Relative Auslander Correspondence
via
Exact Categories

Haruhisa Enomoto
Student ID: 321601059
Advisor: Prof. Osamu Iyama

Graduate School of Mathematics
Nagoya University

Contents
Abstractiv

Acknowledgementv

Notation and conventions

vi

Chapter 1. Introduction
1.1. Auslander correspondence
1.2. Relative representation theory
1.3. Exact categories

1
1
2
3

Chapter 2. Morita-type characterizations of exact categories
2.1. Introduction
2.2. Exact categories with enough projectives
2.3. Exact categories with enough projectives and injectives
2.4. Exact categories with enough projectives and injectives and higher kernels
2.5. Applications to artin R-algebras

5
5
7
14
19
23

Chapter 3. Classifications of exact structures and CM-finite algebras
3.1. Introduction
3.2. Classifying exact structures via Serre subcategories
3.3. Exact categories of finite type3.4. Classifications of CM-finite algebras

30
30
33
40
46

Appendix A. Some results needed in Chapter 2
A.1. The Auslander-Buchweitz theory for exact categories
A.2. Constructions of exact structures

51
51
52

Appendix B. The AR theory for exact categories over a noetherian ring
B.1. Existence of AR conflations
B.2. AR conflations, the AR duality and dualizing varieties

55
55
56

Bibliography

59

iii

Abstract
In this thesis, we discuss the representation theory of algebras by using exact categories andgive some new results. This thesis is mainly based on the author’s papers [En1, En2]. The author
prepared this thesis as both Self Designed Study Report / Research report and Small Group class’
Report of his master thesis in Graduate School of Mathematics, Nagoya University.
This thesis consists of three chapters together with some appendices. In Chapter 1, we explain
the general idea behind this thesis, including Auslander correspondence and exact categories. Then
we raise two motivating questions, Problem A and B. In Chapter 2, we investigate Problem A,
that is, a Morita-type characterizations of exact categories. This chapter is based on the author’s
published paper [En1]. In Chapter 3, we study Problem B, that is, to give an Auslander-type
correspondence of exact categories. Using the result in the previous chapter, we obtain a newclassification result of CM-finite algebras in terms of exact structures. This chapter is based on
the author’s paper [En2].

iv

Acknowledgement
First and foremost, I would like to thank my advisor Osamu Iyama, for his support and many
helpful comments over the past three years. Without his guidance, my work would not be possible. When I was an undergraduate, I preferred viewing mathematics from a highly abstractcategorical perspective, and would not be willing to know about down-to-earth objects. Since myundergraduate seminar, he has been teaching me kindly a lot of technique of actual computation,mathematical intuition and examples, along with a good deal of inspiring questions and suggestions. Thanks to his generous supervision, I recognized the importance of relating abstract things
to concrete phenomena, which is the guiding principle of this thesis and my future research.
I would also like to express my gratitude to all the administrative staffs of Graduate Schoolof Mathematics for the financial support and the resources.
In addition, I thank all the “habitants” of the study room A-333, especially H. Inoue, forproviding me with a lot of funny and relaxing time over the past six(!) years since I entered thisuniversity. Talking and discussion with them has certainly be stimulating and encouraging myinterest in mathematics, 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 M. Tobisawa,
who has healed my mind when I was somewhat depressive, and who has given me the hope andstrength to live.

v

Notation and conventions
Throughout this thesis, we adopt the following notation and conventions.
(On categories)• All categories are skeletally small unless otherwise stated, that is, the isomorphism classesof objects form a set.
• All subcategories are full and closed under isomorphisms.
• Let C and C 0 be categories, D a subcategory of C and F : C → C 0 a functor. We denote
by F (D) the essential image of D under F , that is, the subcategory of C 0 consisting of
all objects isomorphic to the images of objects in C under F . We simply call F (D) theimage of D under F .
(On additive categories)• All functors between additive categories are additive.
• For a subcategory D of an additive category C, we write add D for the subcategory of C
consisting of all objects which are summands of finite direct sums of objects in D. For
an object G in C, we simply write add G for add{G}.
• We say that an additive category C is of finite type if there exists an object G such thatC = add G. In this case, such G is called an additive generator of C.
• When we consider modules over rings or categories, we always mean right modules.
(On exact categories) We refer to [Bü, Ke] for the basics on exact categories.
• When we consider an extension-closed subcategory C of an exact category E, we always
regard C as the exact category with the exact structure inherited from those of E.
• As for exact categories, we use the terminologies inflations, deflations and conflations.
• We often denote by  an inflation and by  a deflation.
• We say that a functor F : E → E 0 between exact categories is an exact equivalence if F
is an equivalence of categories, F is exact and F reflects exactness.
• For an exact category E, we denote by P(E) (resp. I(E)) the subcategory of E consistingof all projective (resp. injective) objects. We say that E has enough projectives P (resp.
enough injectives I) if E has enough projective objects and P = P(E) (resp. enoughinjective objects and I = I(E)).

vi

CHAPTER 1

Introduction
Representation theory studies how an algebraic object acts on vector spaces. It dates back to
Frobenius, who studied finite groups by using their representations. Since then, finite groups, Liegroups and Lie algebras have played leading roles in the representation theory. It was not until
around 1970 that the representation theory of algebras, the main topic of this thesis, has attracted
an attention, undoubtedly due to the influential work of Maurice Auslander. He developed themethod of analyzing modules (= representation) over algebras, and it enabled us to visualize thewhole structure in a quite combinatorial and explicit manner. His work consists of the beautiful
combination of abstract categorical techniques and concrete description of representations.
In this chapter, we will give a rough explanation of the motivations of representation theoryof algebras, with emphasis on the Auslander correspondence. Then we will see how the notionof exact categories fits in with them when considering relative representation theory. The generalreferences of representation theory of algebras are [ARS, ASS]. For basics on exact categories, see[Bü, Ke].
1.1. Auslander correspondence
Let Λ be a ring. Then the purpose of the representation theory of Λ is to understand thestructure 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. Thenfor a noetherian ring Λ, the category mod Λ of finitely generated right modules is a target of ourinvestigation, which has a nice property such as being abelian. If in addition Λ is finite-dimensional
over a field, then the classical Krull-Schmidt theorem holds, that is, every object in mod Λ uniquely
decomposes into a finite direct sum of indecomposable Λ-modules. Therefore the investigation ofmod Λ is completely reduced to that of indecomposable objects in mod Λ.
Let us consider the simplest case, where there are only finitely many indecomposable objects inmod Λ up to isomorphism. Such an algebra is called representation-finite, and has been investigatedintensively. As for this class, the following result due to Auslander [Au2] is important, which serves
as a leading model for this work.
Theorem 1.1.1 (Auslander correspondence). Let k be a field. Then there exists a bijection
between the following classes (up to some equivalence).
(1) Representation-finite finite-dimensional k-algebras Λ.
(2) Abelian Hom-finite k-categories E with finitely many indecomposable objects.
(3) Finite-dimensional k-algebras Γ satisfying gl.dim Γ ≤ 2 ≤ dom.dim Γ.
We omit some unexplained terminologies (see [Au2, ARS]). This relates the study of representation-finite algebras Λ to another class of algebras Γ, and the representation theory of Λis encoded in the structural theory of Γ. The correspondence between (1) and (3) is usuallyreferred to as an Auslander correspondence, however we include (2) here to help understand what
is happening under this correspondence in detail. Let us divide this correspondence into two parts,(1)-(2) and (2)-(3), and explain the idea and philosophy behind them.
1.1.1. Bijection between (1) and (2): the Morita theory. The map from (1) to (2)can be easily described: it sends Λ to the category mod Λ, which is by definition of finite type.
This map is an injection since Λ can be recovered from the category mod Λ (this point will become
more delicate when we will consider the relative version later).
12

1. INTRODUCTION

The crucial point here is the surjectivity of the map. This follows from the following wellknown variants of the classical Morita theory.
Proposition 1.1.2. The following are equivalent for a k-category E.
• E is a Hom-finite abelian k-category with a projective generator.
• There is a finite-dimensional algebra Λ such that E is equivalent to mod Λ.
This gives the categorical characterizations of module categories, which is what the Morita
theory is mainly aiming at. For a general account of the Morita theory, we refer the reader to[AF].
1.1.2. Bijection between (2) and (3): Auslander algebras. More mysterious and subtle
is the relation between (2) and (3). Actually this part stems from a general consideration about
categories with finitely many indecomposables, as we shall explain.
First recall that an additive category E is of finite type if it has an object G satisfying E =add G, where add G is the subcategory of E consisting of all direct summands of finite direct sumsof G. Under mild assumption, it is equivalent to that E has finitely many indecomposables. For
such categories, the following observation is crucial, which was essentially proved by Auslander(e.g. [Au2]).
Proposition 1.1.3. There exists a bijection between the following classes (up to some equivalence).
(2)0 Additive categories E of finite type.
(3)0 Rings Γ.
We only give the construction of maps here. For an additive category E of finite type, choose
an additive generator G of E and put Γ := EndE (G). Conversely for a ring Γ, put E := proj Γ, thecategory of finitely generated projective right Γ-modules. This category has an additive generatorΓΓ , hence is of finite type.
This says that considering categories of finite type is equivalent to considering algebras, andwe call an algebra Γ corresponding to the category E is an Auslander algebra of E. This is aquite interesting and suggestive phenomenon, because E and Γ are of a rather different nature apriori: One is a categorical, hence abstract, object, which we usually deal with by using conceptual
categorical argument, whereas the other is kind of concrete and down-to-earth object. Here thequestion naturally arises: What is the relation between categorical properties of E and homological(= representation theoretic) behavior of Γ? Indeed the bijection of (2) and (3) claims that theablianness of E can be checked by the homological condition on Γ.
Combining this with the previous observation, the general idea behind Auslander correspondence is to seek the relation between the following for a given category E.
(1) How E is actually realized as a category which naturally appears in the representation
theory of algebras.
(2) Categorical properties E has.
(3) Homological behavior of the Auslander algebra Γ of E.
1.2. Relative representation theory
The main motivation of this project is to explore the appropriate analogue of Auslander’sobservation in the framework of exact categories, so that we can deal with relative representation
theory.
1.2.1. What is the relative representation theory? As we explained earlier, the main
purpose of the representation theory of an algebra Λ is to investigate the category mod Λ. However,
it is often the case that more explicit and interesting results are available if we restrict the class ofmodules we consider. Such a strategy is what we call the relative representation theory of algebras.
To demonstrate the validity and the origin of the relative theory, let us consider the case ofcommutative rings. First consider Λ = Z and the category mod Z of finitely generated abeliangroups. Although the whole structure of mod Z is controlled by the structure theorem of finitely

1.3. EXACT CATEGORIES

3

generated abelian groups, we obtain a simpler result if we focus on torsionfree abelian groups:
they are nothing but finite direct sums of Z itself. More generally, for a Dedekind domain R, thestructure of finitely generated torsionfree modules over R are well-understood.
Dedekind domains are one-dimensional regular rings, so let us next move to the higherdimensional situation. For a noetherian local ring R, an R-module M is called maximal CohenMacaulay if the depth of M is equal to dim R, and denote by CM R the category of such modules.
This category is known to be more tractable than mod R in various aspects, and the investigationof CM R is now one of the main streams of the representation theory of commutative rings, called
Cohen-Macaulay representation theory (see [LW, Yo]). Here rings R such that CM R is of finite
type, called CM-finite ring, has played the central role. It is known that this theory has a close
connection with algebraic geometry and singularity theory. For example, CM-finite Gorensteinrings nicely correspond to simple singularities of type ADE.
Let us consider the non-commutative analogue of this CM theory. For a nice class of finitedimensional algebras, called Iwanaga-Gorenstein, there exists an appropriate analogue of CM Λ
available. This class of algebras are ubiquitous in the representation theory, which generalize
self-injective algebras and algebras with finite global dimension. Therefore, for such an algebraΛ, we are led to investigate CM Λ more deeply instead of mod Λ. What we should do first is,
as in the “absolute” case, the investigation of algebras such that CM Λ has only finitely many
indecomposables, that is, CM-finite ones.
1.2.2. Relative Auslander correspondence fails. Taking the previous argument in account, when considering relative representation theory, it is natural to seek the analogue of the
Auslander correspondence for the relative representation-finite algebras such as CM-finite ones.
If we can achieve this, then we will have a bijection between the following:(1) CM-finite Iwanaga-Gorenstein algebras Λ.
(2) Hom-finite k-categories E with finitely many indecomposables which satisfy some categorical properties.
(3) Finite-dimensional k-algebras satisfying some homological properties.
The maps are defined (if possible) as follows: For Λ in (1), we define E in (2) to be CM Λ, andthe map between (2) and (3) is a restriction of Proposition 1.1.3. Actually this kind of strategy
works for some classes of relative theory, e.g. for one-dimensional orders and 1-Iwanaga-Gorenstein
algebras, see [ARo] and [Iy3] for the details. Also it is asked in [Che1, Problem D] whether thiskind of bijection exists for CM-finite algebras.
However, in general, it often happens that the map from (1) to (2) is not injective. For
example, there exist non-Morita-equivalent Iwanaga-Gorenstein algebras Λ and Λ0 such that CM Λ
and CM Λ0 are equivalent. The problem is that the algebra Λ cannot be recovered from theadditive structure of the representation category such as CM Λ, unlike mod Λ. Now we are now
in position to introduce the main subject of this thesis, exact categories, which provide us withenough information on CM Λ to reconstruct Λ.
1.3. Exact categories
Quillen introduced the concept of exact categories in [Qu] with an application to higher algebraic K-theory in mind. Since its appearance, this concept has turned out to be quite useful andhas been made use of in many branches of mathematics, including representation theory, algebraic
topology and functional analysis. To our purpose, exact categories are very essential since they
provide an appropriate framework and suitable language for relative representation theory, as weshall see.
An exact category is a generalization of abelian categories. More precisely, 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 any extension-closed subcategories ofabelian categories with the natural structure of exact categories. We refer the reader to Definition
3.2.9 and [Bü] for the precise definition. Since most of the relative representation categories like4

