Λ. Then E is left
Schur if and only if E is closed under extensions, kernels and images. In particular, we have thebijection between the following two sets:(1) mbrick Λ, the set of monobricks in mod Λ.
(2) The set of subcategories of mod Λ closed under extensions, kernels and images.
The maps are given by Filt and sim.
Proof. We use the result in [Eno4, Corollary 5.19]: every torsion-free class in mod Λ satisfies
the Jordan-Hölder type property. We refer the reader to [Eno4] for the details on this property.
By Proposition 5.2.8, we only have to show that every left Schur subcategory E of mod Λ isclosed under kernels and images. By Theorem 5.2.11, there is a monobrick M satisfying E =Filt M. Consider the cofinal closure M of M and put F := F(E). Then we have M = sim F ⊃ M
by Proposition 5.3.13. Let f : X → Y be a map in E. Since F is closed under kernels and images
in mod Λ, we obtain the following short exact sequence in F:
0

Ker f

X

Im f

0.

Since F satisfies the Jordan-Hölder property, we can speak of composition factors inside F. Since
X is in E = Filt M, all the composition factors of X inside F belongs to M by M ⊂ sim F.
Therefore, all the composition factors of Im f and Ker f must be in M, since the above short
exact sequence is a conflation in F. This implies Im f and Ker f belong to Filt M = E.
Our next aim is to give a combinatorial classification of monobricks for Nakayama algebras.
The following basic observation on quotient algebras and monobricks is useful. Recall that fora two-sided ideal I of a finite-dimensional algebra Λ, we have the natural fully faithful functor
mod(Λ/I) ,→ mod Λ, and its essential image consists of Λ-modules M satisfying M I = 0. By this,we may identify mod(Λ/I) with the subcategory of mod Λ consisting of such modules.
Proposition 5.6.2. Let Λ be a finite-dimensional algebra and I its two-sided ideal. Then by
identifying mod(Λ/I) with the subcategory of mod Λ, we havembrick(Λ/I) = mod(Λ/I) ∩ mbrick Λ
Proof. This follows from the fact that the natural functor mod(Λ/I) ,→ mod Λ is fully faithfuland that a morphism in mod(Λ/I) is an injection in mod(Λ/I) if and only if so is in mod Λ.

By this, the classification of monobricks over Λ/I is reduced to that of Λ in principle. Keeping
this in mind, it suffices to consider the following two Nakayama algebras.
Definition 5.6.3. Let n be a positive integer. Then we define two algebras An and Bn as
follows:(1) An is the path algebra of the following quiver.
1

2

···

n

112

5. MONOBRICKS(2) Bn is the quotient of the path algebra of the following by the ideal generated by all thepaths of lengths n.
1
n

2

n−1

3
···

Note that we have the natural identification Bn /hen i ∼= An−1 , where en is the primitive
idempotent of Bn corresponding to n.
The following shows that to classify monobricks over Nakayama algebras, it suffices to consider
An and Bn .
Proposition 5.6.4. Let Λ be a basic connected Nakayama algebra with # sim(mod Λ) = n.
(1) If the quiver of Λ is acyclic, then Λ ∼= An /I for some I, thus mbrick Λ ⊂ mbrick An holds.
(2) If the quiver of Λ is cyclic, then there exist a Nakayama algebra B 0 and two algebra
surjections B 0  Λ and B 0  Bn such that mbrick Λ ⊂ mbrick Bn holds inside mbrick B 0 .
Proof. (1) This is well-known, e.g. [ASS, Theorem V.3.2](2) The existence of a Nakayama algebra B 0 such that Λ and Bn are quotients of B 0 is
obvious (consider the path algebra of the cyclic quiver and annihilate sufficiently large paths),
thus it suffices to see that every brick M in mod B 0 is contained in mod Bn . This is clear since if
indecomposable module M does not belong to Bn , then it is easily checked that M has non-zero
endomorphism which is not an isomorphism.
To deal with modules over An and Bn , we will use the following combinatorial description.
Definition 5.6.5. Let n be a positive integer.
• We put [n] := {1, 2, . . . , n}.
• For two elements i, j in [n], we calculate i + j ∈ [n] and i − j ∈ [n] modulo n, for example,
n + 1 = 1 and 1 − 1 = n.
• An arc on [n] is an element of [n] × [n].
• An admissible arc on [n] is an arc (i, j) satisfying i < j.
• For an arc α = (i, j) on [n], we call i its starting point and j its ending point.
• The socle series of an arc α = (i, j) on [n] is a sequence of elements in [n] defined by(i, i + 1, . . . , j − 1).
• An arc diagram D on [n] is a set of arcs, that is, a subset of [n] × [n].
• An arc diagram D is admissible if every arc in D is admissible.
We represent arcs and arc diagrams on [n] as follows: Consider the Euclidean plane R2 andput i on (i, 0) + Z(n, 0) for each i ∈ [n]. Then for an arc α = (i, j), we draw “arcs” in the upper
half-plane which connect each i and the first j which appear right to this i.
For example, the following is an arc diagram D = {(1, 1), (2, 3), (3, 2)} on [3]. These three arcs
have socle series (1, 2, 3), (2), (3, 1) respectively.

···

•
3

3

•
1

1

•
2

2

•
3

3

•
1

1

•
2

2

•
3

3

•
1

···

As this figure, it is convenient to draw a dashed line on the x-axis and label each line segment asabove, so that the socle series of α is the sequence of labels surrounded by α. Also, we often draw
an admissible arc diagram by omitting the repeated part, for example, the following is a picture

5.6. MONOBRICKS OVER NAKAYAMA ALGEBRAS

113

of an admissible arc diagram {(1, 2), (1, 4), (3, 4)} on [4].

• 1 • 2 • 3 •1
2
3
4
We say that a sequence (n1 , . . . , na ) is a partial sequence of a sequence (m1 , . . . , mb ) if there
is some integer i with 1 ≤ i ≤ b − a + 1 satisfying n1 = mi , n2 = mi+1 , . . . , na = mi+a−1 holds.
For example, (3, 1), (2) and (2, 3, 1) are subsequences of (2, 3, 1), but (1, 2), (2, 1) and (3, 1, 2) arenot.
Definition 5.6.6. We say that a pair {α, β} of two different arcs α and β on [n] is a weakly
non-crossing pair if either of the following conditions is satisfied:• The socle series of α is a partial sequence of that of β.
• The socle series of β is a partial sequence of that of α.
• The socle series of α and β are disjoint, that is, there exists no element in [n] whichappears in the both socle series.
Moreover, for a weakly non-crossing pair {α, β}, we define the following.
(1) It is a mono-crossing pair if α and β have the same starting point.
(2) It is a epi-crossing pair if α and β have the same ending point.
(3) It is a non-crossing pair if it is neither mono-crossing nor epi-crossing.
We say that {α, β} is a strictly crossing pair if they are not weakly non-crossing.
Intuitively, a pair of two arcs is weakly non-crossing if they do not cross in the half-planemodel except at their starting points or ending points, and is non-crossing if in addition they do
not share neither the stating points nor the ending points.
Example 5.6.7. Consider an arc diagram {(1, 1), (2, 3), (3, 1), (3, 2)} on [3]:• 3 • 1 • 2 • 3 • 1 •
3
1
2
3
1
2
Then the crossing relations between four arcs are as follows:···NC(1, 1)
EC

NC(3, 1)

•
3

3

•
1

···
···

(2, 3)
SC

MC

2

NC

(3, 2)

Here NC, EC, MC and SC means non-crossing, epi-crossing, mono-crossing and strictly crossing
respectively.
Remark 5.6.8. Suppose that α = (a, b) and β = (c, d) are distinct admissible arcs. Then it isstraightforward to see that {α, β} is strictly crossing if and only if a < c < b < d or c < a < d < b.
Definition 5.6.9. Let n be a positive integer and D an arc diagram on [n].
(1) D is non-crossing if every distinct pair of arcs in D is a non-crossing pair.
(2) D is mono-crossing if every distinct pair of arcs in D is either a mono-crossing or a
non-crossing pair.
Now let us return to the algebraic side.
Definition 5.6.10. Let n be a positive integer and α = (i, j) be an arc on [n]. Then wedenote by Mα the unique indecomposable Bn -module satisfying soc Mα = i and top Mα = j − 1.
If α is admissible, that is, i < j, then we may regard Mα as an An−1 -module by the surjection
Bn  Bn /hen i ∼= An−1 .

114

5. MONOBRICKS

Now by the standard description of indecomposable modules and morphisms between themover Nakayama algebras (e.g. [ASS, Theorem V.3.5]), it is easy to show the following.
Proposition 5.6.11. Let n be a positive integer. Then the assignment α 7→ Mα induces abijection between the set of arcs on [n] and brick Bn , and a bijection between the set of admissible
arcs on [n] and brick An−1 . Moreover, the following hold for two arcs α and β on [n].
(1) {α, β} is a non-crossing pair if and only if {Mα , Mβ } is a semibrick.
(2) {α, β} is a mono-crossing pair if and only if {Mα , Mβ } is a monobrick and not a semibrick.
Therefore, we have the following bijections, where MD := {Mα | α ∈ D} for an arc diagram D:{mono-crossing arc diagrams on [n]}{non-crossing arc diagrams on [n]}

M(−)
∼

∼

mbrick Bn

sbrick Bn

and{mono-crossing admissible arc diagrams on [n]}{non-crossing admissible arc diagrams on [n]}

M(−)
∼

∼

mbrick An−1

sbrick An−1 .

By combining this with Theorem 5.6.1, the problem of classifying all the subcategories closed
under extensions, kernels and images is reduced to purely combinatorial problem, namely, classifying all the mono-crossing (admissible) arc diagrams on [n].
In the rest of this section, we give an explicit formula on # mbrick An and # mbrick Bn . Notethat the formula on # sbrick An and # sbrick Bn is given by Asai [Asa1, Lemmas 3.4, 3.7]: 
12n# sbrick An−1 =(the n-th Catalan number, [OEIS, A000108])
n+1 n
 
2n
# sbrick Bn =([OEIS, A000984])nHere ni denotes the binomial coefficient. We can also compute # sbrick An−1 using Proposition
5.6.11. Non-crossing admissible arc diagrams on [n] clearly correspond to the classical non-crossing
partition on [n] (see [Rin2, N.4.1] for the detail). Therefore, its number is equal to the number of
non-crossing partitions, which is well-known to be equal to the Catalan number.
Our enumeration of monobricks is based on the same idea: find a bijection between theset of mono-crossing arc diagrams and some combinatorial sets, whose number has already been
computed by combinatorialists.
The following is our enumerative result.
Theorem 5.6.12. Let n be a positive integer. Then the following equalities hold.
 

n
X
1
n n+i
, the n-the large Schröder number.
# mbrick An = [OEIS, A006318](n) =
i+1 i
i
i=0
(5.6.1)n Xn−1 n+i# mbrick Bn = [OEIS, A002003](n) = 2.
(5.6.2)ii
i=0Proof. By Proposition 5.6.11, it suffices to count the numbers of(1) mono-crossing admissible arc diagrams on [n + 1], and(2) mono-crossing arc diagrams on [n].
(1) We will show that the number of mono-crossing admissible arc diagrams on [n] is equal tothe (n − 1)-the large Schröder number. The outline of the enumeration is as follows: we will show

5.6. MONOBRICKS OVER NAKAYAMA ALGEBRAS

115

that mono-crossing admissible arc diagrams are in bijection with non-crossing linked partitions
introduced in [Dyk], whose number was known to be the large Schröder number.
A non-crossing linked partition of [n] is a set π of non-empty subsets of [n] satisfying thefollowing conditions.
S(NCL1) [n] = E∈π E holds.
(NCL2) For every E, F ∈ π with E 6= F , there exists no a < b < c < d satisfying a, c ∈ E andb, d ∈ F .
(NCL3) We have #(E ∩ F ) ≤ 1 for every distinct E, F ∈ π, and if j ∈ E ∩ F , then either
j = min E, #E > 1 and j 6= min F hold, or the converse j = min F , #F > 1 andj 6= min E hold. In particular, if E ∩ F 6= ∅, then #E, #F > 1 holds.
Then the number of non-crossing linked partitions of [n] is equal to the (n − 1)-the large Schrödernumber by [Dyk]. We will prove the equality (5.6.1) by constructing a bijection from the set of
non-crossing linked partitions of [n] to the set of mono-crossing admissible arc diagrams. Ourconstruction is essentially the same as the graphical presentation given in [CWY].
Let π be a non-crossing linked partition of [n]. For each E ∈ π and j ∈ E with j 6= min E, wedraw an arc (min E, j). By this, we obtain an admissible arc diagram Dπ .
We claim that Dπ is actually a mono-crossing arc diagram. Let {α, β} be a pair of arcs in Dπwith α 6= β. By (NCL2) and Remark 5.6.8, this pair is weakly non-crossing. Thus it suffices toshow {α, β} is not epi-crossing. Assume that {α, β} is epi-crossing, then α and β have the sameending point, that is, we can write as α = (iα , j) and β = (iβ , j) with iα 6= iβ . By the constructionof Dπ , there is E and F in π satisfying {iα , j} ⊂ E, {iβ , j} ⊂ F , iα = min E and iβ = min F .
Then E 6= F holds by iα 6= iβ . Now we have j ∈ E ∩ F but j 6= min E, min F , which contradicts
to (NCL3). Therefore, {α, β} is not epi-crossing, thus Dπ is a mono-crossing arc diagram.
Conversely, let D be a mono-crossing admissible arc diagram on [n]. For each i in [n], define
Ei ⊂ [n] as follows:
{i} ∪ {j | (i, j) ∈ D} if there is some arc starting at i,
Ei := {i}if there is no arc either starting or ending at i,∅otherwise
Note that i = min Ei holds if Ei 6= ∅, thus Ei ’s are pairwise distinct.
Put πD := {Ei | 1 ≤ i ≤ n, Ei 6= ∅}. We claim that πD is a non-crossing linked partition of[n]. Clearly πD satisfies (NCL1). Assume that πD does not satisfy (NCL2), that is, there is some
Ei1 , Ei2 ∈ πD and a, c ∈ Ei1 , b, d ∈ Ei2 satisfying a < b < c < d. Then we have i1 < b < c < d by
i1 = min Ei1 . We consider two cases i1 < i2 and i2 < i1 .
If i1 < i2 , then i1 < i2 ≤ b < c < d holds by i2 = min Ei2 . Now c ∈ Ei1 and d ∈ Ei2 imply(i1 , c) ∈ D and (i2 , d) ∈ D. From this, (i1 , c) and (i2 , d) is strictly crossing by i1 < i2 < c < d,which is a contradiction. If i2 < i1 , then i2 < i1 < b < c holds. Now c ∈ Ei1 and b ∈ Ei2 imply(i1 , c) ∈ D and (i2 , b) ∈ D. Since these two arcs are strictly crossing, this is a contradiction. Thus
(NCL2) holds.
Next we will show that πD satisfies (NCL3). Suppose that Ei1 ∩ Ei2 6= ∅ for i1 6= i2 andtake j ∈ Ei1 ∩ Ei2 . If #Ei1 = 1, then j = i1 and there is no arc either starting or ending at i1 .
However, i1 ∈ Ei2 and i1 6= i2 implies that (i2 , i1 ) ∈ D, which is a contradiction. Thus #Ei1 > 1
and #Ei2 > 1 hold. Now if j 6= i1 and j 6= i2 , then (i1 , j), (i2 , j) ∈ D holds. This is a contradiction
since these two arcs are epi-crossing. Thus either j = i1 or j = i2 holds. In the former case, wehave j = i1 = min Ei1 and j = i1 6= i2 = min Ei2 , and in the latter we have j = min Ei2 andj 6= min Ei1 . Therefore (NCL3) is satisfied.
Now we have shown that πD is a non-crossing linked partition of [n] if D is a mono-crossing
admissible arc diagram. It is quite straightforward to see that D = DπD holds for a mono-crossing
admissible arc diagram, so we omit the proof.
Finally we show that π = πDπ holds for a non-crossing linked partition π of [n]. Let E ∈ π,
and we will show E = Ei for i := min E. We consider two cases:(Case 1): E = {i}. Suppose that there is some arc (i, j) in Dπ . By the construction of Dπ ,there is some F ∈ π with i = min F and j ∈ F . This contradicts to (NCL3) since i ∈ E ∩ F

116

5. MONOBRICKS

and #E = 1. It follows that there is no arc starting at i. Similarly, suppose that there is somearc (j, i) in Dπ . Then there is some F ∈ π with j = min F and i ∈ F . This contradicts to(NCL3) by i ∈ E ∩ F and #E = 1. Therefore, there is no arc either starting or ending at i, hence
Ei = {i} = E holds.
(Case 2): #E > 1. In this case, there is some arc starting at i in Dπ . By construction, E ⊂ Ei
holds. Conversely, take j ∈ Ei . Then (i, j) ∈ Dπ holds, thus there is some F ∈ π with i = min F
and j ∈ F . Since i ∈ E ∩ F and i = min E = min F , we must have E = F by (NCL3). Thus
j ∈ F = E holds, hence E = Ei .
We have shown π ⊂ πDπ . Conversely, take Ei ∈ πDπ . We consider two cases.
(Case 1): Ei = {i}. In this case, by construction, there is no arc either starting or ending at
i in Dπ . This means that there is no F ∈ π with #F > 1 which contains i. Thus {i} ∈ π should
hold by (NCL1), that is, Ei ∈ π.
(Case 2): #Ei > 1. By construction, there is some arc (i, j) in Dπ , thus there is some E ∈ πsatisfying i = min E and j ∈ E. It suffices to show E = Ei . If j 0 ∈ E with j 0 6= i, then (i, j 0 ) ∈ Dπ
holds by construction. Thus j 0 ∈ Ei holds, and we obtain E ⊂ Ei . Conversely, suppose j 0 ∈ Ei
with j 0 6= i. Then (i, j 0 ) ∈ Dπ , so there is some E 0 ∈ π with i = min E 0 and j 0 ∈ E 0 . Theni ∈ E ∩ E 0 satisfies i = min E = min E 0 , which implies E = E 0 by (NCL3). Thus j 0 ∈ E 0 = E
holds. Therefore, we have Ei = E ∈ π.
Hence we obtain π = πDπ , which completes the proof.
(2) We will show the equality (5.6.2) by calculating the generating function using (5.6.1). Put
an := # mbrick An , bn := # mbrick Bn and consider the following generating functions.
f (t) :=
g(t) :=

∞
X
n=1
∞
X

an tm ,
bn tn .

n=1

By using (5.6.1), it is known that the following hold (see e.g. [Bru, Theorem 8.5.7]):

f (t) =1−t−√

1 − 6t + t2
2

To compute g(t), we claim the following
Pnrelation between an and bn .
(Claim): The equality bn = an + i=1 i ai an+1−i holds.
Proof of (Claim): Let MDn be the set of mono-crossing arc diagram on [n], so #MDn = bn .
For 0 ≤ i ≤ n, we define MDn (i) as follows: MDn (0) consists of D ∈ MDn such that there is no
arc in D whose socle series contains n, and for 1 ≤ i ≤ n, MDn (i) consists of D ∈ MDn there is an
arc in D whose socle series contains n, and the minimum length of such arcs is i. Here the lengthof the arc (i, j) is defined to be j − i ∈ [n]. Then clearly we have the following decomposition, andwe will count the number of MDn (i).