1. INTRODUCTION

CM Λ are extension-closed in the module category, we can (and should) regard them as exact
categories, not just as additive ones.
1.3.1. Motivating problems and main results. Return to our original situation. The
problem we have encountered is that Λ cannot be recovered from the additive structure of CM Λ.
As you might guess, Λ can be reconstruct from CM Λ together with the exact structure on it (seee.g. Proposition 3.4.2 for the detail)Let us consider the relative representation-finite algebras in the same way as we did for theclassical Auslander correspondence, but in the framework of exact categories. The bijections wewould like to establish is of the following form:(1) CM-finite Iwanaga-Gorenstein algebras Λ.
(2) Hom-finite exact k-categories E with finitely many indecomposables satisfying some categorical properties.
(3) Finite-dimensional k-algebras together with some homological information.
The first thing we have to do is the Morita-type theory concerning the bijection between (1)and (2), which can be summarized as follows:
Problem A. Characterize when a given exact category E is exact-equivalent to CM Λ forsome Iwanaga-Gorenstein algebras. More generally, describe relationship between the categoricalproperties of an exact category E and its actual representation-theoretic realization.
Some scattered results were available on this problem, e.g. [Che2, KIWY, Ka], but no generalwork was available. In Chapter 2 of this thesis, this problem is tackled with in full generality,
where we consider more wider case than CM category of Iwanaga-Gorenstein algebras. For thecase of Iwanaga-Gorenstein algebras, we can give a quite simple answer to it as follows:
Theorem (= Corollary 2.4.13). Let k be a field and E a Hom-finite exact k-category. Thenthe following are equivalent.
• There exists a finite-dimensional Iwanaga-Gorenstein k-algebra Λ such that E is exact
equivalent to CM Λ.
• E is idempotent complete Frobenius, and has a projective generator and higher kernels.
Next let us consider a bijection between (2) and (3). To establish this, it seems to be necessary
to give an analogue of Proposition 1.1.3 for exact categories of finite type. Since categories of finite
type are the same thing as algebras by Proposition 1.1.3, we should discuss how the structure ofexact categories fits in with this Auslander-algebra construction:
Problem B. Let E be a category of finite type and Γ its Auslander algebra. How can weclassify all the possible exact structures on E by means of homological properties of Γ?
This is quite a natural question, but there was no research on this subject. We give an answer
to this problem in Chapter 3 as follows.
Theorem (= Theorem 3.3.7). Let k be a field, E a Hom-finite idempotent complete k-categoryand Γ its Auslander algebra. Then there exists a bijection between the following.
• Exact structures on E.
• Sets of isomorphism classes of simple Γ-modules satisfying the 2-regular condition.
Finally, by combining Problem A and B, we obtain the somewhat explicit classification of
CM-finite Iwanaga-Gorenstein algebras, see Theorem H. This answered an open question raised
in [Che1, Problem D] in the case of Iwanaga-Gorenstein algebras.

CHAPTER 2

Morita-type characterizations of exact categories
In this chapter, we mainly discuss about Problem A in the first chapter. Using the Moritatype embedding, we show that any exact category with enough projectives has a realization as a(pre)resolving subcategory of a module category. When the exact category has enough injectives,
the image of the embedding can be described in terms of Wakamatsu tilting (=semi-dualizing)
subcategories. If moreover the exact category has higher kernels, then its image coincides withthe category naturally associated with a cotilting subcategory up to summands. We apply theseresults to the representation theory of artin algebras. In particular, we show that the ideal quotientof a module category by a functorially finite subcategory closed under submodules is a torsionfreeclass of some module category.
2.1. Introduction
Since Quillen introduced exact categories in [Qu], many branches of mathematics have madeuse of this concept. One of the most important classes of exact categories is given by a cotilting module U over a ring as the associated Ext-orthogonal category ⊥ U , which forms an exact
category with enough projectives and injectives. This class of exact categories is fundamental in
the representation theory of algebras, such as the tilting theory in derived categories as well as
Cohen-Macaulay representations. In this chapter, we give characterizations of such kinds of exact
categories as ⊥ U among all exact categories by investigating the relationship between the following(= the detailed version of Problem A).
(1) Categorical properties of exact categories, e.g. enough projectives, enough injectives,
Frobenius, idempotent complete, having (higher) kernels, abelian, · · · .
(2) Representation theoretic realizations of exact categories, e.g. mod Λ for a ring Λ, Extorthogonal category ⊥ U for a cotilting module U , the exact category XW associated witha Wakamatsu tilting module W , their resolving subcategories, · · · .
As a consequence, we deduce several known results including [Che2, KIWY, Ka]. Our approach is based on the Morita-type embedding: For a skeletally small exact category E withenough projectives, the category P of projective objects in E gives a fully faithful exact functor(Proposition 2.2.1)
P : E → Mod P,
X 7→ E(−, X)|P .
This gives a realization of E as a subcategory of the module category.
First we apply this functor P to show the following basic observation, where the category
mod C consists of finitely presented C-modules in a stronger sense, see Definition 2.2.2.
Proposition C. (=Proposition 2.2.8) Let E be a skeletally small exact category. The following are equivalent.
(1) E is idempotent complete and has enough projectives.
(2) There exists a skeletally small additive category C such that E is exact equivalent to someresolving subcategory of mod C.
When E has enough projectives and injectives, we show that the image of all injective objects
under the above embedding P forms a special subcategory, which we call a Wakamatsu tilting
subcategory. This is a categorical analogue of a Wakamatsu tilting module introduced in [Wa] as
a common generalization of a tilting module and a cotilting module. It is also known as a semidualizing module [Chr, ATY], which is a certain analogue of a dualizing complex of Grothendieck56

2. MORITA-TYPE CHARACTERIZATIONS OF EXACT CATEGORIES[Ha]. Any Wakamatsu tilting subcategory W of a module category mod C gives rise to an exact
subcategory XW of mod C which has enough projectives and injectives, see Proposition 2.3.2. The
category XW for the special case W = proj C is nothing but the category GP C of Gorenstein
projective C-modules (see Definition 2.3.7).
Using these concepts, one can embed exact categories with enough projectives and injectives
as follows.
Theorem D. (=Theorem 2.3.3) Let E be a skeletally small exact category. The following are
equivalent.
(1) E is idempotent complete and has enough projectives and injectives.
(2) There exist a skeletally small additive category C and a Wakamatsu tilting subcategory
W of mod C such that E is exact equivalent to some resolving-coresolving subcategory of
XW .
As an application, we characterize when a given exact category is exact equivalent to one of
the three important cases mod C, XW or GP C for some additive category C and some Wakamatsu
tilting subcategory W, see Theorem 2.2.15, 2.3.10 and 2.3.12 respectively.
Next we consider a special case when W is a cotilting subcategory of mod C. The category
XW coincides with the Ext-orthogonal subcategory ⊥ W in this situation, and we give a simple
characterization when E and XW are exact equivalent. We extend the notion of n-kernels [Ja] ton ≥ −1 (see Definition 2.4.5), and prove the following main result.
Theorem E. (=Theorem 2.4.11) Let E be a skeletally small exact category and n be an integern ≥ 0. The following are equivalent.
(1) E is idempotent complete, has enough projectives and injectives and has (n − 1)-kernels.
(2) There exist a skeletally small additive category C with weak kernels and an n-cotilting
subcategory W of mod C such that E is exact equivalent to XW .
This theorem provides us with a concrete method to prove that a given exact category isequivalent to those associated with a cotilting module. Using this, we prove some results on artin
algebras over a commutative artinian ring R.
Theorem F. (=Theorem 2.5.1) Let Λ be an artin R-algebra and M ∈ mod Λ, and put C :=Sub M . Consider the quotient category E = (mod Λ)/[C]. Then the following hold.
(1) E admits an exact structure in which E has enough projectives and injectives and has
0-kernels.
(2) Suppose that ind(τ − C) \ ind C is a finite set. Then there exist an artin R-algebra Γ anda 1-cotilting module U ∈ mod Γ such that E is exact equivalent to a torsionfree class⊥U ⊂ mod Γ.
A key ingredient of the proof of Theorem F is the theory of relative homological algebra due
to Auslander-Solberg [ASo1, ASo2, ASo3] and its relation to exact categories via [DRSS].
As another application of Theorem E, we deduce the following result by Auslander-Solberg.
Theorem G. (=Theorem 2.5.10) Let Λ be an artin R-algebra and M ∈ mod Λ. Set G :=
Λ ⊕ M , C := DΛ ⊕ τ M , Γ := EndΛ (G) and U := HomΛ (G, C) ∈ mod Γ. Then the following hold.
(1) U is a cotilting Γ-module with id U = 2 or 0.
(2) HomΛ (G, −) : mod Λ → mod Γ induces an equivalence mod Λ ' ⊥ U .
(3) mod Λ admits an exact structure such that projective objects are precisely objects in add G
and the equivalence mod Λ ' ⊥ U is an exact equivalence.
(4) EndΛ (G) and EndΛ (C) are derived equivalent.
This theorem is essentially contained in [ASo2, Proposition 3.26], but we give a simple proof
by using the modified exact structure on mod Λ and Theorem E.
Finally, let us give a brief description of the individual sections. In Section 2, we study
exact categories with enough projectives. We prove Proposition C and give a characterization of2.2. EXACT CATEGORIES WITH ENOUGH PROJECTIVES

7

exact categories of the form mod C for some additive category C. In Section 3, we study exact
categories with enough projectives and injectives. To this purpose, we introduce Wakamatsu tilting
subcategories and prove some properties. Then we prove Theorem D, and give a characterization of
exact categories coming from Wakamatsu tilting subcategories. In Section 4, we introduce cotilting
subcategories, and study their relationship to higher kernels. In Section 5, we apply these results
to the representation theory of artin algebras. In Appendix A.1 in this thesis, we develop the
analogue of Auslander-Buchweitz approximation theory in the context of exact categories, which
we need in Section 4. In Appendix A.2, we collect some results which enable us to construct newexact structures from a given one, which we use in Section 5.
2.2. Exact categories with enough projectives
In this section, we always assume that E is a skeletally small exact category with enoughprojectives P = P(E). We freely use basic properties of the bifunctor ExtiE (−, −) for i ≥ 0, which
can be defined by using projective resolutions.
2.2.1. Morita-type theorem. We start with recalling the notion of modules over a category.
For an additive category C, a right C-module X is a contravariant additive functor X : C op → Abfrom C to the category of abelian groups Ab. We denote by Mod C the (not skeletally small)
category of right C-modules, and morphisms are natural transformations between them. Then the
category Mod C is an abelian category with enough projectives, and projective objects are precisely
direct summands of (possibly infinite) direct sums of representable functors C(−, C) for C ∈ C.
For a skeletally small exact category E with enough projectives P := P(E), we define a functor
P : E → Mod P,

X 7→ E(−, X)|P ,(2.2.1)

which we call the Morita embedding because of the following properties.
Proposition 2.2.1. Let E be a skeletally small exact category with enough projectives P. Thenthe Morita embedding functor P : E → Mod P is fully faithful and exact. Moreover P preserves
projectivity and all extension groups.
Proof. This is well-known and standard. For the convenience of the reader, we shall give a
proof.
We first observe that P is an exact functor. In fact, P is obviously left exact, and for eachprojective object P and each deflation f : Y  Z, the induced map E(P, Y ) → E(P, Z) issurjective by the definition of projectivity. In addition, if P ∈ E is projective, then PP = P(−, P )is a projective P-module. Therefore P sends projective objects to projective modules.
Next we will see that P is fully faithful. Since E has enough projectives, for any object X ∈ E,
there exist conflations X2  P1  X1 and X1  P0  X. By the exactness of P, this gives anexact sequence PP1 → PP0 → PX → 0 in Mod P, and moreover we can check that P1 → P0 → X
is also a cokernel diagram in E. Hence we have the following diagram
0/ E(X, Y )/ E(P0 , Y )/ E(P1 , Y )

o P

0


/ (Mod P)(PX, PY )/ (Mod P)(PP0 , PY )/ (Mod P)(PP1 , PY )P

o P

whose rows are exact and the second and third vertical morphisms are isomorphisms by the
Yoneda lemma. It follows that E(X, Y ) → (Mod P)(PX, PY ) is also an isomorphism, thus P isfully faithful.
Note that P sends a projective resolution of X to a projective resolution of PX, since it preserves projectivity and exactness. Therefore P induces an isomorphism ExtiE (X, Y ) ∼= ExtiMod P (X, Y )for all i ≥ 0 because P is fully faithful.
However, P is far from essentially surjective because Mod P is too large. This leads us to thefollowing definition of a subcategory of Mod P. Suppose that C is an additive category. Recall thata right C-module M is called finitely generated if there exists an epimorphism C(−, C)  M for8