MDn =

n
G

MDn (i),i=0

For i = 0, clearly elements in MDn (0) are precisely mono-crossing admissible arc diagrams on[n]. Thus #MDn (0) = an holds.
Let 1 ≤ i ≤ n. There are i arcs whose socle series contain n and whose lengths are i, thatis, (n − i + 1, 1), (n − i + 2, 2), . . . , (n, n + i). Elements in MDn (i) contains precisely one such arc,
since any two such arcs are strictly crossing. Fix one such arc α = (j, j + i), and we will countthe number of elements in MDn (i) which contains α. Let D be such an element. Then consider
the restriction of D to the part {j, j + 1, . . . , j + i}, more precisely, consider the set of arcs whose

5.7. EXAMPLES OF COMPUTATIONS

117

socle series are partial sequences of that of α.
α

···

•
j

j

j+1

•
•
j+1 j+2

···

•
j+i

···

By shifting −i, these arcs except α gives mono-crossing admissible arc diagram on [i] (note that
it is not [i − 1], since arcs cannot share the endpoint with α). Conversely, any mono-crossing
admissible arc diagram on [i] can be occur in this way by shifting +i.
In a similar way, consider the set of arcs whose socle series are disjoint from that of α. By
shifting −(j + i), these arcs gives mono-crossing admissible arc diagram on [n − i + 1], and viceversa. Therefore, there are i · ai · an−i+1 possible arc diagrams in MDn (i).

Now, by using (Claim), we obtain the following equality.
g(t) =

∞
X

bn tn

n=1

= (a1 + 1a1 a1 )t + (a2 + 1a1 a2 + 2a2 a1 )t2 + (a3 + 1a1 a3 + 2a2 a2 + 3a3 a1 )t3 + · · ·= (a1 t + a2 t2 + a3 t3 + · · · ) · (1 + a1 + 2a2 t + 3a3 t2 + · · · )df (t))= f (t) · (1 +√ dt
1 − t − 1 − 6t + t2 1
3−t
√
=
·
+1
2
2
1 − 6t + t2


1
1+t
√
=
−1
2
1 − 6t + t2
Since this coincides with the generating function of [OEIS, A002003], we have done.

Remark 5.6.13. In the paper [Eno7], we will compute the number of monobricks in mod An
by using the completely different method. In fact, the number of subcategories in mod kQ closed
under extensions, kernels and images (thus left Schur) is determined in [Eno7] for a Dynkin quiver
Q. If we choose Q to be an An quiver with the linear orientation, An is nothing but kQ which is
Nakayama, thus the number of monobricks in mod An is equal to the number of such subcategories
by Theorem 5.6.1. Moreover, in [Eno7], it is shown that the number appeared in the right handside of (5.6.1) for a fixed i is equal to the number of monobricks with i elements.
5.7. Examples of computations
In what follows, we fix an algebraically closed field k. For several finite-dimensional algebras
Λ, we show the lists of monobricks and left Schur subcategories, and the behavior of the maps
W : SchurL Λ  wide Λ and F : SchurL Λ  torf Λ in terms of their counterparts max : mbrick Λ sbrick Λ and (−) : mbrick Λ  mbrickc.c. Λ.
Example 5.7.1. Let Q be a quiver 1 ← 2 ← 3, then the AR quiver of mod kQ is given in
Table 1. By Theorem 5.6.12, we have # mbrick kQ = 22, the third Schröder number. There are
1 + 6 monobricks M with #M ≤ 1, namely, an empty set, and a singleton consisting of eachindecomposable kQ-modules.
In Table 2, we list the remaining 15 monobricks, together with their poset structure. For
example, the notation 1 < 21, 3 means that this poset consists of the disjoint union of two chains1 < 21 and 3. For each monobrick M, we write the corresponding left Schur subcategory Filt M in
the AR quiver, where the black vertices are M, and the white are the rest. If Filt M is not a wide
subcategory, then we write the monobrick corresponding to W(Filt M), which is equal to max M
by Theorem 5.4.5. Similarly, if Filt M is not a torsion-free class, then we write the monobrickcorresponding to F(Filt M), which is equal to the cofinal closure M by Theorem 5.3.15.

118

5. MONOBRICKS

Table 1. The Auslander-Reiten quiver of mod k[1 ← 2 ← 3]
3
2
1
2
1

1

3
2

2

3

Table 2. Monobricks M over k[1 ← 2 ← 3] with #M ≥ 2
wide?

max M

torsion-free?

M

No

2
1

Yes

itself

1<2

No

3
2
1

No

1 < 21 < 2

1, 2

Yes

itselfYes

itself1, 32

Yes

itselfNo

1, 2 < 32

1, 3

Yes

itselfYes

itself

2<3
2
1
1

No

3
2
1

No

1 < 21 < 2

2, 3
1

Yes

itselfNo

1 < 21, 3

2, 2

Yes

itselfNo

1 < 2, 2

2 < 32

No

3
2

Yes

itself2, 3

Yes

itselfYes

itself

1 < 21 < 2

No

3
2
1

Yes

itself

1 < 21, 3

No

2, 3
1

Yes

itself

1 < 2, 2

No

2, 2

Yes

itself1, 2 < 32

No

1, 32

Yes

itself1, 2, 3Yes

itselfYes

itselfM (as a poset)
1 < 21
3
1

3
1

3
1

3
1

left Schur subcats

3
1

3
1

3
1

3
1

5.7. EXAMPLES OF COMPUTATIONS

119

Now let us see some specific examples of computation of max M and M. For a given monobrickM, it is easy to describe its poset structure (we have L ≤ M in M whenever there is a nonzero map L → M ). Thus its maximal element max M can be easily computed. For example,
consider M = {1, 21, 3}. Then since we have an injection 1 ,→ 21 and there are no other non-zero
homomorphism, its poset structure is 1 < 21, 3, hence max M = {21, 3}.
The computation of M is a little harder than max M, but not so difficult. Recall from
Corollary 5.3.9 that M consists of all bricks N which satisfies the following two conditions:(1) N is a submodule of some M ∈ M.
(2) Every map N → M 0 with M 0 ∈ M is either zero or an injection.
Thus, to compute M, first list up all submodules of elements in M which are bricks and not in
3
M, then check whether the condition (2) above holds. For example, let M = {2, 2}. Then proper1
submodules which are bricks are exactly 21 and 1. However, there is a non-zero non-injection 21  2,
thus we exclude 21. In this way we obtain M = M ∪ {1}.
Next consider the path algebra of an A3 quiver with another orientation.
Example 5.7.2. Let Q be a quiver 1 → 2 ← 3. There are 1 + 6 monobricks M with #M ≤ 1,
namely, an empty set, and a singleton consisting of each indecomposable kQ-modules. It turns
out that # mbrick Λ = 26, which is different from 1 ← 2 ← 3. This means that the number of left
Schur subcategories (or monobricks) depends on the orientation of the quiver.
In Table 4, we list the remaining 19 monobricks and their maximal elements and cofinal
closures. Wide subcategories are categories in which max M is “itself,” and torsion-free classes
are categories in which M is “itself.” In this case, there are several examples which are not closed
under direct summands, kernels or images. Subcategories E with (*) are not closed under direct
summands (hence is closed under neither images nor kernels), and the white vertices in E indicate
indecomposables of add E which do not belong to E. The only one category with (**) is closed
under images, thus closed under direct summands, but is not closed under kernels. The remaining
categories are all closed under kernels and images, and there are 22 such subcategories. This
number coincides with the previous example, and see the next remark for the explanation.
Remark 5.7.3. In [Eno7], it is shown that the number of subcategories of mod kQ which areclosed under extensions, kernels and images does not depend on the orientation of the underlying
graph for a Dynkin quiver Q, although the number of monobricks does depend as we have seen.
In particular, if Q is of type An , then the number of such subcategories is equal to the n-th large
Schröder number by Theorem 5.6.12.
The next example is non-hereditary case, which already appeared in the introduction.
Example 5.7.4. Let Λ be any Nakayama algebra whose quiver is 1  2. Then there are four
bricks in mod Λ, namely, brick Λ = {1, 2, 12, 21}. By using this (and without any consideration ofother modules), we obtain the list of monobricks Table 5.
Finally, we consider representation-infinite case.
Example 5.7.5. Let Q be a 2-Kronecker quiver, that is, Q = [1 ⇔ 2]. Then a complete
classification of indecomposable kQ-module is known, see e.g. [ARS, Section VIII.7]. By using
this, we obtain the following three classes of bricks.
(1) Indecomposable preprojective modules {P1 , P2 , P3 , . . . }.
(2) Regular simple modules {Rλ }λ∈P1 (k) .
(3) Indecomposable preinjective modules {I1 , I2 , I3 , . . . }.
Here P1 = P (1), P2 = P (2), P3 = τ − P1 , P4 = τ − P2 , P5 = τ − P3 , . . . and I1 = I(2), I2 = I(1), I3 =
τ I1 , I4 = τ I2 , . . . , where P (i) (resp. I(i)) is the indecomposable projective (resp. injective) module
corresponding to the vertex i, and τ is the Auslander-Reiten translation.
To classify monobricks over kQ, we need to know the lists of pairs (B1 , B2 ) of bricks such that
there is a non-zero non-injection from B1 to B2 , and pairs such that there is an injection but nonon-zero non-injection from B1 to B2 . This is summarized in Figure 1, where B1B2 (resp.
B1 ,→ B2 ) indicates that there is a non-zero non-injection B1 → B2 (resp. an injection).