2. MORITA-TYPE CHARACTERIZATIONS OF EXACT CATEGORIES

some C ∈ C. By the Yoneda lemma, finitely generated projective C-modules are precisely direct
summands of representable functors.
Definition 2.2.2. Let C be a skeletally small additive category.
(1) We denote by proj C the full subcategory of Mod C consisting of all finitely generated
projective C-modules.
(2) We denote by mod C the full subcategory of C-modules X ∈ Mod C such that there exists
an exact sequence in Mod C of the form
· · · → P2 → P 1 → P0 → X → 0(2.2.2)where Pi is in proj C for each i ≥ 0.
In the same way, for a ring Λ we define the subcategories proj Λ and mod Λ of Mod Λ.
Note that the notation mod C often stands for weaker notions, e.g. the category of finitely
presented C-modules. In Proposition 2.2.7 below, we characterize when these notions coincide.
Let us recall the following classes of subcategories of exact categories.
Definition 2.2.3. Let E be an exact category with enough projectives and C a full subcategoryof E which is closed under isomorphisms.
(1) We say that C is resolving if C satisfies the following conditions.
(a) C contains all projective objects in E.
(b) C is closed under extensions, that is, for each conflation L  M  N , if both L
and N are in C, then so is M .
(c) C is closed under kernels of deflations, that is, for each conflation L  M  N , ifboth M and N are in C, then so is L.
(d) C is closed under summands.
Dually we define coresolving subcategories for an exact category with enough injectives.
(2) C is called thick if it is closed under extensions, kernels of deflations, cokernels of inflations
and summands.
In addition to these classical concepts, we need the following weaker notion when we later
investigate exact categories which are not idempotent complete.
Definition 2.2.4. Let E be an exact category with enough projectives and C a full subcategoryof E closed under isomorphisms. We call C preresolving if C satisfies the following conditions.
(a) add C contains all projective objects in E.
(b) C is closed under extensions.
(c) For each X in C, there exists a conflation ΩX  P  X in E such that all terms are in
C and P is projective in E.
Dually we define precoresolving subcategories for an exact category with enough injectives.
We point out that these subcategories are actually exact subcategories of E, because they areclosed under extensions.
The following lemma gives the relationship between resolving and preresolving subcategories.
Lemma 2.2.5. Let E be an exact category with enough projectives P. Then a subcategory C of
E is resolving if and only if it is preresolving and closed under summands.
Proof. The “only if” part is clear. Now we suppose that C is preresolving and closed undersummands. Since add C = C in this case, it suffices to show that C is closed under kernels ofdeflations. Let X  Y  Z be a conflation in E in which Y and Z are in C. By our assumption
for C, there exists a conflation ΩZ  P  Z with ΩZ in C and P in C ∩ P. We then have thefollowing pullback diagram.
ΩZ
ΩZ




//P
/E
X /
X /


/Y


//Z

2.2. EXACT CATEGORIES WITH ENOUGH PROJECTIVES

9

Because C is closed under extensions, the middle column shows that E is in C. Furthermore themiddle row splits because P is projective in E. From this X is a summand of E ∈ C, which showsthat X is actually in C since C is closed under summands.
We give basic properties of mod C.
Proposition 2.2.6. Let C be a skeletally small additive category. Then mod C is a thicksubcategory of Mod C. In addition, mod C has enough projectives, and its projective objects are
precisely objects in proj C.
Proof. It follows from the horseshoe lemma that mod C is closed under extensions. First we
will show that mod C is closed under cokernels of monomorphisms. Let 0 → X → Y → Z → 0 be
an exact sequence in Mod C and suppose that Y is in mod C. By the definition of mod C, there isan exact sequence 0 → ΩY → P → Y → 0 such that P is in proj C and ΩY is in mod C. We thenhave the following commutative exact diagram(2.2.3)

0

0


ΩY


ΩY

0


/ ΩZ


/P

/Z

/0

0


/X


/Y

/Z

/0


0


0

whose rows and columns are exact. If X is also in mod C, we have that ΩZ is also in mod C since
mod C is closed under extensions. Then it follows that Z is in mod C by the middle row. Hence
mod C is closed under cokernels of monomorphisms.
Next we show that mod C is closed under summands. Let Z ∈ Mod C be a summand ofY ∈ mod C. We claim that there exists an exact sequence 0 → ΩZ → P → Z → 0 such that P isin proj C and ΩZ is a summand of some object in mod C, which inductively implies that mod C isclosed under summands. We have a split exact sequence 0 → X → Y → Z → 0 with Y in mod C,
so we can use the diagram (2.2.3) again and obtain the exact sequence 0 → ΩZ → P → Z → 0. By
taking the direct sum of 0 → Z = Z and the first column of (2.2.3), we obtain an exact sequence
0 → ΩY → ΩZ ⊕ Z → X ⊕ Z → 0. Since Y ∼= X ⊕ Z and ΩY are in mod C and mod C is closed
under extensions, it follows that ΩZ ⊕ Z is in mod C. Thus ΩZ is a summand of some object inmod C.
By the same argument as in the proof of Lemma 2.2.5, one can show that mod C is closed
under kernels of epimorphisms, which completes the proof that mod C is thick in Mod C. The laststatement of the proposition is obvious by the construction of mod C.
As a corollary, we obtain the following relation between mod C and the category of finitely
presented C-modules (cf. [Au1, Proposition 2.1]). Recall that C is said to have weak kernels if for
C(−,g)

any morphism f : C1 → C0 in C, there exists a morphism g : C2 → C1 such that C(−, C2 ) −−−−→
C(−,f )

C(−, C1 ) −−−−→ C(−, C0 ) is exact in Mod C.
Proposition 2.2.7. Let C be a skeletally small additive category. Then the category mod C
coincides with the category of finitely presented C-modules if and only if C has weak kernels. In
this case mod C is an abelian category.
Proof. Suppose that C has weak kernels and X is a finitely presented C-module. Then we
have an exact sequence C(−, C1 ) → C(−, C0 ) → X → 0 for some objects C0 and C1 in C. By the
Yoneda lemma, the morphism C(−, C1 ) → C(−, C0 ) is induced from a morphism C1 → C0 . Since C
has weak kernels, it follows that this morphism has a series of weak kernels · · · → C2 → C1 → C0 ,

10

2. MORITA-TYPE CHARACTERIZATIONS OF EXACT CATEGORIES

which yields an exact sequence · · · → C(−, C2 ) → C(−, C1 ) → C(−, C0 ) → X → 0. Therefore X is
in mod C.
Conversely, suppose that all finitely presented C-modules are in mod C. Let C1 → C0 be amorphism in C. Consider the exact sequence 0 → X → C(−, C1 ) → C(−, C0 ) → Z → 0 in Mod C.
Then Z is in mod C by our assumption. Since mod C is thick in Mod C by Proposition 2.2.6, wehave X ∈ mod C. In particular we have an exact sequence C(−, C2 ) → C(−, C1 ) → C(−, C0 ) since
X is finitely generated, which gives a weak kernel of C1 → C0 .
Finally we show that mod C is abelian if these conditions are satisfied. It is easy to see thatthe category of finitely presented C-modules is closed under cokernels in Mod C, thus we only haveto show that mod C is closed under kernels. However this is clear since mod C is thick in Mod C by
Proposition 2.2.6.

Now we can prove that E can be embedded into mod P as a (pre)resolving subcategory.
Proposition 2.2.8. Let E be a skeletally small exact category with enough projectives P andP : E → mod P the Morita embedding (2.2.1). Then P(E) is a preresolving subcategory of mod P
and E is exact equivalent to P(E). It is resolving in mod P if and only if E is idempotent complete.
Proof. Since E has enough projectives and P is an exact functor by Proposition 2.2.1, one
can easily check that P(E) is contained in mod P. To show that P(E) is preresolving in mod P, we
check the conditions of Definition 2.2.4.
(a): Clearly add P(E) contains all projective objects in mod P, since projective objects in
mod P are direct summands of representable functors of the form P(−, P ) = PP for P ∈ P.
(b): We show that P(E) is closed under extensions. Suppose that 0 → PL → X → PM → 0
is an exact sequence in mod P. Then there exists a conflation ΩM  P  M in E, and sending
this by P gives us the following commutative diagram
0

/ PΩM

/ PP

/ PM

/0

0


/ PL


/X

/ PM

/0

(2.2.4)

with exact rows. The vertical morphisms exist by the projectivity of PP . Then we obtain thecorresponding morphism ΩM → L in E because P is fully faithful. By taking pushout, we get thefollowing commutative diagram
/P
//M
ΩM /

L /


/N

//M

in E. Because P is exact, P sends this diagrams to the diagram isomorphic to (2.2.4). This provesthat X ∼= PN .
(c): Let M be any object in E. Since there exists a conflation ΩM  P  M with P in P,
we have an exact sequence 0 → PΩM → PP → PM → 0 in mod P. Since PP is projective, thecondition (c) holds.
Next we show that E is exact equivalent to P(E). We have to confirm that for morphisms
L → M → N in E, if 0 → PL → PM → PN → 0 is exact, then L → M → N is a conflation. First
note that L → M → N is a kernel and cokernel pair because P is fully faithful and 0 → PL →PM → PN → 0 is exact. Take a deflation P  N in E with P projective. Since PP is projective,
PP → PN factors through PM  PN . Because P is fully faithful, this implies that P  N also
factors through M → N . This shows that the composition P → M → N is a deflation in E, and
M → N has a kernel L → M . By [Ke, Proposition A.1.(c)], we have that M → N is a deflation,
which shows that L → M → N is a conflation in E.
Finally we check the last statement. If E is idempotent complete, then clearly so is P(E).
This shows that P(E) is closed under summands in mod P. Thus E is resolving by Lemma 2.2.5.
Conversely, suppose that P(E) is resolving. Since mod P is idempotent complete by Proposition
2.2.6, the assumption that P(E) ⊂ mod P is closed under summands clearly implies that P(E) is
idempotent complete, so is E.
2.2. EXACT CATEGORIES WITH ENOUGH PROJECTIVES

11

This proposition gives a simple criterion for a morphism in an exact category with enoughprojectives to be a deflation.
Corollary 2.2.9. Let E be a skeletally small idempotent complete exact category with enoughprojectives, and let f : X → Y be a morphism in E. Then f is a deflation in E if and only if forany projective object P ∈ E, the induced map E(P, f ) : E(P, X) → E(P, Y ) is surjective.
Proof. The “only if” part is clear. Suppose that E(P, X) → E(P, Y ) is surjective for any
projective object P . This is clearly equivalent to that PX → PY is surjective. Since both
P(E) ⊂ mod P and mod P ⊂ Mod P are subcategories closed under kernels of epimorphisms, so isP(E) ⊂ Mod P. Thus we obtain an exact sequence 0 → PZ → PX → PY → 0 in P(E). Since P
reflects exactness and is fully faithful by Proposition 2.2.8, we have a conflation Z  X  Y ,which shows that f is a deflation.

As a consequence, we give a correspondence between (pre)resolving subcategories of module
categories and exact categories with enough projectives. To state it accurately, we need some
preparation.
Let us recall the following classical Morita theorem.
Definition-Proposition 2.2.10. Let C and D be skeletally small additive categories. The
following are equivalent.
(1) There exists an equivalence Mod C ' Mod D.
(2) There exists an exact equivalence mod C ' mod D.
(3) There exists an equivalence proj C ' proj D.
In this case, we say that C and D are Morita equivalent. For example, C and proj C are always
Morita equivalent. Moreover, if C and D are both idempotent complete, then these are Morita
equivalent if and only if C ' D.
Proof. The last assertion and the equivalence of (1) and (3) are classical, and we refer thereader to [Au3, Proposition 2.6] for the proof.
(1) ⇒ (2): The equivalence Mod C ' Mod D induces an equivalence proj C ' proj D sincefinitely generated projective is a categorical notion. It follows from the definition of mod C that it
also induces an exact equivalence mod C ' mod D.
(2) ⇒ (3): Projective objects in the exact category mod C are precisely objects in proj C by
Proposition 2.2.6. Then the assertion follows immediately.
Note that C is idempotent complete if and only if proj C ' C. Also observe that any additive
functor F : C → D induces an additive functor proj F : proj C → proj D.
Our aim is to classify exact categories with enough projectives such that the subcategories
of projective objects are (Morita) equivalent to a fixed category C. To this purpose, we have to
introduce the appropriate notion of equivalence between these categories.
Definition 2.2.11. Suppose that E and E 0 are skeletally small exact categories with enoughprojectives. Assume that P(E) and P(E 0 ) are Morita equivalent to C via equivalences F : proj C 'proj P(E) and F 0 : proj C ' proj P(E 0 ). We say that pairs (E, F ) and (E 0 , F 0 ) are C-equivalent if
there exists an exact equivalence G : E ' E 0 such that the following diagram commutes up to
natural isomorphism
proj P(E) oF'proj C

where ι and ι0 are the inclusions.

/ proj P(E 0 )proj ι0proj ι


proj E

F0''
proj G/ proj E 0

12

2. MORITA-TYPE CHARACTERIZATIONS OF EXACT CATEGORIES

Note that if C, E and E 0 are idempotent complete, then the above diagram can be replaced by
the following one.
P(E) o

F
'

C

F0
'

/ P(E 0 )
ι0

ι


E

'
G


/ E0

Now we are in position to state the following Morita-type theorem. This theorem says that exact categories with enough projectives with a fixed category C of projective objects are completelyclassified by resolving subcategories of mod C.
Theorem 2.2.12. Let C be a skeletally small additive category.
(1) There exists a bijection between the following two classes.
(a) C-equivalence classes of pairs (E, F ) where E is a skeletally small exact category withenough projectives P such that P is Morita equivalent to C.
(b) Preresolving subcategories of mod C.
It sends (E, F ) in (a) to the image of E → mod P ' mod C, and the inverse map sends Ein (b) to (E, id) for the identity functor id : proj C = proj C.
(2) If C is idempotent complete, the bijection of (1) restricts to a bijection between the following.
(a) C-equivalence classes of pairs (E, F ) where E is a skeletally small idempotent complete exact categories with enough projectives P such that P is equivalent to C.
(b) Resolving subcategories of mod C.
To prove this, we need the following preparation.
Lemma 2.2.13. Pairs (E, F ) and (E 0 , F 0 ) are C-equivalent if and only if the following diagram
commutes up to natural isomorphism(−)◦F

mod P(E)O'

0/ mod C o (−)◦F mod P(E 0 )'OP(2.2.5)

P

'

E

G

/ E0

0

where we identify mod P (resp. mod P ) with mod(proj P) (resp. mod(proj P 0 )).
Proof. The assertion follows immediately from the following general fact. Let A and B be
additive categories and K, L : A ⇒ B fully faithful functors. Consider the composition K 0 , L0 :B → Mod B ⇒ Mod A, where B → Mod B is the Yoneda embedding and Mod B ⇒ Mod A are(−) ◦ K and (−) ◦ L. Then K and L are isomorphic if and only if K 0 and L0 are isomorphic. The
details are left to the reader.
Proof of Theorem 2.2.12. (1): The map from (a) to (b) is well-defined by Proposition
2.2.8 and Lemma 2.2.13.
To see that the map from (b) to (a) is well-defined, it suffices to show that the subcategory P
of projective objects in E is Morita equivalent to C. Since E is a preresolving subcategory of mod C,
it easily follows that P is Morita equivalent to P(mod C) = proj C, which is Morita equivalent toC.
These maps are easily seen to be inverse to each other.
(2): Suppose that C is idempotent complete. If E is idempotent complete exact category withenough projectives, then P(E) is also idempotent complete. By Definition-Proposition 2.2.10, one
can see that the bijection in (1) restricts to the one in (2).

Restring this theorem to the case of rings, one can obtain the following result, whose details
are left to the reader. When E has enough projectives P and P = add P for an object P in E, wecall P a projective generator.
Corollary 2.2.14. Let Λ be a ring.

2.2. EXACT CATEGORIES WITH ENOUGH PROJECTIVES

13(1) There exists a bijection between the following two classes.
(a) (proj Λ)-equivalence classes of pairs (E, F ) where E is a skeletally small exact category
with a projective generator P such that EndE (P ) is Morita equivalent to Λ.
(b) Preresolving subcategories of mod Λ.
(2) The bijection of (1) restricts to a bijection between the following.
(a) (proj Λ)-equivalence classes of pairs (E, F ) where E is a skeletally small idempotent
complete exact category with a projective generator P such that EndE (P ) is isomorphic to Λ.
(b) Resolving subcategories of mod Λ.
2.2.2. A characterization of mod P. In general mod P has a lot of resolving subcategories.
In this subsection, we shall characterize when P(E) and mod P coincide. As an application, we will
give a criterion for a given exact category to be equivalent to mod C for some additive category C.
fg
Let us introduce some terminologies. We say that a complex A −→ B −→ C in an additivef ◦(−)g◦(−)

category C is C(D, −)-exact for a subcategory D of C if C(D, A) −−−−→ C(D, B) −−−−→ C(D, C) isexact for all D ∈ D. Dually we define the C(−, D)-exactness in the obvious way. We say that acomplex Xn → · · · → X1 → X0 in an exact category E decomposes into conflations if there exist
a commutative diagram in E
An+1

P' P'

Xn NN
N& &

An

8 8 A2 M& M
&
qqq
/ ···
/ X1
MMM
q8 q8
&&

A1

/ X0
8qq8

8 8 A0qqq

where An+1  Xn  An , · · · , A1  X0  A0 are conflations.
Theorem 2.2.15. Let E be a skeletally small exact category.
(1) The following conditions are equivalent.
(a) There exists a skeletally small additive category C such that E is exact equivalent tomod C.
(b) E has enough projectives P, idempotent complete, and for any E(P, −)-exact complex
f3

f2

f1

· · · −→ P2 −→ P1 −→ P0
with all terms in P, the morphism f1 : P1 → P0 can be factored as a deflation
followed by an inflation.
(2) The following conditions are equivalent.
(a0 ) There exists a skeletally small additive category C with weak kernels such that E isexact equivalent to mod C.
(b0 ) E has enough projectives, idempotent complete, and any morphism in P can befactored as a deflation followed by an inflation.
Proof. (a) ⇒ (b): We may assume that E = mod C. Then mod C is idempotent complete andhas enough projectives by Proposition 2.2.6. Note that in mod C, the notion of (mod C)(proj C, −)exactness is equivalent to exactness in Mod C by the Yoneda Lemma. Suppose that there exists
an exact sequence
f3

f2

f1

· · · −→ P2 −→ P1 −→ P0
in Mod C whose all terms are in proj C. Then the cokernel of f1 is clearly in mod C. Since mod C ⊂Mod C is resolving and Coker f1 is in mod C, we must have Im fi is in mod C for all i ≥ 1. This
clearly implies that the above complex decomposes into conflations in mod C. Especially, f1 isdecomposed into P1  Im f1  P0 .
(a)0 ⇒ (b)0 : Let f : P1 → P0 be a morphism in proj C. Then Coker f is finitely presented,
hence is in mod C by Proposition 2.2.7. Since mod C is resolving in Mod C, the same argument asabove implies that P1  Im f  P0 is the desired factorization in mod C.
(b) ⇒ (a): Recall that the Morita embedding P realizes E as a resolving subcategory ofmod P by Proposition 2.2.8. We must check that P : E → mod P is essentially surjective. Let

14

2. MORITA-TYPE CHARACTERIZATIONS OF EXACT CATEGORIESX be an arbitrary object in mod P. By the definition of mod P we have an exact sequence· · · → PP2 → PP1 → PP0 → X → 0 with Pi in P for all i ≥ 0. By (b), the corresponding P1 → P0
in P factorizes into a deflation followed by an inflation in E. Then it immediately follows that thecokernel of PP1 → PP0 is in P(E), which proves that P is essentially surjective.
(b)0 ⇒ (a)0 : We only have to check that P has weak kernels. By (b)0 , for any P1 → P0 in
P there exist conflations X1  P1  X and X  P0  X 0 . Moreover since E has enoughprojectives, there exists a conflation X2  P2  X1 . It is clear that the composition P2  X1 
P1 is a weak kernel of P1 → P0 .

2.3. Exact categories with enough projectives and injectives
In this section, we study exact categories with both enough projectives and enough injectives.
We will show that an arbitrary exact category with enough projectives and injectives can berealized as a preresolving-precoresolving subcategory of the exact category XW , the exact category
associated to a Wakamatsu tilting subcategory W of a module category. We freely use results inthe previous section.
2.3.1. Wakamatsu tilting subcategories. Wakamatsu introduced a generalization of theclassical concept of tilting modules, called Wakamatsu tilting modules, which have possibly infinite
projective dimensions [Wa, MR]. They are also called semi-dualizing modules by some authors[Chr, ATY]. With a Wakamatsu tilting module W , we can associate an exact category XW withenough projectives and injectives. Typical examples are the category of Gorenstein projective
Λ-modules GP Λ and the Ext-orthogonal category ⊥ U for a cotilting module U . In what follows, we introduce a categorical analogue of Wakamatsu tilting modules, called Wakamatsu tilting
subcategories.
In this subsection, we denote by E an exact category with enough projectives. Let ⊥ W denotethe category consisting of objects X in E satisfying Ext>0
E (X, W) = 0.
For a subcategory W of E, we first introduce the following subcategory XW of E in which objects in W behaves like an injective cogenerator, see Proposition 2.3.2. We say that a subcategory
W of an exact category E is self-orthogonal if Ext>0
E (W, W) = 0.
Definition 2.3.1. Let E be an exact category with enough projectives and W an additive
subcategory of E.
(1) We denote by XW the full subcategory of E consisting of all objects X 0 ∈ ⊥ W suchthat there exist conflations X 0  W 0  X 1 , X 1  W 1  X 2 , · · · with W i ∈ W andX i ∈ ⊥ W for i ≥ 0.
(2) We say that W is a Wakamatsu tilting subcategory if it satisfies the following conditions.
(a) W is self-orthogonal.
(b) XW contains all projective objects in E.
Note that the subcategory P(E) of E consisting of all projective objects is always Wakamatsu
tilting in E. A Λ-module W for a ring Λ is said to be a Wakamatsu tilting module or a semidualizing module if add W is a Wakamatsu tilting subcategory of mod Λ. Our definition coincides
with the usual one.
Basic properties of the category XW are as follows.
Proposition 2.3.2. Let E be an exact category with enough projectives and W an additive
self-orthogonal subcategory of E.
(1) XW is closed under extensions, kernels of deflations and summands in E and it is thickin ⊥ W.
If moreover W is Wakamatsu tilting, then the following hold.
(2) XW is a resolving subcategory of E.
(3) XW has enough projectives and injectives.
(4) P in XW is projective in XW if and only if P is projective in E.
(5) I in XW is injective in XW if and only if I is in add W.

2.3. EXACT CATEGORIES WITH ENOUGH PROJECTIVES AND INJECTIVES

15

Proof. The same proof of Proposition 5.1 in [AR3] applies. For the convenience of the reader,
we shall give a proof.
(1): We first show that XW is closed under extensions. Suppose that A  B  C is a
conflation in E with A and C in XW . It is easy to see that ⊥ W is closed under extensions,thus we have B is in ⊥ W. By the definition of XW , there exist conflations A  W  A1 andC  W 0  C 1 such that W and W 0 are in W and A1 and C 1 are in XW ,Consider the pushout diagram
A /


W /

/B


/U


A1


A1

//C
//C

in E. Since C is in XW ⊂ ⊥ W, the middle row splits. Thus we may assume that U = W ⊕ C. We
then have the commutative diagram.
B /B // W ⊕C/ W ⊕ W0

/ / A1


//D


C1


C1

where the middle columns is a direct sum of W = W → 0 and C  W 0  C 1 . Since A1 and C 1
are in XW , if follows that D is an extension of objects in XW . By considering the middle row, one
may proceed this process to see that B is in XW .
We can prove that XW is closed under kernels of deflations and summands by the same
argument as the proof in Proposition 2.2.6, so we leave it to the reader.
Next we show that XW is a thick subcategory of ⊥ W. Obviously it suffices to see that XW
is closed under cokernels of inflations in ⊥ W. Let A  B  C be a conflation with all terms in⊥
W and assume that A and B are in XW . We have a conflation A  W  A0 with W in W and
A0 in XW . Then consider the following pushout diagram.
A /


W /

/B


/D


A0


A0

//C
//C

Because A0 and B are in XW and XW is closed under extensions, D must be in XW . On the other
hand, the middle row splits because C is in ⊥ W, which implies that C is a summand of D. Since
XW is closed under summands, it follows that C is in XW , which shows that XW ⊂ ⊥ W is thick.
(2)-(5): If W is in addition Wakamatsu tilting, then by definition all projective objects are in
XW , which clearly implies that XW is resolving in E. The remaining assertions are obvious fromthe definition.

2.3.2. Morita-type theorem. We assume that E is a skeletally small exact category withenough projectives P = P(E) and enough injectives I = I(E). The following result gives an
explicit description of the image of the Morita embedding functor P : E → mod P (2.2.1) in termsof a Wakamatsu tilting subcategory.
Theorem 2.3.3. Let E be a skeletally small exact category with enough projectives P andenough injectives I. For the Morita embedding P : E → mod P, we set W = P(I). Then W is a
Wakamatsu tilting subcategory of mod P, and P(E) is a preresolving-precoresolving subcategory of
XW (Definition 2.2.4). Moreover it is resolving-coresolving if and only if E is idempotent complete.

16

2. MORITA-TYPE CHARACTERIZATIONS OF EXACT CATEGORIESProof. First we see that P(E) is contained in XW . Since E has enough injectives I, for any
object X ∈ E, there exist conflations X  I 0  X 1 , X 1  I 1  X 2 , · · · with I i injective forall i. Applying P, we obtain conflations PX  PI 0  PX 1 , PX 1  PI 1  PX 2 , · · · with PI i in
W for all i. Recall that P preserves all extension groups by Proposition 2.2.1, thus P(E) ⊂ ⊥ W
holds. Therefore, the conflations show that PX is in XW .
Next we show that W is a Wakamatsu tilting subcategory of mod P. Since P(E) ⊂ XW ⊂ ⊥ W
holds, W is clearly self-orthogonal and P(P) is contained in XW . On the other hand, XW is closed
under direct sums and summands by Proposition 2.3.2. Thus it follows that proj P = add P(P) ⊂XW , which implies that W is Wakamatsu tilting.
It remains to prove that P(E) is a preresolving-precoresolving subcategory of XW . Since
P(E) ⊂ mod P and XW ⊂ mod P are extension-closed subcategories, it follows that P(E) is closed
under extensions in XW . By Proposition 2.3.2(4)(5), projective (resp. injective) objects in XW are
precisely objects in proj P (resp. add W). Thus add P(E) contains both projective and injective
objects in XW . Moreover the images of projective and injective resolutions in E under P : E → XW
yield desired conflations in Definition 2.2.4(c). Thus P(E) is preresolving-precoresolving in XW .
If E is idempotent complete, then P(E) is closed under summands in mod P, which implies
that P(E) is resolving and coresolving in XW by Lemma 2.2.5. Conversely, suppose that P(E)is resolving or coresolving in XW . Since mod P is idempotent complete and XW is closed undersummands, P(E) is clearly idempotent complete, hence so is E.
Note that in the case of Frobenius categories, Theorem 2.3.3 was shown in [Che2, Theorem
4.2]
Conversely, any preresolving-precoresolving subcategories of XW clearly have enough projectives and injectives. Hence one obtains the following classification of exact categories with enoughprojectives and injectives. First we modify the notion of the C-equivalence defined in Definition
2.2.11.
Definition 2.3.4. Suppose that E and E 0 are skeletally small exact categories with enoughprojectives and injectives. Assume that P(E) and P(E 0 ) are Morita equivalent to C via equivalences
F : proj C ' proj P(E) and F 0 : proj C ' proj P(E 0 ), and that I(E) and I(E 0 ) are Morita equivalent
to W via equivalences H : proj W ' proj I(E) and H 0 : proj W ' proj I(E 0 ) We say that pairs(E, F, H) and (E 0 , F 0 , H 0 ) are (C, W)-equivalent if there exists an exact equivalence G : E ' E 0
which makes the following diagrams commute up to natural isomorphism
proj P(E) oF'proj C/ proj P(E 0 )proj ι0proj ι


proj E

F0''
proj G/ proj E 0

proj I(E) oH'proj W/ proj I(E 0 )proj ι0proj ι


proj E

H0''
proj G/ proj E 0

where ι and ι0 are the inclusions.
Theorem 2.3.5. Let C be an additive category and W a Wakamatsu tilting subcategory in
mod C.
(1) There exists a bijection between the following two classes.
(a) (C, W)-equivalence classes of pairs (E, F, H) where E is a skeletally small exact category with enough projectives P and enough injectives I such that P is Morita
equivalent to C and I is Morita equivalent to W.
(b) Preresolving-precoresolving subcategories of XW .
Moreover, any exact categories with enough projectives and injectives occur in this way.
(2) Suppose that C and W are idempotent complete. Then the bijection of (1) restricts to abijection between the following.
(a) (C, W)-equivalence classes of pairs (E, F, H) where E is a skeletally small idempotent
complete exact category with enough projectives P and enough injectives I such that
P is equivalent to C and I is equivalent to W.
(b) Resolving-coresolving subcategories of XW