120

5. MONOBRICKS

Table 3. The Auslander-Reiten quiver of mod k[1 → 2 ← 3]
3
2

1
13
2

2
1
2

3

Table 4. Monobricks M over k[1 → 2 ← 3] with #M ≥ 2
E

M

max M

M

1
2

itselfM
2 < 123
2

2 < 32
2 < 123
2, 1
2, 3
1, 3
2 2

M

1, 3

itselfitself

3>2<1
2
2

1, 3
2 2

itselfN

itselfitself2 < 12 < 123

(∗)

13
2

N

2 < 32 < 123

(∗)

13
2

N

2 < 12, 3

1, 3
2

itself2 < 32, 1

3, 1
2

itself13
2

N

itselfitself13
2

itselfitselfitself

1 < 132
2

max M

itself

13
2

(∗)

E

itself
3>2<1
2
2

13
2

itself3 < 13 > 1
22
2

3 < 13
2
2

13
2

(∗∗)

itself1, 2, 3
2 < 12, 3

3
2

2

<

3, 1
2

itself2 < 32, 1

<

N =

<

itself

13
2

<

1, 3
2

1
2

Table 5. Monobricks over cyclic Nakayama algebras with 2 simples
M
∅
1
2
1
2

max M
itselfitselfitselfitselfM
itselfitselfitself2 < 12M2
1

1 < 21
2 < 12
1, 2

max M
itself2
1
1
2

itselfM
1 < 21
itselfitselfitselfAny other pairs can be deduced from the composition of arrows in Figure 1. Since there arelots of monobricks, we only consider cofinally closed monobricks. This is enough for classifying
monobricks since a set of bricks is a monobrick if and only if it is a subset of some cofinally closed
monobricks by Corollary 5.3.17.
The following are the list of all cofinally closed monobricks, or the list of simple objects in all
torsion-free classes.
(M1) {P1 , P2 , . . . , Pi } for some i.

5.7. EXAMPLES OF COMPUTATIONS

121

Figure 1. Structures of bricks in mod k[1 ⇔ 2]
P1
P2
P3
···
any Rλ
I1

I2

···

I3

(M2) {P1 , P2 , P3 , . . . }.
(M3) {P1 } ∪ {Rλ }λ∈X for any non-empty subset X ⊂ P1 (k).
(M4) {P1 } ∪ {Rλ }λ∈P1 (k) ∪ {Ii } for i ≥ 2.
(M5) {I1 }.
In this list, finite monobricks are (M1), (M3) for a finite set X and (M5). The poset structure is
as follows.
(M 1)
P1 < P2 < · · · < Pi

(M 2)

(M 3)

(M 4)

P1 < P2 < · · ·

Rλ
..
.
Rλ0

Rλ
..
.
Rλ0

P1

P1
X

(M 5)
Ii

I1

P1 (k)

Table 6. The poset structure of each monobrickUsing this, we can easily compute W(F) for each torsion-free class, since W(F) is equal toFilt(max(sim F)) by Theorem 5.4.5. Moreover, max(sim F) is nothing but the brick labels startingat F (Remark 5.4.6), we can compute the brick labels (c.f. [DIRRT, Example 3.6]). This can besummarized as follows.
(M 3)(M 4), (M 5)(M 1) (M 2){Pi }∅{Rλ }λ∈X{Ii }Table 7. The maximal element of each monobrick, or all the semibricks

Since max : mbrickc.c. Λ → sbrick Λ is surjective by Proposition 5.5.1, this table can also beseen as a table of all semibricks.
We remark that if X consists of one element in (M3), then the monobrick is isomorphic to
P1 < P2 as posets, although the former corresponds to non-functorially finite torsion-free classbut the latter to functorially finite.

CHAPTER 6

Rigid modules and ICE-closed subcategories in quiver
representations
This chapter is based on [Eno7].
We introduce image-cokernel-extension-closed (ICE-closed) subcategories of module categories.
This class unifies both torsion classes and wide subcategories. We show that ICE-closed subcategories over the path algebra of Dynkin type are in bijection with basic rigid modules, and thatthe number does not depend on the orientation of the quiver. We give an explicit formula of thisnumber for each Dynkin type, and in particular, it is equal to the large Schröder number for type
A case.
6.1. Introduction
Let Λ be an artin algebra and mod Λ the category of finitely generated right Λ-modules. There
are several kinds of subcategories of mod Λ which have been investigated in the representation
theory of algebras, e.g. wide subcategories, torsion classes, torsion-free classes, and so on. These
subcategories are defined by the property that they are closed under certain operations: e.g. taking
kernels, cokernels, images, extensions, submodules, or quotients.
In this paper, we propose a new class of subcategories of mod Λ, ICE-closed subcategories,
which is a subcategory closed under Images, Cokernels and Extensions. Typical examples of ICEclosed subcategories are torsion classes and wide subcategories, but there are more than them.
Recently, there are lots of studies on the classification of nice subcategories in terms of nicemodules. One of the most prominent results is the τ -tilting theory established in [AIR], which givesa bijection between functorially finite torsion classes and certain class of modules called support τ tilting modules. Actually, it is a generalization of the Ingalls-Thomas bijection [IT], which classifies
functorially finite torsion classes over hereditary algebras by support tilting modules.
The aim of this paper is to provide a similar classification of ICE-closed subcategories overhereditary algebras. More precisely, we show that such subcategories are in bijection with rigid
modules, modules without self-extensions. The main result is summarized as follows:
Theorem R (= Theorem 6.2.3). Let Q be a Dynkin quiver. Then there is a bijection between
the following two sets:(1) The set of ICE-closed subcategories of mod kQ.
(2) The set of isomorphism classes of basic rigid kQ-modules.
For an ICE-closed subcategory C, the corresponding rigid kQ-module is the basic Ext-projective
generator P (C) of C, and for a rigid kQ-module U , the corresponding ICE-closed subcategory isgiven by the category cok U consisting of cokernels of maps in add U .
In addition, we show that a subcategory closed under cokernels and extensions is automaticallyclosed under images, thus is ICE-closed. Actually, this theorem holds for any representation-finite
hereditary artin algebras, and can be generalized to representation-infinite case by restricting theclass of ICE-closed subcategories, see Theorem 6.2.3 for the precise statement.
In the appendix, we will count the number of ICE-closed subcategories by using several resultsin other papers. The result is summarized as follows.
Theorem S. Let Q be a Dynkin quiver. Then the number of ICE-closed subcategories only
depends on the underlying Dynkin graph, not on the choice of an orientation (Theorem 6.A.3).
122

6.2. BASIC DEFINITIONS AND THE MAIN RESULT

123

Moreover, we have an explicit formula of this number for each Dynkin type, and if Q is of type
An , then it is equal to the n-th large Schröder number (Corollay 6.A.15).
We expect that there is a hidden theory which generalizes this paper to non-hereditary case,as [AIR] generalizes [IT].
Organization. This paper is organized as follows. In Section 6.2, we give basic definitions
and state a main result Theorem 6.2.3. In Section 6.3, we give a proof of Theorem 6.2.3. In
Section 6.4, we consider the relation between ICE-closed subcategories, torsion classes and wide
subcategories in detail via rigid modules. In the appendix, we count the number of ICE-closed
subcategories for each Dynkin type.
Conventions and notation. Throughout this paper, all subcategories are assumed to be
full and closed under isomorphisms, direct sums and direct summands. An artin R-algebra is an
R-algebra over a commutative artinian ring R which is finitely generated as an R-module. We
often omit the base ring R, and simply call it an artin algebra.
For an artin algebra Λ, we denote by mod Λ the category of finitely generated right Λ-modules.
All modules are finitely generated right modules unless otherwise stated. For a collection C of Λmodules, we denote by add C the subcategory of mod Λ consisting of direct summandsLof finitendirect sums of objects in C. A module M is called basic if there is a decomposition M = i=1 Mi
such that each Mi is indecomposable and pair-wise non-isomorphic. For a module M , we denoteby |M | the number of non-isomorphic indecomposable direct summands of M .
6.2. Basic definitions and the main result
In this section, we give basic definitions and introduce some notation, and state our main result.
First of all, recall that a module M ∈ mod Λ over an artin algebra Λ is rigid if Ext1Λ (M, M ) = 0
holds. Then we define several conditions on the subcategory of mod Λ.
Definition 6.2.1. Let Λ be an artin algebra and C a subcategory of mod Λ.
(1) C is closed under extensions if for every short exact sequence in mod Λ
0

L

M

N

0,

we have that L, N ∈ C implies M ∈ C(2) C is closed under quotients (resp. submodules) if for every short exact sequence in mod Λ
0

L

M

N

0,

we have that M ∈ C implies N ∈ C (resp. L ∈ C).
(3) C is closed under cokernels (resp. images) if for every map f : M → N with M, N ∈ C,
we have Coker f ∈ C (resp. Im f ∈ C).
(4) C is a torsion class if C is closed under quotients and extensions.
(5) C is a wide subcategory if C is closed under kernels, cokernels and extensions.
(6) C is image-cokernel-extension-closed, abbreviated by ICE-closed, if C is closed underimages, cokernels and extensions.
(7) C is cokernel-extension-closed, abbreviated by CE-closed, if C is closed under cokernels
and extensions.
Then clearly all torsion classes and wide subcategories are (I)CE-closed, thus ICE-closed
subcategories can be seen as a generalization of these two classes.
To an extension-closed subcategory of mod Λ, we can associate a rigid module by taking theExt-progenerator.
Definition 6.2.2. Let Λ be an artin algebra and C an extension-closed subcategory of mod Λ.
(1) An object X ∈ C is Ext-projective in C if Ext1Λ (X, C) = 0 holds. We denote by P(C) the
subcategory of C consisting of all the Ext-projective objects in C.