2.3. EXACT CATEGORIES WITH ENOUGH PROJECTIVES AND INJECTIVESProof. The proof of Theorem 2.2.12 applies.

17One can conclude the following results on a ring immediately.
Corollary 2.3.6. Let Λ be a ring and W a Wakamatsu tilting Λ-module.
(1) There exists a bijection between the following two classes.
(a) (proj Λ, add W )-equivalence classes of pairs (E, F, H) where E is a skeletally small
exact category with a projective generator P and an injective cogenerator I such that
EndE (P ) is Morita equivalent to Λ and EndE (I) is Morita equivalent to EndΛ (W ).
(b) Preresolving-precoresolving subcategories of XW .
Moreover, any exact categories with projective generators and injective cogenerators occur
in this way.
(2) The bijection of (1) restricts to a bijection between the following.
(a) (proj Λ, add W )-equivalence classes of pairs (E, F, H) where E is a skeletally small
idempotent complete exact category with a projective generator P and an injective
cogenerator I such that EndE (P ) is isomorphic to Λ and EndE (I) is isomorphic toEndΛ (W ).
(b) Resolving-coresolving subcategories of XW .
Next we consider the case of Frobenius categories. Recall that an exact category E is called
Frobenius if it has enough projectives and injectives, and projective objects and injective objects
coincide. A typical example of a Frobenius category is given by the following.
Definition 2.3.7. For a skeletally small additive category C, we denote by GP C the exact
subcategory XW of mod C for the Wakamatsu tilting subcategory W = proj C of mod C. Modules
in GP C are called Gorenstein projective. Similarly for a ring Λ, we denote by GP Λ the exact
subcategory XΛ of mod Λ.
It follows from Proposition 2.3.2 that GP C is a Frobenius category. We remark that different
names such as Cohen-Macaulay or totally reflexive are used for GP C.
Using the bijection of Theorem 2.3.5, one can easily show the following results about Frobenius
categories, where we use the notion of C-equivalence defined in Definition 2.2.11.
Theorem 2.3.8. Let C be a skeletally small additive category.
(1) There exists a bijection between the following two classes.
(a) C-equivalence classes of pairs (E, F ) where E is a skeletally small Frobenius category
such that P(E) is Morita equivalent to C.
(b) Preresolving-precoresolving subcategories of GP C.
(2) Suppose that C is idempotent complete. Then the bijection of (1) restricts to a bijection
between the following.
(a) C-equivalence classes of pairs (E, F ) where E is a skeletally small idempotent complete Frobenius category such that P(E) is equivalent to C.
(b) Resolving-coresolving subcategories of GP C.
Corollary 2.3.9. Let Λ be a ring.
(1) There exists a bijection between the following two classes.
(a) (proj Λ)-equivalence classes of pairs (E, F ) where E is a skeletally small Frobenius
category E with a projective generator P such that EndE (P ) is Morita equivalent toΛ.
(b) Preresolving-precoresolving subcategories of GP Λ.
(2) The bijection of (1) restricts to a bijection between the following.
(a) (proj Λ)-equivalence classes of pairs (E, F ) where E is a skeletally small idempotent
complete Frobenius category E with a projective generator P such that EndE (P ) is
isomorphic to Λ.
(b) Resolving-coresolving subcategories of GP Λ.

18

2. MORITA-TYPE CHARACTERIZATIONS OF EXACT CATEGORIES

2.3.3. A characterization of XW . Let E be a skeletally small exact category with enoughprojectives P and enough injectives I. We have the Morita embedding P : E → XW for W = P(I)by Theorem 2.3.3. The following theorem characterizes when P(E) and XW (or GP P) actuallycoincide.
Theorem 2.3.10. Let E be a skeletally small exact category. Then the following are equivalent.
(1) There exist a skeletally small additive category C and a Wakamatsu tilting subcategory
W of mod C such that E is exact equivalent to XW .
(2) E is idempotent complete and has enough projectives P and enough injectives I, and any
E(P, −)-exact and E(−, I)-exact complex· · · → P 2 → P1 → P 0 → I 0 → I 1 → I 2 → · · ·with Pi in P and I i in I for i ≥ 0 decomposes into conflations.
To prove this, we need the following technical lemma.
Lemma 2.3.11. Let C be a skeletally small additive category and W an additive self-orthogonal
subcategory of mod C. Suppose that · · · → M2 → M1 → M0 → M−1 → M−2 → · · · is a complex in⊥
W which decomposes into conflations Xi+1  Mi  Xi in mod C for all i ∈ Z. If this complex
is (mod C)(−, W)-exact, then Xi is in ⊥ W for all i ∈ Z.
Proof. Put F := mod C for simplicity. Since Mi ’s are in ⊥ W, we have an exact sequence
F(Mi , W) → F(Xi+1 , W) → Ext1F (Xi , W) → 0.
by the long exact sequence of Ext. We first show that Ext1F (Xi , W) vanishes. Consider thefollowing commutative diagram.
0/ F(Xi+1 , W)O

/ F(Mi+1 , W)

/ F(Mi+2 , W)

F(Mi , W)/ F(Mi+1 , W)

/ F(Mi+2 , W)

Since given complex is F(−, W)-exact, the bottom row is exact. One can easily show that Mi+2 →Mi+1 → Xi+1 is a cokernel diagram, which implies that the top row is also exact. Thus we
see that F(Mi , W) → F(Xi+1 , W) is surjective, which shows that Ext1F (Xi , W) = 0. Since
ExtjF (Xi , W) = Ext1F (Xi+j−1 , W) = 0 for j ≥ 1 by the dimension shift, it follows that Xi is in⊥W.

Proof of Theorem 2.3.10. (1) ⇒ (2): We may assume that E = XW , and in this case I
coincides with add W by Proposition 2.3.2(5). Suppose that
· · · → P2 → P 1 → P0 → I 0 → I 1 → I 2 → · · ·is a complex in XW satisfying the condition. Since E(P, −)-exactness is equivalent to exactness in
Mod P, we can decompose it into short exact sequences Xi+1  Pi  Xi and X i  I i  X i+1
for i ≥ 0 with X0 = X 0 . Since the X0 is obviously in mod C and mod C ⊂ Mod C is thick, all Xi
and X i are in mod C for i ≥ 0.
By Lemma 2.3.11, X i is in ⊥ W for all i ≥ 0. Thus the conflations X i  I i  X i+1 ,i+1X I i+1  X i+2 , · · · imply that X i is in XW for i ≥ 0. Since XW ⊂ mod C is resolving, Xi
is also in XW for i ≥ 0, which proves that this complex decomposes into conflations in XW .
(2) ⇒ (1): Suppose that (2) holds. Since the Morita embedding P : E → XW in Theorem
2.3.3 is fully faithful for W = P(I), it suffices to show that it is essentially surjective. Let X be
an object in XW . Since X is in mod P and in XW , there exists a complex· · · → P2 → P1 → P 0 → I 0 → I 1 → I 2 → · · ·(2.3.1)iwith Pi in P and I in I for i ≥ 0 satisfying the following properties:· · · → PP2 → PP1 → PP0 → PI 0 → PI 1 → PI 2 → · · ·ii+1(2.3.2)⊥is exact, X is the cokernel of PP1 → PP0 and all the kernels of PI → PI
are in W for i ≥ 0.
Decompose (2.3.2) into short exact sequences Xi+1  PPi  Xi and X i  PI i  X i+1 in Mod P

2.4. EXACT CATEGORIES WITH ENOUGH PROJECTIVES AND INJECTIVES AND HIGHER KERNELS 19

for i ≥ 0 with X0 = X = X 0 . Since X is in XW ⊂ mod P and mod P is thick in Mod P, it follows
that Xi and X i are in mod P for all i ≥ 0. Moreover, X i is clearly in XW for i ≥ 0. We havethat Xi is in XW for i ≥ 0 since XW is a resolving subcategory of mod P. It follows that (2.3.2) is(mod P)(P(P), −)-exact and (mod P)(−, W)-exact, which shows that (2.3.1) is E(P, −)-exact andE(−, I)-exact. Hence by (2) the complex (2.3.1) can be decomposed into conflations in E. Since
P is an exact functor, it immediately follows that X is in P(E).
In the case of Frobenius categories, this gives an internal characterization of categories ofGorenstein projective modules GP C.
Corollary 2.3.12. Let E be a skeletally small exact category. The following are equivalent.
(1) There exists a skeletally small additive category C such that E is exact equivalent to GP C.
(2) E is an idempotent complete Frobenius category, and any E(P, −)-exact and E(−, P)exact complexes· · · → P2 → P1 → P0 → P−1 → P−2 → · · ·in P decomposes into conflations in E.
Note that this class of Frobenius categories E such that the image of the Morita embedding
P : E → mod P coincides with GP P are called standard in [Che2]. The above corollary gives anintrinsic characterization of standard Frobenius categories. Also the category XW for a Wakamatsu
tilting Λ-module W and the category GP Λ for a ring Λ can be characterized by similar conditions
to Theorem 2.3.10(2) and Corollary 2.3.12(2), which we leave to the reader.
2.4. Exact categories with enough projectives and injectives and higher kernels
In this section we study the special class of Wakamatsu tilting subcategories, cotilting subcategories, and study its relationship with higher kernels. More precisely, we show that an exact
category E with enough projectives and injectives is equivalent to XW for a cotilting subcategory
W of a module category if and only if E has higher kernels.
2.4.1. Cotilting subcategories. First we introduce the notion of n-cotilting subcategories,
which is a generalization of cotilting modules, as we will see in Proposition 2.4.3 below.
Definition 2.4.1. Let E be an exact category with enough projectives and W an additive
subcategory of E. We say that W is an n-cotilting subcategory if it satisfies the following conditions.
(1) For every W ∈ W, we have id W ≤ n, that is, Ext>nE (−, W ) = 0.
(2) W is self-orthogonal, that is, Ext>0
(W,
W)
=
0.
E
(3) The categories ⊥ W and XW coincide (see Definition 2.3.1 for the category XW ).
Note that an n-cotilting subcategory is always Wakamatsu tilting.
Next we study basic properties of cotilting subcategories. For a subcategory X of an exact
category E, we denote by Xbn the subcategory of E consisting of all objects M such that there
exists a complex which decomposes into conflations in E
0 → Xn → Xn−1 → · · · → X0 → M → 0,
where Xi is in X for 0 ≤ i ≤ n. We write Xb for the subcategory of E whose objects are those in
Xbn for some n. We set Xb0 = X and Xbn = 0 for n < 0.
Proposition 2.4.2. Let E be an exact category with enough projectives and W an additive
self-orthogonal subcategory of E. For an integer n ≥ 0, the following are equivalent.
(1) W is an n-cotilting subcategory of E.
n
d
(2) X
W = E.
(3) XW contains all projective objects, and for any complex which decomposes into conflations
0 → Xn → Xn−1 → · · · → X0 → M → 0,
if Xi is in⊥

W for 0 ≤ i < n, then Xn is in XW .

20

2. MORITA-TYPE CHARACTERIZATIONS OF EXACT CATEGORIESProof. (1) ⇒ (3): All projective objects are clearly contained in ⊥ W, thus in XW . Let
0 → Xn → Xn−1 → · · · → X0 → M → 0
be a complex which decomposes into conflations with Xi is in ⊥ W for 0 ≤ i < n. The dimension
shift argument shows that ExtE>0 (Xn , W) = Ext>nE (M, W) = 0 since id W ≤ n. Hence Xn is in⊥
W = XW , which shows (3).
(3) ⇒ (2): This is clear since E has enough projectives and all projectives are in XW .
nd(2) ⇒ (1): For any M in E = X
W , there exist conflations
Xn  Xn−1  Mn−1 ,Mn−1  Xn−2  Mn−2 ,··· ,M1  X0  M(2.4.1)The dimension shift argument shows that Ext>nE (M, W) =where Xi is in XW for 0 ≤ i ≤ n.
Ext>0
E (Xn , W) = 0, which proves that id W ≤ n.
It remains to check that ⊥ W = XW . If M is in ⊥ W, then all terms in (2.4.1) are in ⊥ Wbecause ⊥ W is resolving in E. On the other hand, according to Proposition 2.3.2, the category
XW is closed under cokernels of inflations in ⊥ W. This clearly implies that Mn−1 , · · · , M1 , M are
in XW , thus we have ⊥ W = XW .
Let us describe the relation between our cotilting subcategories and classical cotilting modules.
The notion of cotilting modules over artin R-algebras is the dual notion of tilting modules, andboth are widely studied in the representation theory of algebras. Let Λ be an artin R-algebra andU a finitely generated Λ-module. Then U is called a cotilting module if it satisfies the following
conditions.
(1) id UΛ is finite.
(2) U is self-orthogonal.
\(3) DΛ belongs to add
U.
The following statement illustrates the relation between our definition and the classical one.
Proposition 2.4.3. Let Λ be an artin R-algebra and W a subcategory of mod Λ satisfying
add W = W. For an integer n ≥ 0, the following are equivalent.
(1) W is an n-cotilting subcategory of mod Λ.
(2) There exists a cotilting Λ-module U with id U ≤ n such that W = add U .
Proof. We refer the reader to [AR3, Theorem 5.4] for the proof.



2.4.2. Higher kernels and the main result. Our aim is to characterize the exact categoryof the form ⊥ W = XW for some n-cotilting subcategory W of a module category mod C. In thissubsection we assume that E is a skeletally small exact category with enough projectives P andenough injectives I. Recall that we have the Morita embedding (2.2.1) P : E → XW with W = P(I)by Theorem 2.3.3.
The following result gives a criterion for this embedding P : E → XW to be dense up tosummands. A similar method was used in the proof of [KIWY, Theorem 2.7]. We simplify theproof by using the analogue of Auslander-Buchweitz approximation for exact categories, which werefer to Appendix A.1.
Proposition 2.4.4. Let E be a skeletally small exact category with enough projectives P andenough injectives I. Consider the Morita embedding P : E → mod P and put W := P(I). Ifnd = mod P holds for some n ≥ 0, then W is an n-cotilting subcategory of mod P and we have
P(E)XW = add P(E). In this case, XW = P(E) holds if and only if E is idempotent complete.
n[ = mod P implies
Proof. Since we have P(E) ⊂ XW by Theorem 2.3.3, the equality P(E)nd
XW = mod P. Thus W is n-cotilting by Proposition 2.4.2. On the other hand, recall that P(E)is preresolving in mod P and has enough injectives W. By Proposition 2.3.2, XW is resolving in
mod P and has enough injectives add W. Applying Corollary A.1.4 to X = XW and X 0 = P(E),
we have XW = add P(E). The last assertion is clear.
2.4. EXACT CATEGORIES WITH ENOUGH PROJECTIVES AND INJECTIVES AND HIGHER KERNELS 21

To investigate further properties of cotilting subcategories, we extend the notion of n-kernels[Ja] to n ≥ −1, which simplify our results below.
Definition 2.4.5. Let C be an additive category.
(1) Let n be an integer n ≥ 1. We say that C has n-kernels if for any morphism f : X → Y in C,
there exists a complex
fn
f2
f1
f
0 → Xn −→ · · · −→ X1 −→ X −→Yin C such that the following diagram is exact.
C(−,fn )C(−,f2 )C(−,f1 )C(−,f )

0 → C(−, Xn ) −−−−−→ · · · −−−−−→ C(−, X1 ) −−−−−→ C(−, X) −−−−→ C(−, Y )(2) Suppose that C is in addition an exact category.
(a) We say that C has 0-kernels if every morphism f in C can be factored as a deflation
followed by a monomorphism, that is, f = ig holds for a deflation g and a monomorphismi.
(b) We say that C has (−1)-kernels if every morphism can be factored as a deflation followedby an inflation.
Dually we define the notion of n-cokernels for n ≥ −1.
Note that having n-kernels implies having m-kernels for m ≥ n ≥ −1. Also we point out thatan exact category C has (−1)-kernels if and only if C is abelian and its exact structure is the usual
exact structure on abelian categories.
We remark that having higher kernels is almost equivalent to the finiteness of the global
dimension, as the following classical proposition shows.
Proposition 2.4.6. Let C be a skeletally small idempotent complete additive category withweak kernels and n an integer n ≥ 1. Then the following hold.
(1) C has n-kernels if and only if the global dimension of mod C is at most n + 1.
(2) Suppose that EndC (M ) is right coherent for an object M ∈ C. Then add M has n-kernels
if and only if the right global dimension of EndC (M ) is at most n + 1.
The following proposition gives a necessary condition for E to be equivalent to XW for ann-cotilting subcategory W of a module category mod C.
Proposition 2.4.7. Let C be a skeletally small additive category with weak kernels and W an
n-cotilting subcategory of mod C for n ≥ 0. Then XW has (n − 1)-kernels.
Proof. Recall that XW = ⊥ W since W is n-cotilting. Also note that mod C is abelian by
Proposition 2.2.7 since C has weak kernels. If n = 0, that is, XW = mod C, then XW clearly has(−1)-kernels since mod C is abelian.
Let f : X → Y be a morphism in XW . Then we have conflations M1  X  M0 andM0  Y  M−1 in mod C such that f is the composition X  M0  Y . Suppose that n = 1. It>1⊥follows that Ext>0
mod C (M0 , W) = Extmod C (M−1 , W) = 0 since id W ≤ 1. Thus M0 is in W = XW .
⊥⊥Since W is resolving in mod C, it follows that M1 is also in W. Consequently X  M0 is a
deflation in ⊥ W, which shows that XW has 0-kernels.
Next let us consider the case n ≥ 2. Note that XW is contravariantly finite in mod C by
Corollary A.1.5. This gives a right XW -approximation X1 → M1 . Since XW contains all projective
objects, this morphism is an epimorphism. Hence we obtain a conflation M2  X1  M1 in mod C.
Repeat this construction until we get Mn−1  Xn−2  Mn−2 . (If n = 2, we interpret X0 = X.)>n
By the dimension shift argument, we have Ext>0
mod C (Mn−1 , W) = · · · = Extmod C (M−1 , W) = 0,
which implies Mn−1 ∈ XW . Consider the complex
0 → Mn−1 → Xn−2 → · · · → X1 → X → Y
in XW . Since the morphism Xi  Mi in each conflation Mi+1  Xi  Mi is a right XW approximation for i ≥ 1, it is easy to see that
0 → XW (−, Mn−1 ) → · · · → XW (−, X1 ) → XW (−, X) → XW (−, Y )is exact. Hence XW has (n − 1)-kernels.

22

2. MORITA-TYPE CHARACTERIZATIONS OF EXACT CATEGORIES

Surprisingly, having n-kernels is not only necessary but also sufficient for an exact category
to be equivalent to XW for an n-cotilting subcategory W of mod C, as the following propositionshows. This was proved in [KIWY, Theorem 2.7] for the case of Frobenius categories.
Proposition 2.4.8. Let E be a skeletally small exact category with enough projectives P andenough injectives I. Consider the Morita embedding (2.2.1) P : E → mod P. Suppose that there
exists a subcategory M of E which contains P and has (n−1)-kernels for n ≥ 0. Then the following
hold.
(1) W := P(I) is an n-cotilting subcategory of mod P.
(2) The equality add P(E) = XW holds. Moreover P(E) = XW if and only if E is idempotent
complete.
(3) P has weak kernels.
Proof. Since M has weak kernels and every object in M has a deflation from some object
in P, it is easy to see that P has weak kernels. Thus (3) holds.
n[ = mod P to prove (1) and (2). Let X be
By Proposition 2.4.4, we only have to show P(E)any object in mod P. Fix a morphism P1 → P0 in P such that PP1 → PP0 → X → 0 is exact.
First suppose that n ≥ 2. Then by taking the (n − 1)-kernel of P1 → P0 inside M, we get acomplex
0 → M n → · · · → M 2 → P1 → P 0
in M. Since M contains P, it easily follows that
0 → PMn → · · · → PP1 → PP0 → X → 0n[ = mod P.
is a complex which decomposes into conflations in mod P. This gives P(E)The case n = 0 and 1 are quite similar and left to the reader.

This immediately gives the following characterization of cotilting subcategories of a module
category mod C amongst Wakamatsu tilting subcategories.
Corollary 2.4.9. Let C be a skeletally small additive category and W a Wakamatsu tilting
subcategory of mod C. Then the following are equivalent for n ≥ 0.
(1) C has weak kernels and W is n-cotilting.
(2) XW has (n − 1)-kernels.
(3) XW has a subcategory M which contains all projective objects and has (n − 1)-kernels.
Proof. (1) ⇒ (2): This is Proposition 2.4.7.
(2) ⇒ (3): This is trivial.
(3) ⇒ (1): This is obvious from Proposition 2.4.8, since the Morita embedding P can be
identified with the natural inclusion XW → mod C in this case.

As an application, we immediately get the following information about global dimensions.
Corollary 2.4.10. Let Λ be an artin R-algebra and W a Wakamatsu tilting Λ-module which
is not a cotilting module. Then for any M ∈ XW such that Λ ∈ add M , the global dimension ofEndΛ (M ) is infinite.
Proof. Let W be a Wakamatsu tilting Λ-module and suppose that there exists a Λ-module
M ∈ XW such that Λ ∈ add M and the global dimension of EndΛ (M ) is finite. Then by Proposition
2.4.6, add M has n-kernels for some n ≥ 0. Applying Corollary 2.4.9 to M = add M , we have that
M must be a cotilting module.

Summarizing these results, we obtain the internal characterizations of exact categories associated with cotilting subcategories.
Theorem 2.4.11. Let E be a skeletally small exact category. For an integer n ≥ 0, thefollowing are equivalent.
(1) There exist a skeletally small additive category C with weak kernels and an n-cotilting
subcategory W of mod C such that E is exact equivalent to XW .

2.5. APPLICATIONS TO ARTIN R-ALGEBRAS

23(2) E is idempotent complete, has enough projectives and injectives and has (n − 1)-kernels.
(3) E is idempotent complete, has enough projectives and injectives and E has a subcategory
M such that M contains P(E) and M has (n − 1)-kernels.
By restricting our attention to artin R-algebras, we have the following criterion.
Corollary 2.4.12. Let E be a skeletally small Hom-finite exact R-category. For an integern ≥ 0, the following are equivalent.
(1) There exist an artin R-algebra Λ and a cotilting Λ-module U with id U ≤ n such that E
is exact equivalent to ⊥ U .
(2) E is idempotent complete, has a projective generator P and has enough injectives, andhas (n − 1)-kernels.
(3) E is idempotent complete, has a projective generator P and has enough injectives, and E
has a subcategory M such that M contains P and M has (n − 1)-kernels.
(4) E op satisfies one of the conditions (1)-(3).
Proof. The conditions (1)-(3) are equivalent by Theorem 2.4.11 since the notion of n-cotilting
subcategories in mod Λ coincides with usual cotilting modules by Proposition 2.4.3 for an artin
R-algebra Λ. It remains to show (4). It suffices to check that (1) is self-dual. Suppose that E = ⊥ U
for a cotilting Λ-module with id U ≤ n. The Brenner-Butler theorem implies that Γ U is a cotilting
Γ-module for Γ := EndΛ (U ) and HomΛ (−, U ) gives an exact duality E = ⊥ (UΛ ) → ⊥ (Γ U ). Thus
E op is exact equivalent to ⊥ (Γ U ), so E op satisfies (1).

Next we specialize these results to Frobenius categories. Recall that a two-sided noetherianring Λ is called Iwanaga-Gorenstein if the right and left injective dimensions of Λ are finite. It
was shown in [Za, Lemma A] that id Λ Λ = id ΛΛ holds for an Iwanaga-Gorenstein ring Λ. We call
Λ an n-Iwanaga-Gorenstein if id ΛΛ ≤ n. If Λ is an artin R-algebra, then Λ is Iwanaga-Gorenstein[
if and only if ΛΛ is a cotilting module, or GP
Λ = mod Λ (see [AR3, Proposition 6.1] or [AR2,Proposition 4.2]).
We have the following characterization of GP Λ for an Iwanaga-Gorenstein ring Λ. Note that
this result was mentioned in [Ka, Proposition 4].
Corollary 2.4.13. Let E be a skeletally small exact category. For an integer n ≥ 0, thefollowing are equivalent.
(1) There exists an n-Iwanaga-Gorenstein ring Λ such that E is exact equivalent to GP Λ.
(2) E is idempotent complete and Frobenius, has a projective generator P such that EndE (M )is noetherian, and has both (n − 1)-kernels and (n − 1)-cokernels.
(3) E is idempotent complete and Frobenius, has a projective generator P such that EndE (P )is noetherian, and has a subcategory M such that M contains P and M has both (n−1)kernels and (n − 1)-cokernels.
Proof. (1) ⇒ (2): We may assume that E = GP Λ for an Iwanaga-Gorenstein ring Λ. By
Theorem 2.4.11, E has (n − 1)-kernels. To show that E has (n − 1)-cokernels, we only have to notethat HomΛ (−, Λ) : GP Λ → GP Λop gives a duality.
(2) ⇒ (3): This is trivial.
(3) ⇒ (1): Theorem 2.4.11 gives id ΛΛ ≤ n and E ' GP Λ. Since (3) is self-dual, we haveid Λ Λ ≤ n, which implies Λ is n-Iwanaga-Gorenstein.

2.5. Applications to artin R-algebras
In the previous section, we gave a necessary and sufficient condition for an exact category E
to be exact equivalent to ⊥ U for some cotilting module U . As we have seen in Theorem 2.4.11,
it suffices to check that E has enough projectives and injectives, and has higher kernels. In thissection, we apply this criterion to the representation theory of artin algebras. We always denoteby Λ an artin R-algebra in this section.