124

6. RIGID MODULES AND ICE-CLOSED SUBCATEGORIES IN QUIVER REPRESENTATIONS(2) C has enough Ext-projectives if for every object X ∈ C, there is a short exact sequence
0

Y

P

X

0

with P ∈ P(C) and Y ∈ C.
(3) An object P ∈ C is an Ext-progenerator if add P = P(C) and C has enough Extprojectives.
Now we are ready to state our main result. Throughout this paper, we will use the following
notations for an artin algebra Λ.
• rigid Λ denotes the set of isomorphism classes of basic rigid Λ-modules.
• ice Λ denotes the set of ICE-closed subcategories of mod Λ.
• icep Λ denotes the set of ICE-closed subcategories of mod Λ with enough Ext-projectives.
• For an extension-closed subcategory C of mod Λ with an Ext-progenerator, we denote by
P (C) the unique basic Ext-progenerator of C.
• For a Λ-module U , we denote by cok U the subcategory of mod Λ consisting of cokernelsof maps in add U .
• For a collection U of Λ-modules, we denote by Fac U (resp. Sub U) the subcategory ofmod Λ consisting of quotients (resp. submodules) of objects in add U.
Theorem 6.2.3. Let Λ be a hereditary artin algebra. Then we have the following bijections
rigid ΛcokP

icep Λ.

Moreover, if Λ is representation-finite, then every CE-closed subcategory is automatically ICEclosed, and icep Λ = ice Λ holds.
This bijection extends a bijection between support (τ -)tilting modules and functorially finite
torsion classes given in [IT] or [AIR] in the following sense. Let Λ be a hereditary artin algebra.
It is known that a torsion class is functorially finite if and only if it has enough Ext-projectives.
Then the following diagram commutes, and the horizontal maps are bijective.
rigid Λcok
P

stilt ΛFac

icep Λ

f-tors ΛP

Here the bottom bijections were those given in [IT] or [AIR]. See Proposition 6.4.4 for the detail.
Remark 6.2.4. ICE-closed subcategories generalize both torsion classes and wide subcategories. Another generalization of these two classes was introduced in [Eno6], right Schur subcategories (actually the dual was studied in the paper). Every ICE-closed subcategory is a right
Schur, but the converse does nod hold in general. If Λ is Nakayama, then these coincides by [Eno6,
Theorem 6.1]. In [Eno6], we classify right Schur subcategories in any length abelian category byusing simple objects in them. This is in contrast with our use of Ext-projectives.
6.3. Proof of the main theorem
In this section, we give a proof of Theorem 6.2.3. Throughout this section, we denote by Λ ahereditary artin algebra.
First we give a map P (−) : icep Λ → rigid Λ.
Proposition 6.3.1. Let C be an CE-closed subcategory of mod Λ. Then the following hold.
(1) Every Ext-projective object in C is rigid.
(2) There are only finitely many indecomposable Ext-projective objects in C up to isomorphism.
(3) If C has enough Ext-projectives, then it has an Ext-progenerator P (C), and C = cok P (C)holds.

6.3. PROOF OF THE MAIN THEOREM

125

Proof. (1) Clear from definition.
(2) We claim that there are at most |Λ| indecomposable Ext-projectives in C. Let M1 , . . . , Mm
be pairwise non-isomorphic Ext-projectives in C. Then clearly M := M1 ⊕ · · · ⊕ Mm is basic rigid,
or partial tilting since Λ is hereditary. Then by taking the Bongartz completion, there is a Λmodule N such that M ⊕ N is a basic tilting Λ-module (see [ASS, Lemma VI.2.4] for the detail).
It follows that m = |M | ≤ |M ⊕ N | = |Λ|.
(3) Since C has enough projectives, (2) implies that C has an Ext-progenerator P (C). Since
C is closed under cokernels and P (C) ∈ C, clearly C ⊃ cok P (C) holds. Conversely, we haveC ⊂ cok P (C) since C has enough Ext-projectives.

The following lemma is essential in our proof. This says that add U is closed under images if
Λ is herediatry and U is rigid.
Lemma 6.3.2. Let Λ be a hereditary artin algebra and U a rigid Λ-module. Then Fac U ∩Sub U = add U holds.
Proof. Clearly we have add U ⊂ Fac U ∩ Sub U . Conversely, let X ∈ Fac U ∩ Sub U . Take aleft (add U )-approximation ϕ : X → U X with U X ∈ add U , which is an injection by X ∈ Sub U .
Then we have the following commutative exact diagram in mod Λ:
U0

0

X

ϕ

UX

C

0

By applying Hom(−, U ), we obtain an exact sequence
HomΛ (U X , U )(−)◦ϕ

Ext1Λ (C, U )HomΛ (X, U )Ext1Λ (U X , U ).

Since ϕ is a left (add U )-approximation, (−) ◦ ϕ is a surjection. In addition, Ext1Λ (U X , U ) vanishes
since U is rigid, hence Ext1Λ (C, U ) = 0. On the other hand, since Λ is hereditary, we have an exact
sequence
Ext1Λ (C, U0 )Ext1Λ (C, X)

0.

Since we have Ext1Λ (C, U0 ) = 0, we obtain Ext1Λ (C, X) = 0. It follows that the short exact sequence
0 → X → U X → C → 0 splits, which implies X ∈ add U .

Our next aim is to show that cok U is ICE-closed if U is rigid. We will make use of thesubcategory XU associated to U , which was introduced by Auslander-Reiten [AR2].
Definition 6.3.3. Let Λ be an artin algebra and U a Λ-module with Ext>0Λ (U, U ) = 0. Then
we denote by XU a subcategory of mod Λ consisting of modules X such that there is an exact
sequence
f2

f1

f0

· · · −→ U1 −→ U0 −→ X → 0>0with ExtΛ (U, Im fi ) = 0 for all i ≥ 0.
We borrow the following lemma from [AR2].
Lemma 6.3.4 ([AR2, Proposition 5.1]). Let Λ be an artin algebra and U a Λ-module with
Ext>0
Λ (U, U ) = 0. Then XU is closed under extensions, has an Ext-progenerator U , and is closed
under mono-cokernels, that is, for every short exact sequence
0

L

M

N

0

in mod Λ, if L and M belong to XU , then so does N .
The following is basic properties of XU in our hereditary setting. In particular, we have
XU = cok U for a rigid module U over a hereditary algebra Λ.

126

6. RIGID MODULES AND ICE-CLOSED SUBCATEGORIES IN QUIVER REPRESENTATIONS

Proposition 6.3.5. Let Λ be a hereditary artin algebra and U a rigid Λ-module.
(1) The following are equivalent for X ∈ mod Λ.
(a) X belongs to XU .
(b) X belongs to cok U .
(c) There is an short exact sequence of the following form with U1 and U0 in add U .
0

U1

U0

X

0

(2) cok U is an ICE-closed subcategory of mod Λ.
(3) cok U has an Ext-progenerator U .
Proof. (1) Clearly (a) implies (b) by XU ⊂ cok U . Also the implication (b) ⇒ (c) follows
immediately from Lemma 6.3.2 since Λ is hereditary and U is rigid, and Lemma 6.3.4 shows (c)⇒ (a).
(2) By Lemma 6.3.4, we only have to show that XU = cok U is closed under images, because
this will immediately imply that XU is closed under cokernels since XU is closed under monocokernels.
Take any ϕ : X → Y with X, Y ∈ XU . Since X ∈ XU ⊂ Fac U , we may assume X ∈ add U toshow Im ϕ ∈ XU . Since Y belongs to XU , there is a short exact sequence 0 → U1 → U0 → Y → 0
with U1 , U0 ∈ add U by (1). By taking pullback, we obtain the following exact commutative
diagram.
0
U1
E
X
0
p.b.
0

U1

Z

Im ϕ

0

Y

0

p.b.
0

U1

U0

Now we have Ext1Λ (X, U1 ) = 0 by X ∈ add U , which implies that the top short exact sequence

splits. Thus E ∼= U1 ⊕ X ∈ add U and Z ∈ Fac U ∩ Sub U hold. Therefore we get Z ∈ add U by
Lemma 6.3.2. Then the middle horizontal short exact sequence implies Im ϕ ∈ XU .
(3) Obvious from the short exact sequence in (1)(a).

Now we are ready to prove Theorem 6.2.3.
Proof of Theorem 6.2.3. Proposition 6.3.1 gives a map P (−) : icep Λ → rigid Λ, and Proposition 6.3.5 gives a map cok : rigid Λ → icep Λ. These propositions also show that these maps aremutually inverse to each other.
Finally we prove statements for the representation-finite case. Let Λ be a representation-finite
hereditary artin algebra, that is, mod Λ has finitely many indecomposables up to isomorphism.
Then [Eno2, Corollary 3.15] implies that every subcategory of mod Λ closed under extensions hasenough Ext-projectives. In particular, we have ice Λ = icep Λ. Moreover, if C is a CE-closed
subcategory of mod Λ, then Proposition 6.3.1 implies that C = cok P (C) holds. Since P (C) is rigid,
C is automatically closed under images by Proposition 6.3.5.

6.4. Maps to torsion classes and wide subcategories
The class of ICE-closed subcategories contain both the classes of torsion classes and wide
subcategories, so it is natural to ask the relation between these three classes. The aim of thissection is to introduce two natural maps from the set of ICE-closed subcategories to the set of
torsion classes and wide subcategories, and to investigate these maps via rigid modules.
Let us introduce some notation. For an artin algebra Λ, we denote by tors Λ (resp. f-tors Λ)
the set of torsion classes (resp. functorially finite torsion classes) in mod Λ. Similarly, we denote bywide Λ (resp. f-wide Λ) the set of wide subcategories (resp. functorially finite wide subcategories)
in mod Λ.
First, we construct two maps T : ice Λ  tors Λ and W : ice Λ  wide Λ which are the identities
on tors Λ and wide Λ respectively.