24

2. MORITA-TYPE CHARACTERIZATIONS OF EXACT CATEGORIES

2.5.1. The category (mod Λ)/[Sub M ] as a torsionfree class. Recall that a subcategory
F of mod Λ is said to be a torsionfree class if F is closed under extensions and submodules. It isclassical that for a cotilting module U ∈ mod Λ with id U ≤ 1, the subcategory ⊥ U of mod Λ is atorsionfree class, see [ASS].
To state our main theorem in this subsection, let us recall the definition of ideal quotients ofadditive categories. Let C be an additive category and D an additive subcategory of C. Denote by[D] the ideal of C consisting of all morphisms in C which factor through some objects in D. Thenthe ideal quotient C/[D] is an additive category whose objects are those in C and whose morphisms
are given by(C/[D])(M, N ) := C(M, N )/[D](M, N )for M and N in C. Also see Definition A.1.1 for the notion of functorially finiteness.
Theorem 2.5.1. Let Λ be an artin R-algebra and C a functorially finite subcategory of mod Λ
which is closed under submodules. Put E := mod Λ and denote by π : E  E/[C] the natural
quotient functor. Then the following hold.
(1) There exists an exact structure on E/[C] induced by C-conflations (see Appendix A.2 fordetails) such that E/[C] has 0-kernels.
(2) The exact category E/[C] has enough projectives P and injectives I.
(3) P (resp. I) is precisely the essential image of add(Λ, τ − C) (resp. add(DΛ, τ C)) under π.
(4) P is of finite type if and only if I is of finite type.
(5) Suppose that (4) holds. Let M (resp. N ) be an object satisfying P = add M (resp.
I = add N ) and put Γ := EndE/[C] (M ). Then (E/[C])(M, N ) is a 1-cotilting Γ-moduleand E/[C] is exact equivalent to a torsionfree class ⊥ U .
We omit the dual result for the case when C is closed under quotients. Note that this wasproved in [Iy3, Proposition 5.5] in case C = mod(Λ/I) for some two-sided ideal I and (mod Λ)/[C]is of finite type.
Remark 2.5.2. Let C be a subcategory of mod Λ which is closed under submodules. By [ASm,
Proposition 4.7], the following are equivalent.
(1) C is functorially finite.
(2) C is contravariantly finite.
(3) C = Sub M for some M ∈ mod Λ.
Here Sub M is the subcategory consisting of all modules cogenerated by M , in other words, thesmallest additive subcategory containing M which is closed under submodules.
To begin the proof, the following elementary observation is useful. In what follows, we denoteby π(f ) : L → M in E/[C] the image of f : L → M in E under the natural functor π : E  E/[C].
Proposition 2.5.3. Let Λ be a right noetherian ring. The following are equivalent for an
additive subcategory C of mod Λ.
(1) C is closed under quotient modules.
(2) For every X in mod Λ, there exists a right C-approximation CX  X which is an injection.
(3) C is contravariantly finite, and epimorphisms are preserved by the natural functor π :mod Λ  (mod Λ)/[C].
L
Proof. (1) ⇒ (2): Let X be in mod Λ. Consider the direct sum C,f C → X where C runs
over isomorphism classes of objects in C and f runs over all morphisms f : C → X. Write CX
for its image. Since Λ is assumed to be right noetherian, CX is a quotient module of finite direct
sum of objects in C, which implies that CX is also in C. It is easy to see that CX  X is a rightC-approximation.
fg(2) ⇒ (3): Contravariantly finiteness is obvious. Let 0 → L −→M −→ N → 0 be a short exact
sequence in mod Λ. We will see that π(g) is epic. So let ϕ : N → X be a morphism in mod Λ
such that π(ϕg) = π(ϕ)π(g) = 0 in (mod Λ)/[C]. By the definition of the quotient category, itfollows that ϕg factors through some object in C. It follows that ϕg factors through the right2.5. APPLICATIONS TO ARTIN R-ALGEBRAS

25

C-approximation i : CX  X. Thus there exists a morphism h : M → CX which makes thefollowing diagram commute.
0

/L

f

/M
h



CX

g

/N

/0

ϕ

/ i


/X

Since ihf = ϕgf = 0 and i is monic, we have hf = 0. Thus there exists a morphism k : N → CX
such that h = kg. Then ikg = ih = ϕg holds, which implies that ik = ϕ since g is epic. Therefore
π(ϕ) = 0 holds in E/[C], which proves that π(g) is epic.
(3) ⇒ (1): Suppose that C is in C and f : C  X is a surjection. It follows from (3) that
π(f ) : C → X is epimorphism in E/[C]. However C is a zero object in E/[C], thus X ∼= 0, that is,
X ∈ C.

Note that if Λ is a right artinian ring, then the dual of Proposition 2.5.3 holds.
To prove Theorem 2.5.1, we have to introduce a new exact structure given by C-conflations
on mod Λ, which induces the desired exact structure on (mod Λ)/[C]. For the notion of (−, C)conflations, (C, −)-conflations and C-conflations, we refer the reader to Appendix A.2.
Now let us prove Theorem 2.5.1. We divide the proof into several propositions. For the rest of
this subsection, we always assume that Λ is an artin R-algebra and that C is a functorially finite
subcategory of E := mod Λ which is closed under submodules.
Proposition 2.5.4. Endow E with an exact structure EC . Then E/[C] naturally inherits anexact structure from EC as in Proposition A.2.3.
Proof. Denote by π : E  E/[C] the natural functor. According to Proposition A.2.3, it
suffices to show that the image of every C-inflation under π is a monomorphism and the imageof every C-deflation is an epimorphism. The assertion for C-inflations follows from the dual of
Proposition 2.5.3.
Let 0 → L → M → N → 0 be a C-conflation. We show that M → N is an epimorphism
in E/[C]. Suppose that N → X is a morphism in E such that M → N → X is zero. Then thecomposition M → N → X factors through the left C-approximation M → C M . This yields thefollowing commutative diagram
/L
/M
/N
/0
0
0


/K


/ CM


/X

with exact rows. Since C is closed under submodules, K is in C. Because L  M  N is (−, C)conflation, L → K factors through L → M . Then it is routine to check that N → X factors
through C M → X, thus N → X is zero in E/[C] as desired.
Proposition 2.5.5. The exact category E/[C] has 0-kernels.
Proof. We have to show that every morphism π(f ) : X → Y in E/[C] can be factored as a
C-deflation followed by a monomorphism.
Let f : X → Y be a morphism in E = mod Λ. Then we have the factorization X  Im f  Y .
Note that Im f → Y is monic by the dual of Proposition 2.5.3. Thus it suffices to show that
X → Im f can be factored as a C-deflation followed by a monomorphism. Therefore we may
assume that f is a surjection.
Let CY → Y be a right C-approximation. Consider the pullback diagram
/K
/E
/ CY
/0
0
0

/K


/X


/Y

in E. Then the right square gives a short exact sequence
0 → E → X ⊕ CY → Y → 0

/0

26

2. MORITA-TYPE CHARACTERIZATIONS OF EXACT CATEGORIES

in E. Since CY → Y is a right C-approximation, clearly this sequence is a (C, −)-conflation. To
transform it into a C-conflation, take a left C-approximation E  C E and consider the pushout0/E/ X ⊕ CY ca

/Y

/0

0


/ CE


/F

/Y

/0

b

(2.5.1)acin E. The left square gives a short exact sequence
0 → E → C E ⊕ X ⊕ CY → F → 0(2.5.2)in E, which is a (−, C)-conflation since E  C E is a left C-approximation. Since ca is a (C, −)deflation, a diagram chase for (2.5.1) shows that (2.5.2) is also a (C, −)-conflation. Thus (2.5.2)
is a C-conflation. Because C E and CY is zero in E/[C], we obtain a deflation π(g) : X  F in
EC /[C], where we write a = [g, h].
By the commutativity of the right square in (2.5.1), we have π(f ) = π(c)π(g). Since π(g) is a
deflation, it suffices to show that π(c) : F → Y is monic in E/[C].
Let ϕ : Z → F be a morphism such that the composition cϕ factors through some objects in
C. Since CY → Y is a right C-approximation, it follows that cϕ factors through CY → Y . Thus
we have a map d : Z → CY such that cϕ = ca ◦ t [0, d]. This implies that there exists a map
e : Z → C E such that ϕ − a ◦ t [0, d] = be, hence ϕ = a ◦ t [0, d] + be. Since the images of t [0, d] andb under π are zero in E/[C], it follows that π(ϕ) = 0 in E/[C], which shows that π(c) is monic. 
Now we are in position to finish the proof of Theorem 2.5.1.
Proof of Theorem 2.5.1. (1): The assertion follows from Proposition 2.5.4 and 2.5.5.
(2): We basically follow the idea of [ASo1]. It is well-known that a short exact sequence
0 → L → M → N → 0 is (−, X)-exact if and only if (τ − X, −)-exact (see, e.g. [ARS, Corollary4.4]). Hence this sequence is C-exact if and only if it is (add(C, τ − C), −)-exact. It was shown in[ASo1, Theorem 1.14] that this exact structure has enough projectives and injectives if and only if
add(C, τ − C) is functorially finite. Since C is functorially finite in mod Λ, it follows that so is τ − C.
Thus it is clear that add(C, τ − C) is functorially finite. Therefore EC has enough projectives andinjectives. By Theorem A.2.3, EC /[C] also has enough projectives and injectives, thus (2) holds.
(3): Note that projective (resp. injective) objects in EC are precisely objects in add(Λ, C, τ − C)(resp. add(DΛ, C, τ C)). Thus Theorem A.2.3 immediately gives (3).
(4): This is clear from (3).
(5): By using the Morita embedding HomE/[C] (M, −) : E/[C] → mod Γ, the assertion is immediate from Corollary 2.4.12.

As an immediate consequence, we have the following.
Corollary 2.5.6. Let Λ be an artin R-algebra and S a semisimple module in mod Λ. Put
E := (mod Λ)/[add S] and Γ := EndE (Λ ⊕ τ − S). Then U := E(Λ ⊕ τ − S, DΛ ⊕ τ S) ∈ mod Γ is a
1-cotilting module and we have an equivalenceE(Λ ⊕ τ − S, −) : E ' ⊥ U.

(2.5.3)If S is simple projective and not injective, then E is shown to be equivalent to the subcategoryof mod Λ consisting of modules M such that HomΛ (M, S) = 0. In this case, Γ is an APR tilt ofΛ at S and U is the corresponding APR cotilting Γ-module (see [ASS, Example 2.8(c)]) and theabove equivalence (2.5.3) coincides with the classical equivalence [ASS, Theorem 3.8].
Historically, the notion of APR tilts was generalized to tilting modules, which induce anembedding of a subcategory of mod Λ into another module category by the Brenner-Butler theorem.
On the other hand, this corollary (and our main theorem) gives an embedding of a certain quotientcategory of mod Λ into another module category. Thus our results can be regarded as anothergeneralization of APR tilts.

2.5. APPLICATIONS TO ARTIN R-ALGEBRAS

27

Example 2.5.7. Let Λ be a self-injective Nakayama algebra of Loewy length four given by
the cyclic quiver with three vertices over a field k. Let S be a simple module corresponding thewhite dot in the following Auslander-Reiten quiver of mod Λ
3

1

2

4
where we identify two vertical arrows. Then obviously C := add S is closed under submodules.
The full translation subquiver consisting of the shaded areas gives the Auslander-Reiten quiver of(mod Λ)/[C].
Then M in Theorem 2.5.1 corresponds to the numbered vertices and Γ is given by the quiver
1 oFF ε 2O h
FF
FF γ
δ F"
3

α

(

4

β

and relation βα = γα = δγε = βε = αβ − εδγ = 0. The following diagram gives the AuslanderReiten quiver of mod Γ
3124
where we identify two vertical arrows. The shaded area corresponds to the subcategory ⊥ U ofmod Γ in Theorem 2.5.1. The above two shaded regions illustrate the equivalence (mod Λ)/[C] '⊥U.
Remark 2.5.8. We point out that the assumption C is closed under submodules in Theorem
2.5.1 cannot be replaced by C is closed under images. The simple example is as follows. Let Λ
be a path algebra of the quiver 1 ← 2 ← 3 ← 4 ← 5 over a field k. Put M := P (4)/S(1) where
P (4) is the indecomposable projective module corresponding to 4 and S(1) is the simple module
corresponding to 1. Then C := add M is easily checked to be closed under images. On the other
hand, one can easily check that (mod Λ)/[C] does not have 0-kernels, e.g. by using Proposition
2.4.6.
2.5.2. The category mod Λ as an exact category. We assume that Λ is an artin Ralgebra. The category mod Λ has various exact structures given in Definition A.2.1. An explicit
classification was given in [Bua, Proposition 3.3.2].
It is natural to ask which exact structure has enough projectives or which has a projective
generator. The answer of this was essentially given in [ASo1] in terms of relative homologicalalgebra. We call a subcategory M of mod Λ a generating subcategory if proj Λ ⊂ M = add M
holds. Dually we define a cogenerating subcategory. A Λ-module M is called a generator (resp.
cogenerator ) if add M is generating (resp. cogenerating).
Proposition 2.5.9. Let Λ be an artin R-algebra.
(1) There exists a bijection between exact structures on mod Λ with enough projectives andcontravariantly finite generating subcategories of mod Λ. It is given by sending an exact
structure on mod Λ to the category of all projective objects, and the inverse map is given
by sending a generating subcategory M to (mod Λ)(M,−) .
(2) The exact structure on mod Λ has a projective generator if and only if it has an injective
cogenerator. If G is a projective generator in this exact structure, then C := DΛ ⊕ τ M
is an injective cogenerator.

28

2. MORITA-TYPE CHARACTERIZATIONS OF EXACT CATEGORIES(3) The bijection of (1) restricts to a bijection between exact structures with projective generators and isomorphism classes of basic generators of mod Λ.
Proof. This follows directly from results in [ASo1, Theorem 1.15] and [DRSS, Proposition
1.7] (or Theorem A.2.2).
Hence the nontrivial generators yield nontrivial embeddings of mod Λ into another module
category. Our main results in this direction is the following. This theorem was essentially provedin [ASo2, Proposition 3.26] by using the method of relative cotilting theory. We give an alternative
proof of this theorem by using our results.
Theorem 2.5.10. Let Λ be an artin R-algebra and G be a generator of mod Λ. Put C :=DΛ ⊕ τ G, Γ := EndΛ (G) and U := HomΛ (G, C) ∈ mod Γ. Then the following hold.
(1) U is a cotilting Γ-module with id U = 2 or 0. The case id U = 0 occurs only when G is
projective.
(2) HomΛ (G, −) : mod Λ → mod Γ induces an equivalence mod Λ ' ⊥ U .
(3) mod Λ admits an exact structure such that projective objects are precisely objects in add G
and the equivalence mod Λ ' ⊥ U is an exact equivalence.
(4) EndΛ (G) and EndΛ (C) are derived equivalent.
We need the following preparation.
Lemma 2.5.11. Let A be an abelian category. Suppose that A is endowed with some exact
structure, which we denote by AE . If AE has 0-kernels, then it coincides with the standard exact
structure on A.
Proof. It suffices to show that any epimorphism in AE is a deflation. Let f : Y → Z be an
epimorphism. Since AE has 0-kernels, there is a factorization f = ig such that g is a deflationand i is a monomorphism. Since f is an epimorphism, so is i. Thus i is an isomorphism since A
is abelian. Therefore f is a deflation.
Proof of Theorem 2.5.10. Let G be a generator and put E := mod Λ. Then by Theorem
2.5.9, E(G,−) has a projective generator G and an injective cogenerator C.
To use Corollary 2.4.12, we observe that E(G,−) has 1-kernels, which is immediate since mod Λ
has kernels. Hence (2)-(4) hold by Corollary 2.4.12. The remaining statement of (1) easily follows
from Lemma 2.5.11.
As an application of Theorem 2.5.10, we obtain the following result, which says that every
module category of an artin R-algebra sits inside the module category of an artin R-algebra withfinite global dimension, with a little modification of its exact structure.
Corollary 2.5.12. Let Λ be an artin R-algebra. Then there exist an artin R-algebra Γ withfinite global dimension and a cotilting Γ-module U with id U = 2 or 0 such that mod Λ ' ⊥ U .
Proof. There exists a generator G of mod Λ whose endomorphism ring has a finite global
dimension, see [Au2, Section 3]. Thus the assertion follows from Theorem 2.5.10.