6.4. MAPS TO TORSION CLASSES AND WIDE SUBCATEGORIES

127

Definition 6.4.1. Let Λ be an artin algebra and C an ICE-subcategory of mod Λ.
(1) T(C) denotes the smallest torsion class containing C.
(2) W(C) is a subcategory of C defined as follows:
W(C) = {W ∈ C | Ker ϕ ∈ C for any map ϕ : C → W with C ∈ C}
Clearly T(T ) = T for T ∈ tors Λ and W(W) = W for W ∈ wide Λ. It is non-trivial that W
actually defines a map W : ice Λ → wide Λ, as we shall see below.
Proposition 6.4.2. Let Λ be an artin algebra and C an ICE-closed subcategory of mod Λ.
Then W(C) is a wide subcategory of mod Λ.
Proof. This a special case of [Eno6, Theorem 4.5], but here we will give a proof using thegeneral result in [KaSc, Exercise 8.23]. According to it, we say that an object X ∈ mod Λ isC-coherent if X ∈ Fac C and Ker ϕ ∈ C for every map C → X with C ∈ C. Since C is closed undercokernels, it is easy to check that every C-coherent object belongs to C, namely, W(C) coincides
with the category of C-coherent objects. Then since C is extension-closed, [KaSc, Exercise 8.23]
implies that W(C) is a wide subcategory of mod Λ.
If Λ is hereditary, then T is equal to Fac, as the following general proposition shows.
Proposition 6.4.3 (c.f. [IT, Proposition 2.13]). Let Λ be a hereditary artin algebra and C an
extension-closed subcategory of mod Λ. Then Fac C = T(C) holds, namely, Fac C is a torsion class.
Proof. We only have to show that Fac C is closed under extensions. Take a short exact
sequence0LM
N0with L, N ∈ Fac C, and take surjections πL : CL  L and πN : CN  N . Since Λ is hereditary,
the induced map Ext1Λ (N, CL ) → Ext1Λ (N, L) is a surjection. Thus we have the following exact
commutative diagram, where we in addition take pullback along πN .
0

CL

F

CN
p.b.

0

CL

πN

E

N

0

M

N

0

p.o.

πL

0

0

L

Since C is closed under extensions, we have F ∈ C. Thus we obtain M ∈ Fac C.

Next we will consider counterparts of the maps Fac and W in terms of rigid modules. Let Λ
be a hereditary artin algebra. We denote by stilt Λ the set of isomorphism classes of basic support
tilting Λ-modules. If U a rigid Λ-module, then Fac U is closed under extension by [AS, Proposition
5.5, Corollary 5.9], thus it is a torsion class. By the result of [IT] or [AIR], there is a unique basic
support tilting module U satisfying Fac U = Fac U . We call U the co-Bongartz completion of U .
Now the following proposition can immediately follows from definition, so we omit the proof.
Proposition 6.4.4. Let Λ be a hereditary artin algebra. Then the following diagram commutes, and the dashed maps are given by taking the co-Bongartz completion.
stilt Λ1

rigid Λ

Fac
P
cok
P

Fa
c

(−)

stilt ΛFacP

f-tors Λ
icep Λ

1Fac

f-tors Λ

Next we will investigate the map W : icep Λ → wide Λ, which is more non-trivial than T. To
do this, we introduce the Fac-minimality, covers and split projectives, following [AS].

128

6. RIGID MODULES AND ICE-CLOSED SUBCATEGORIES IN QUIVER REPRESENTATIONSDefinition 6.4.5. Let P be a subcategory of mod Λ. We say that P is Fac-minimal if there
is no proper subcategory P 0 of P satisfying P ⊂ Fac P 0 . We say that U ∈ mod Λ is Fac-minimal
if add U is Fac-minimal.
Note that subcategories are
L required to be closed under direct sums and direct summands. If
U ∈ mod Λ is basic and U = i∈I Ui with each Ui indecomposable,
then U is Fac-minimal if andLonly if there is no proper subset J of I satisfying U ∈ Fac( j∈J Uj ). Fac-minimal basic modules
are called covering-indecomposable modules in [AS].
Definition 6.4.6. Let C be a subcategory of mod Λ. Then an object P in C is split projective
if every surjection C  P in mod Λ with C ∈ C splits. We denote by P0 (C) the subcategory of C
consisting of all split projective objects in C.
It can be shown P0 (C) is closed under direct sums and direct summands. Clearly we have theinclusion P0 (C) ⊂ P(C) for an extension-closed subcategory C of mod Λ.
Next we recall the notion of covers of a category, introduced in [AS].
Definition 6.4.7. Let C be a subcategory of mod Λ and P a subcategory of C.
(1) P is a cover of C if C ⊂ Fac P holds.
(2) P is a minimal cover of C if P is a cover of C and there is no proper subcategory P 0 of
P which is a cover of C.
(3) An object P in C is a (minimal) cover of C if so is add P .
(4) C has a finite (minimal) cover if there is an object P in C which is a (minimal) cover ofC.
We will use some results in [AS] summarized as follows.
Proposition 6.4.8 ([AS, Theorem 2.3, Corollary 2.4]). Let Λ be an artin algebra and C a
subcategory of mod Λ. Then the following hold.
(1) Let P be a cover of C. Then P is a minimal cover of C if and only if P is Fac-minimal
if and only if P = P0 (C). In particular, the minimal cover of C is unique if it exists.
(2) If C has a finite cover, then C has a finite minimal cover. Thus there is a unique basic
Fac-minimal cover P of C up to isomorphism, which satisfies P0 (C) = add P .
By using this, we can define the following operation which yields a Fac-minimal basic module.
Definition 6.4.9. Let Λ be an artin algebra and M ∈ mod Λ. Then the Fac-minimal versionof M is a basic Fac-minimal cover M0 of add M , which is unique up to isomorphism by Proposition
6.4.8.
If M is a cover of a subcategory C of mod Λ, then Proposition 6.4.8 implies that its Fac-minimalversion M0 satisfies add M0 = P0 (C).
Now we will use the following general observation when Ext-projectives coincides with split
projectives, which is of interest in its own.
Proposition 6.4.10. Let Λ be an artin algebra and C an extension-closed subcategory of mod Λwith enough Ext-projectives. Then the following are equivalent:(1) P0 (C) = P(C) holds, that is, every Ext-projective object in C is split projective in C.
(2) P(C) is Fac-minimal.
(3) C is closed under epi-kernels, that is, for every short exact sequence
0

L

M

N

0,

if M and N belong to C, then so does L.
Proof. (1) ⇔ (2): Immediate from Proposition 6.4.8, since P(C) is a cover of C by the enoughExt-projectivity of C.
(1) ⇒ (3): Suppose that we have a short exact sequence 0 → L → M → N → 0 with
M, N ∈ C. Since C has enough Ext-projectives, there exists a short exact sequence 0 → N 0 →6.4. MAPS TO TORSION CLASSES AND WIDE SUBCATEGORIES

129

P → N → 0 with P ∈ P(C) and N 0 ∈ C. Then by taking the pullback, we obtain the following
exact commutative diagram.
0

0

N0

N0

0

L

E

0

L

ϕ

P

0

M

N

0

0

0

Since C is extension-closed, the middle vertical exact sequence implies E ∈ C. Since P ∈ P(C), itis split projective by (1). Thus ϕ splits, hence L is a direct summand of E. This implies L ∈ C
since a subcategory C is assumed to be closed under direct summands.
(3) ⇒ (1): Let P be an Ext-projective object in C, and take any surjection π : C  P withC ∈ C. Then we have a short exact sequence 0 → Ker π → C → P → 0, thus Ker π ∈ C holds by(3). Therefore, since P is Ext-projective, this short exact sequence must split. This shows that P
is split projective.

Now next we consider functorial finiteness of wide subcategories, which is of interest in its
own. In particular, we will show that the functorial finiteness is equivalent to the existence offinite (co)cover, and to the contravariantly (covariantly) finiteness.
The following describes the relation between finite covers and covariantly finiteness.
Lemma 6.4.11 ([AS, Theorem 4.5, Proposition 3.7]). Let Λ be an artin algebra and C a subcategory of mod Λ closed under images. Then the following are equivalent.
(1) C is covariantly finite.
(2) C has a finite cover.
In this case, let Λ → C Λ be a left minimal C-approximation. Then P0 (C) = add C Λ holds.
By using this, we obtain the following characterization of functorially finite wide subcategories.
Proposition 6.4.12. Let Λ be an artin algebra and W a wide subcategory of mod Λ. Thenthe following are equivalent.
(1) W is functorially finite.
(2) W is covariantly finite.
(2)0 W is contravariantly finite.
(3) W has an Ext-progenerator.
(4) W is equivalent to mod Γ for some artin algebra Γ.
Proof. Note that W is closed under images, so we can apply Lemma 6.4.11.
(1) ⇒ (2), (2)0 : Trivial.
(2) ⇒ (3): Take a left minimal W-approximation Λ → P with P in W. Then P is a minimal
cover of Λ with add P = P0 (W) by Lemma 6.4.11. We claim that P is an Ext-progenerator of
C. First, P is Ext-projective in W since it is split projective. Second, since P covers W and W
is closed under kernels, for every object W in W, there is a short exact sequence 0 → W 0 →P 0 → W → 0 with P 0 ∈ add P and W 0 ∈ W. This shows that W has enough Ext-projectives with
P(C) = add P = P0 (C).
(3) ⇒ (4): Let P be an Ext-progenerator of W. Define Γ := EndΛ (P ) and consider the functor
HomΛ (P, −) : mod Λ → mod Γ. Then it easy to see that this induces an equivalence W ' mod Γ
by the standard argument in the Morita theory.

130

6. RIGID MODULES AND ICE-CLOSED SUBCATEGORIES IN QUIVER REPRESENTATIONSF(4) ⇒ (2): Let F : mod Γ ' W be an equivalence. Then the composition mod Γ −→ W ,→mod Λ is exact since W is closed under kernels and cokernels in mod Λ. Thus add(F Γ) is a coverof W, since Γ is a cover of mod Γ and F preserves surjectivity. This implies that W is covariantly
finite by Lemma 6.4.11.
By the dual argument, (4) implies (2)0 . Thus (4) implies (1).

Now let us return to the hereditary setting. The key observation is the following.
Lemma 6.4.13. Let Λ be a hereditary artin algebra and C an ICE-closed subcategory of mod Λwith enough Ext-projectives. Let P be an object in P0 (C) and Q a submodule of P satisfying
Q ∈ C. Then Q is also in P0 (C).
Proof. Since there is an Ext-progenerator of C by Proposition 6.3.1, the category C has afinite cover, thus has a minimal cover P0 (C) by Proposition 6.4.8. Therefore there is a surjectionπ
π : P0  Q with P0 ∈ P0 (C), hence we obtain a short exact sequence 0 → Ker π → P0 −→ Q → 0.
1
1
Now since Λ is hereditary, the map ExtΛ (P, Ker π) → ExtΛ (Q, Ker π) induced by the inclusion
Q ,→ P is surjective. Thus we obtain the following exact commutative diagram.
0
0

Ker π

P0

0
π

Q

0

P

0

p.b.
0

Ker π

E

p

P/Q

P/Q

0

0

Since C is closed under cokernels, we have P/Q ∈ C. Then the middle vertical exact sequence
implies E ∈ C since C is extension-closed. Now p should split since P is split projective in C,
thus the middle horizontal short exact sequence splits. It follows that so does the top horizontal
sequence, hence Q is a direct summand of P0 . Therefore Q ∈ P0 (C) holds.

By using this, we can show the following last result in this section on the relation between
Fac-minimal rigid modules and wide subcategories. We denote by rigid0 Λ the set of isomorphismclasses of basic rigid Λ-modules which are Fac-minimal.
Proposition 6.4.14. Let Λ be a hereditary artin algebra. Then the following diagram commutes and the horizontal maps are bijections, where the map (−)0 is given by taking the Facminimal version.
rigid0 Λ

1

rigid Λ

cok
P
cok
P

(−)0

rigid0 Λ

f-wide Λ

icep Λ

1

W
cok
P

f-wide ΛIn particular, functorially finite wide subcategories are in bijection with basic Fac-minimal rigid
modules.
Proof. It suffices to show the following for U ∈ rigid Λ by Theorem 6.2.3.
(1) cok U is wide if and only if U is Fac-minimal.
(2) U0 is an Ext-progenerator of W(cok U ), where U0 is the Fac-minimal version of U .

6.A. ENUMERATIVE RESULTS

131

By Theorem 6.2.3, the category cok U is an ICE-closed subcategory of mod Λ with P(cok U ) =add U .
(1) It is easy to check that an ICE-subcategory of mod Λ is wide if and only if it is closed
under epi-kernels. Thus the assertion follows from Proposition 6.4.10.
(2) Put C := cok U . Then P0 (C) = add U0 holds by Proposition 6.4.8. First we show P0 (C) ⊂W(C). It suffices to show that every map ϕ : C → P0 with C ∈ C and P0 ∈ P0 (C) satisfies
Ker ϕ ∈ C. Since C is closed under images, we have Im ϕ ∈ C, and then Lemma 6.4.13 shows thatIm ϕ is split projective in C. Thus the induced surjection C  Im ϕ splits, hence Ker ϕ is a direct
summand of C. Thus Ker ϕ ∈ C holds.
Now we have shown U0 ∈ W(C). Moreover, U0 is a cover of W(C) since it is a cover of C, and U0
is split projective in W(C) since it is so in C. Then it is easy to see that U0 is an Ext-progeneratorof W(C) because W(C) is closed under kernels.
It was shown in [IT, Corollary 2.17] that the maps W : f-tors Λ  f-wide Λ : Fac are mutually
inverse bijections (although their definition of f-wide Λ is a bit different from ours). For theconvenience of the reader, we give a short proof of this in our context.
Corollary 6.4.15. Let Λ be a hereditary artin algebra. Then W : f-tors Λ  f-wide Λ : Fac
are mutually inverse bijections between the sets of functorially finite torsion classes and functoriallyfinite wide subcategories.
Proof. Let U be a basic rigid Λ-module. According to Propositions 6.4.4 and 6.4.14, it
suffices to show the following claims.
(1) If U is support tilting, then U = P (Fac U0 ) holds, where U0 is a Fac-minimal version ofU.
(2) If U is Fac-minimal, then U is a Fac-minimal version of P (Fac U ).
(1) By the definition of the Fac-minimal version, we have Fac U0 = Fac U . Thus the claim
follows from U = P (Fac U ), which holds by Theorem 6.2.3.
(2) Since U is Fac-minimal, U is a minimal cover of Fac U . Since P (Fac U ) is a cover of Fac U ,its Fac-minimal version coincides with U .

6.A. Enumerative results
Throughout this appendix, we denote by k a field. In this appendix, we give an explicit
formula of the number # rigidi (kQ) of rigid kQ-modules with i non-isomorphic direct summands
for a Dynkin quiver Q.
In [MRZ, Proposition 6.1], it was shown that this number does not depend on the orientationof Q, but the proof therein relies heavily on cluster combinatorics. In the first subsection, we givea short homological proof of this fact. In the second subsection, we give an explicit formula of# rigidi (kQ) by using the enumerative result on cluster complexes in [Krat].
Let us introduce the set which we want to enumerate in this appendix.
Definition 6.A.1. Let Λ be an artin algebra. For a non-negative integer i, we denote byrigidi Λ the set of isomorphism classes of basic Λ-modules U satisfying |U | = i.
Note that if Λ is hereditary, then rigidi Λ = ∅ unless 0 ≤ i ≤ |Λ| by considering the Bongartz
completion.
6.A.1. Invariance under sink mutation. Let us begin with recalling the mutation of aquiver at a sink.
Definition 6.A.2. Let Q be a quiver. A sink of Q is a vertex v such that there is no arrow
starting at v. For a sink v of Q, the sink mutation µv Q is a new quiver obtained from Q byreversing all arrows which end at v.
Note that v becomes a source in µv Q. It is well-known that if two quivers Q and Q0 have thesame underlying graph which is a tree, then there is a sequence of sink mutations which transforms
Q into Q0 .
Now the following is the main result in this subsection.

132

6. RIGID MODULES AND ICE-CLOSED SUBCATEGORIES IN QUIVER REPRESENTATIONS

Theorem 6.A.3. Let Q be an acylic quiver with n vertices and v a sink of Q. Then for any
0 ≤ i ≤ n, there is a bijection between rigidi (kQ) and rigidi (k(µv Q)).
This result immediately yields the following corollary, since any two Dynkin quivers with thesame underlying graph can be connected via a series of sink mutations.
Corollary 6.A.4 (c.f. [MRZ, Proposition 6.1]). Let Q be a Dynkin quiver. Then # rigidi (kQ)
only depends on the underlying Dynkin graph, not on the choice of an orientation.
In the rest of this subsection, we will give a proof of Theorem 6.A.3.
Definition 6.A.5. Let Q be an acylic quiver and v a sink or a source of Q.
(1) We denote by modv kQ the subcategory consisting of kQ-modules which do not contain
S(v) as a direct summand, where S(v) is the simple module corresponding to v.
(2) For a kQ-module M , we denote by Mv a unique module in modv kQ such that there isa following decomposition for some n ≥ 0.
M∼
= Mv ⊕ S(v)⊕n(3) We define rigidiv (kQ) := rigidi (kQ) ∩ modv (kQ), that is, the set of basic rigid kQ-modules
M with |M | = i such that M does not contain S(v) as an direct summand.
(4) We define rigidihvi (kQ) := rigidi (kQ)\rigidiv (kQ), that is, the set of basic rigid kQ-modules
M with |M | = i such that M contains S(v) as an direct summand.
By definition, we have rigidi (kQ) = rigidiv (kQ) t rigidihvi (kQ). Our strategy is to construct two
bijections rigidiv (kQ) ∼= rigidihvi (k(µv Q)) separately. The first= rigidiv (k(µv Q)) and rigidihvi (kQ) ∼one is established by the reflection functor, and second one by the perpendicular categories.
First, we will use the following property of the classical BGP-reflection functor. For the proof,
we refer the reader to standard textbooks on quiver representation theory such as [ASS].
Proposition 6.A.6. Let Q be an acylic quiver and v a sink of Q. There is a functor
Rv : mod kQ → mod k(µv Q) called the reflection functor, which induces an equivalence modv kQ 'modv k(µv Q). Moreover, this functor induces an isomorphism Ext1kQ (X, Y ) ∼= Ext1k(µv Q) (Rv X, Rv Y )for every X, Y ∈ modv kQ.
This immediately yields the following bijection.
Corollary 6.A.7. Let Q be an acyclic quiver and v a sink of Q. Then we have an bijection∼
Rv : rigidiv (kQ) −→ rigidiv (k(µv Q)) given by U 7→ Rv U .
Next we will construct a bijection rigidihvi (kQ) ∼= rigidi−1 (kQv ), where Qv denotes a quiverobtained by removing v from Q. We will use the following two subcategories determined by Sv .
Definition 6.A.8. Let Λ be an artin algebra and C a collection of objects in mod Λ. Thendefine two subcategories C ⊥0,1 and ⊥0,1 C of mod Λ as follows.
(1) C ⊥0,1 consists of modules X with HomkQ (C, X) = 0 = Ext1kQ (C, X).
(2) ⊥0,1 C consists of modules X with HomkQ (X, C) = 0 = Ext1kQ (X, C).
These subcategories are called right and left perpendicular categories with respect to C. By
using this categories and their relation to (co)localizations, we can prove the key property of themin this case.
Lemma 6.A.9. Let Q be an acylic quiver and v a sink or a source of Q. Then both S(v)⊥0,1
and ⊥0,1 S(v) are wide subcategories of mod kQ, and equivalent to mod kQv .
Proof. The fact that these categories are wide subcategories follows from [GL, Proposition
1.1] since kQ is hereditary, and its proof is quite straightforward, so we omit this.
For the rest, we will use the theory of a recollement. We refer the definitions and details to[Psa]. Let ev be an idempotent corresponding to v, and put e = 1 − ev . The we have the following

6.A. ENUMERATIVE RESULTS

133

recollement diagram, where I is the natural embedding functor.
L

mod kQhei

I

mod kQE

mod e(kQ)eR

Here E := HomkQ (e(kQ), −) = (−)e, and L and R are left and right adjoint functors of E, and bothare fully faithful. Consider the essential image Im I of I. Since E is a localization and a colocaliztion
with respect to the Serre subcategory Im I [Psa, Remark 2.2], we have that Im R and Im L coincidewith the perpendicular category (Im I)⊥0,1 and ⊥0,1 (Im I) respectively [GL, Proposition 2.2]. On
the other hand, Im I consists of modules M with M e = 0, thus Im I = add S(v) holds. Therefore,
we have Im R = S(v)⊥0,1 and Im L = ⊥0,1 S(v). Since L and R are fully faithful, both S(v)⊥0,1
and ⊥0,1 S(v) are equivalent to mod e(kQ)e. Now the assertion holds since we clearly have an
isomorphism of algebras e(kQ)e ∼= kQv .
We remark that this lemma was proved in [CK, Lemma 3.2.5] in a more general setting.
By using this, we obtain the following bijection.
Proposition 6.A.10. Let Q be an acylic quiver and v a sink or a source of Q. Then fori ≥ 1, we have a bijection
rigidihvi (kQ) ∼= rigidi−1 (kQv ).
The map is given by U 7→ (Uv )e ∈ mod e(kQ)e, where e is the same as in the proof of Lemma
6.A.9, and we identify e(kQ)e with kQv .
Proof. We give a proof for the case v is a sink, and the same proof applies for the source caseby using S(v)⊥0,1 instead of ⊥0,1 S(v). First recall that these subcategories are wide subcategories
of mod kQ by Lemma 6.A.9, thus are abelian categories. The key observation is the following
claim.
(Claim): For X ∈ modv (kQ), the following are equivalent:(1) X ⊕ S(v) is rigid.
(2) X ∈ ⊥0,1 S(v) holds, and X is rigid in the abelian category ⊥0,1 S(v).
Proof of (Claim).
(1) ⇒ (2): Since S(v) ⊕ X is rigid, Ext1kQ (X, S(v)) = 0 holds. Moreover, if we have a nonzero map X → S(v), then it must be surjective since S(v) is simple, hence it splits since S(v) is
projective. This contradicts to X ∈ modv kQ, therefore we have HomkQ (X, S(v)) = 0. It follows
that X ∈ ⊥0,1 S(v) holds. Since ⊥0,1 S(v) is a wide subcategory of mod kQ, an Ext1 inside ⊥0,1 S(v)is the same as an Ext1 inside mod kQ. Thus X is rigid in the abelian category ⊥0,1 S(v) since so
is in mod kQ.
(2) ⇒ (1): By the above argument, X is a rigid kQ-module, and Ext1kQ (S(v), X ⊕ S(v))
vanishes since S(v) is projective. Thus X ⊕ S(v) is rigid by X ∈ ⊥0,1 S(v).

By (Claim), the map U 7→ Uv clearly induces the following bijection
rigidi (kQ) ∼= rigidi−1 (⊥0,1 S(v)),
hvi

where the right hand side is the set of basic rigid objects X in the abelian category ⊥0,1 S(v)satisfying |X| = i − 1. Now the assertion holds from Lemma 6.A.9, since ⊥0,1 S(v) is equivalent tomod kQv as abelian categories.

Now we immediately obtain the second bijection.
Corollary 6.A.11. Let Q be an acylic quiver and i a sink of Q. Then there is a bijection
between rigidihvi (kQ) and rigidihvi k(µv Q).
Proof. Note that v is a source of µv Q. Then by Proposition 6.A.10, we have two bijections
between rigidihvi (kQ) ∼= rigidi−1 (kQv ) and rigidihvi k(µv Q) ∼= rigidi−1 k(µv Q)v . Since (µv Q)v = Qv
holds, we obtain a bijection between rigidihvi (kQ) and rigidihvi k(µv Q) by composing the above twobijections.
134

6. RIGID MODULES AND ICE-CLOSED SUBCATEGORIES IN QUIVER REPRESENTATIONS

Now we are ready to prove Theorem 6.A.3.
Proof of Theorem 6.A.3. We have the following equalities by definition.
rigidi (kQ) = rigidiv (kQ) t rigidihvi (kQ)
rigidi (k(µv Q)) = rigidiv (k(µv Q)) t rigidihvi (k(µv Q))
Now we have a bijection rigidiv (kQ) ∼= rigidiv (k(µv Q)) by Corollary 6.A.7, and a bijection rigidihvi (kQ) ∼=irigidhvi (k(µv Q)) by Corollary 6.A.11. Thus by combining these two, we obtain a bijection between
rigidi (kQ) and rigidi (k(µv Q)).

6.A.2. Formula for the number of rigid modules. In this subsection, we give an explicit
formula for # rigidi (kQ) for a Dynkin quiver Q. From now on, we assume that Q is a Dynkinquiver of type Xn ∈ {An , Dn , E6 , E7 , E8 } with n vertices.
Let ∆(Q) be a simplicial complex defined as follows: the set of vertices is rigid(kQ), and an (i−1)-simplex consists of sets of rigid kQ-modules whose direct sum is rigid, or equivalently, belongs
to rigidi (kQ). This complex was introduced by Riedtmann and Schofield [RS]. By definition,# rigidi (kQ) is equal to the number of (i + 1)-faces of ∆(Q), thus the calculation of # rigidi (kQ)
is nothing but that of the face vector of ∆(Q). Although this complex is classical, there seems to
be no papers which contain an explicit formula of # rigidi (kQ).
We give such an formula, by translating our problem to a combinatorial problem on a clustercomplex. Let Φ be the root system of type Xn , and let Φ≥−1 denote the set of almost positive rootsof Φ, that is, positive roots together with negative simple roots. Then the cluster complex ∆(Xn )of type Xn , also known as the generalized associahedron, is a simplicial complex with the vertex
set Φ≥−1 . We refer the reader to [FZ, MRZ] for the details. Then this complex contains ∆(Q) if
Q is bipirtite, that is, every vertex is either a sink or a source. More precisely, the following holds.
Proposition 6.A.12. Let Q be a Dynkin quiver with a bipirtite orientation. Then taking
dimension vectors, we have an embedding ∆(Q) ,→ ∆(Xn ), which induces an isomorphism between∆(Q) and the full subcomplex of ∆(Xn ) spanned by positive roots.
We refer the reader to [MRZ, 4.12] for the proof, and to [BMRRT, Theorem 4.5] for the moretheoretical explanation of this using the cluster category. As a corollary, we have the following
equality.
Corollary 6.A.13. Let Q be a Dynkin quiver. Then # rigidi (kQ) is equal to the number of(i − 1)-faces of ∆(Xn ) which contain no negative simple roots.
Proof. We can transform Q into a bipirtite Dynkin quiver by using sink mutations. Thus
we may assume that Q is bipirtite by Corollary 6.A.4. Then the assertion is immediate from
Proposition 6.A.12.

Now we are ready to show the formula of # rigidi (kQ) by using [Krat], which enumerates thenumber of faces of ∆(Q) satisfying various conditions.
Theorem 6.A.14. Let Q be a Dynkin quiver of type Xn . Then the number # rigidi (kQ) is
equal to (Xn ) in the following list, where ni denotes the binomial coefficient.
 

n n+i
1
(An )
i+1 i
i
 
 





n n+i−2n−1 n+i−31n−1 n+i−2(Dn )+−n−1 i−1
i
i
i
i−1
i−1
(E6 )

(E7 )

i

0
1

1
36

2
300

3
1035

4
1720

5
1368

6
418

i

0
1

1
63

2
777

3
3927

4
9933

5
13299

6
9009

total
4878
7
2431

total
39440

6.A. ENUMERATIVE RESULTS

(E8 )

i

0
1

1
120

2
2135

3
15120

4
54327

5
1083606121555

135

7
71760

8
17342

total
390720

Proof. The computation is achieved by specializing the results [Krat, Theorems FA, FD,
Section 7] to m = 1 and y = 0. More precisely, in [Krat], the m-generalization of cluster complexes
are studied, and m = 1 is the classical case. Then the number of faces of an m-cluster complex
which consists of given numbers of positive roots and negative roots was computed, and y = 0
means that we exclude negative roots.
By our main result, # rigid(kQ) is equal to the number of ICE-closed subcategories of mod kQ.
Pn
Since we have # rigid(kQ) = i=0 # rigidi (kQ), we obtain the following enumeration.
Corollary 6.A.15. Let Q be a Dynkin quiver of type Xn . Then the number of ICE-closed
subcategories in mod kQ is equal to the sum of the numbers given in Theorem 6.A.14 over i =0, 1, . . . , n. In particular, if Q is of type An , then the equality holds,
 

n
X
n n+i
1
,
# ice(kQ) =
i+1 i
i
i=0
where the right hand side is known as the n-th large Schröder number [OEIS, A006318].
Remark 6.A.16. In [Eno6, Theorem 6.13], the author computes # ice(kQ) for a linearlyoriented An quiver by a different method. By combining this with Theorem 6.A.3, we can give
another proof of the fact that # ice kQ is equal to the n-th large Schröder number for a quiver oftype An .

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