Example 2.5.13. Let Λ be a path algebra of the quiver 1 ← 2 ← 3 ← 4 over a field k. The
following diagram is the Auslander-Reiten quiver of mod Λ.

Let G be a generator of mod Λ corresponding to the circles. Then the associated cogenerator C is
the module indicated by the rectangles. Then Γ := EndΛ (G) is given by the quiver
1o
2o
3O o
4
5

2.5. APPLICATIONS TO ARTIN R-ALGEBRAS

29

where the dotted line indicates the zero relation. The Auslander-Reiten quiver of mod Γ is givenby the following diagram.

The shaded area corresponds to the essential image of the embedding mod Λ → mod Γ in Theorem
2.5.10. We keep the shapes of vertices in the quiver. In particular, the direct sum of all rectangles
gives the 2-cotilting module U and mod Λ is equivalent to ⊥ U .

CHAPTER 3

Classifications of exact structures and CM-finite algebras
In this chapter, we deal with Problem B in the first chapter. More precisely, we give a classification of all exact structures on a given idempotent complete additive category. Using this, weinvestigate the structure of an exact category with finitely many indecomposables. We show thatthe relation of the Grothendieck group of such a category is generated by AR conflations. Moreover, we obtain an explicit classification of (1) Gorenstein-projective-finite Iwanaga-Gorenstein
algebras, (2) Cohen-Macaulay-finite orders, and more generally, (3) cotilting modules U with ⊥ U
of finite type.
3.1. Introduction
In the representation theory of finite-dimensional algebras, one of the most important subjects
is to classify certain categories of finite type. Here we say that an additive k-category over a field
k is of finite type if it has only finitely many indecomposable objects up to isomorphism. The aimof this chapter is to give a classification of exact categories of finite type, and thereby provide an
explicit classification of all GP-finite Iwanaga-Gorenstein algebras. Let us explain the motivation
for this.
First we recall how categories of finite type have been studied in the representation theory.
It is well-known that an abelian Hom-finite k-category of finite type is nothing but the categorymod Λ of finitely generated Λ-modules over some representation-finite k-algebra Λ (see [Ga2, 8.2]
or Proposition 3.3.14 below). A classification of such algebras is one of the main problems in
the representation theory of algebras, and has been studied widely by a number of papers, e.g.
[Ga1, Ri, BGRS, GR]. For the case of representation-finite R-orders over a noetherian local ringR, we refer the reader to [Ar, DK, HN, LW, RVdB2, Yo]. Besides abelian categories, triangulated
categories of finite type also has been investigated, e.g. in [Am]. Such triangulated categories
naturally arise in the representation of algebras and in the categorification of cluster algebras.
Among other things, the observation by Auslander [Au2] is of particular importance to us when
we deal with categories of finite type. Let E be a Hom-finite k-category of finite type and consider
the algebra Γ := EndE (M ), where M is a direct sum of all non-isomorphic indecomposables in E.
This Γ is called an Auslander algebra of E, and categorical properties of E should be related tohomological properties of Γ. For example, the condition E being abelian is equivalent to a certain
homological condition of Γ, that is, gl.dim Γ ≤ 2 ≤ dom.dim Γ. This is called the Auslandercorrespondence, and is now the basic and important viewpoint in the representation theory.
However, most of the algebras are representation-wild, and it is hopeless to understand thewhole structure of the module category. Thus nice subcategories of module categories has attracted
much attention. For example, in the representation theory of a commutative noetherian local ringR, the category CM R of Cohen-Macaulay modules has played a central role. For an IwanagaGorenstein ring (two-sided noetherian ring satisfying id(ΛΛ ) = id(Λ Λ) < ∞), we have the natural
notion of Cohen-Macaulay Λ-modules (which we call Gorenstein-projective Λ-modules to avoid
any confusion). In the spirit of Auslander’s observation, it is natural to ask whether there exists
an Auslander-type correspondence for GP-finite Iwanaga-Gorenstein algebras (algebras Λ suchthat GP Λ is of finite type). In this thesis, we give an answer to this problem.
It is well-known that an algebra Γ is the GP-Auslander-algebra of some Iwanaga-Gorenstein
algebra if and only if the global dimension of Γ is finite. In Theorem H, we classify GP-finite
Iwanaga-Gorenstein algebras by algebras with finite global dimension together with their modules
satisfying a certain condition, given as follows.
30

3.1. INTRODUCTION

31

Definition 3.1.1. Let Γ be a two-sided noetherian ring and S a simple right Γ-module. We
say that S satisfies the 2-regular condition if the following conditions are satisfied.
(1) pd SΓ = 2.
(2) ExtiΓ (S, Γ) = 0 for i = 0, 1.
(3) Ext2Γ (S, Γ) is a simple left Γ-module.
This condition is satisfied by the simple modules over commutative regular local rings ofdimension 2, and their non-commutative analogues, e.g. Artin-Schelter regular algebras, CalabiYau algebras, non-singular orders of dimension 2.
We interpret the 2-regular condition in terms of the translation quiver Q(Γ). This is the usual
quiver of Γ together with dotted arrows corresponding to the simple Γ-modules with the 2-regularcondition (Definition 3.3.9). Using these concepts, we obtain the following classification, where aΓ-module is called basic if its indecomposable direct summands are pairwise non-isomorphic.
Theorem H (= Corollary 3.4.9). There exists a bijection between the following for a field k.
(1) Morita equivalence classes of GP-finite Iwanaga-Gorenstein finite-dimensional k-algebrasΛ.
(2) Equivalence classes of pairs (Γ, M ), where Γ is a finite-dimensional k-algebra with finite
global dimension and M is a basic semisimple Γ-module such that every simple summandof M satisfies the 2-regular condition and Homk (M, k) ∼= Ext2Γ (M, Γ) holds as left Γmodules.
(3) Equivalence classes of pairs (Γ, X), where Γ is a finite-dimensional k-algebra with finite
global dimension and X is a union of stable τ -orbits in Q(Γ).
For a pair (Γ, X) in (3), the corresponding algebra Λ in (1) is given by the endomorphism ring of
projective Γ-modules which does not belong to τ -orbits in X.
As Example 3.1.2 illustrates, it provides a systematic method to construct a family of GP-finite
Iwanaga-Gorenstein algebras.
To prove Theorem H, we need the notion of exact categories, which was introduced by Quillen[Qu] as a generalization of abelian categories. It provides an appropriate framework for a relativehomological algebra and has a number of applications in many branches of mathematics, such as
representation theory, algebraic topology and functional analysis. Let us explain why the use ofexact structures is essential in our classification. In the classical situation, we can recover Λ (up toMorita equivalence) only from the additive structure of mod Λ, while this fails to be true for GP Λ;
it often happens that GP Λ ' GP Γ while Λ and Γ are not Morita equivalent. Even in this case, Λ
can be recovered from the exact structure of GP Λ, as the endomorphism ring of the progenerator
of it. In Theorem H, the algebra Γ gives the additive structure of GP Λ, while the module M or
the set X gives the exact structure on it. In this setting, the quiver Q(Γ) with dotted arrow X is
nothing but the Auslander-Reiten quiver of GP Λ.
Example 3.1.2. Let Γ be the algebra given by the quiver in Figure 1, where we identify two
vertical arrows, with commutativity and zero relations indicated by dotted lines. Then Γ has
finite global dimension and the translation quiver Q(Γ) is the same as Figure 1. It has two stableτ -orbits A and B, hence we obtain four GP-finite Iwanaga-Gorenstein algebras, corresponding to
the endomorphism rings of vertices which does not belong to X. Table 1 is an explicit calculationof all GP-finite Iwanaga-Gorenstein algebras Λ such that GP Λ is equivalent to proj Γ.
2(stable τ -orbit A)(stable τ -orbit B)(non-stable τ -orbit C)

1

6
4

3

10
8

7
5

1
11

9

Figure 1. The quiver of Γ (or equivalently, proj Γ).

32

3. CLASSIFICATIONS OF EXACT STRUCTURES AND CM-FINITE ALGEBRASX∅

Quivers for ΛΓ in Figure 1
a

d

A∪B

9

2
e

6 b
c

Relations for Λ
The same relation as Γ.
cba = ed,bac = gf,
f b = ae = acb = dc = f g = cg = de = 0.

f

10

5
g

Obvious commutative relations, and the zero relations of
5 → 7 →9,A
two paths of length two from 11,3
7
113
two paths of length two to 3.
5
9
Obvious commutative relations, and the zero relations of
5 → 8 →10 → 1,2
6
10
1 → 2 →4 → 9,
1
4
8
1
B
two paths of length three and four from 8 to 5 and 4,
5
9
two paths of length two and three from 9 to 5 and 4.
Table 1. All CM-finite Iwanaga-Gorenstein algebras with its CM category proj Γ.
2

6

10

2

To deal with exact structures, we mainly work with idempotent complete additive categories
instead of categories of finite type. For such a category E, we classify all exact structures on E
in terms of its functor category mod E. The precise statement is the following, where C2 (E) is thecategory of E-modules whose projective dimension and grade are equal to 2.
Theorem I (= Theorem 3.2.7). Let E be an idempotent complete additive category. Then
there exists a bijection between the following two classes.
(1) Exact structures on E.
(2) Subcategories D of C2 (E) satisfying the following conditions.
(a) D is a Serre subcategory of mod E.
(b) Ext2E (D, E) is a Serre subcategory of mod E op .
It is surprising to us that such a general description of all exact structures is available. For
example, the existence theorem of the largest exact structure due to [Ru2] easily follows from
Theorem I in the case of idempotent complete categories.
We apply Theorem I to exact categories of finite type, which is our motivating object. When
E is a Hom-finite idempotent complete k-category of finite type, we show that exact structures on
E bijectively correspond to sets of simple Γ-modules satisfying the 2-regular condition, and sets ofdotted arrows of Q(Γ), where Γ is the Auslander algebra of E (Theorem 3.3.7 and Corollary 3.3.10).
We apply this result to obtain Theorem H, and more generally, an Auslander correspondence forcotilting modules U such that ⊥ U is of finite type (Theorem 3.4.8). As another application, we
obtain the following Auslander correspondence for representation-finite R-orders in case dim R ≥ 2,
which improves [Iy4, Theorem 4.2.3].
Theorem J (= Corollary 3.4.11). Let R be a complete Cohen-Macaulay local ring withdim R = d ≥ 2. Then there exists a bijection between the following.
(1) Morita equivalence classes of R-orders Λ such that CM Λ is of finite type.
(2) Equivalence classes of pairs (Γ, e), where Γ is a noetherian R-algebra and e is an idempotent of Γ such that the following conditions are satisfied.
(a) gl.dim Γ = d.
(b) Γe is maximal Cohen-Macaulay as an R-module.
(c) Γ := Γ/ΓeΓ is of finite length over R.
(d) Γ/(rad Γ) is a direct sum of simple Γ-modules satisfying the 2-regular condition.

3.2. CLASSIFYING EXACT STRUCTURES VIA SERRE SUBCATEGORIES

33

Also, we investigate the Grothendieck group K0 (E) of an exact category E of finite type,and show that the relation of K0 (E) is generated by AR conflations under some mild conditions(Corollary 3.3.18). This unifies several known results by [AR2, But, Yo].
This chapter is organized as follows. In Section 2, we state and prove our main classification.
In Section 3, we investigate exact categories of finite type. In Section 4, we apply our previous
results to the study of the representation theory of algebras.
3.2. Classifying exact structures via Serre subcategories
In this section, we give a bijection between exact structures on an idempotent complete additive category E and subcategories of mod E satisfying certain “2-regular condition.” Throughout
this section, we always assume that E is an additive category.
Recall that an additive category E is idempotent complete if every morphism e : X → X in E
satisfying e2 = e has a kernel, or equivalently, a cokernel. For example, every subcategory of anabelian category which is closed under direct sums and direct summands are idempotent complete.
3.2.1. Preliminaries on functor categories. Our strategy to investigate exact structures
on E is to study the module category Mod E over E. Let us recall related definitions.
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 the category of right
E-modules, and morphisms in Mod E are natural transformations between them. This category
Mod E is an abelian category with enough projectives, and projective objects are precisely direct
summands of (possibly infinite) direct sums of representable functors. For simplicity, we put PX :=E(−, X) ∈ Mod E and P X := E(X, −) ∈ Mod E op for any X in E. Note that P(−) : E → Mod E
gives the Yoneda embedding.
We say that a E-module M is finitely generated if there exists an epimorphism PX  M forsome X in E. Throughout this chapter, we often use the fact that E is idempotent complete if
and only if the essential image of the Yoneda embedding P(−) : E → Mod E consists of all finitely
generated projective E-modules. We denote by mod E the category of finitely generated E-modules.
We also denote by mod1 E the category of finitely presented E-modules, that is, the modules M
such that there exists an exact sequence PX → PY → M → 0 for some X, Y in E.
We define a contravariant functor HomE (−, E) : Mod E → Mod E op by the following way:
For M in Mod E, the left E-module HomE (M, E) : E → Ab is the composition of the Yoneda
embedding E → Mod E and (Mod E)(M, −) : Mod E → Ab. Note that HomE (−, E) is a left exact
functor satisfying HomE (P(−) , E) ' P (−) . We denote by ExtiE (−, E) : Mod E → Mod E op the i-thright derived functor of HomE (−, E).
Using these concepts, we can interpret kernel-cokernel pairs in terms of modules over E. Herefg
we say that a complex X −→Y −→ Z in E is a kernel-cokernel pair if f is a kernel of g and g is acokernel of f .
fg

Proposition 3.2.1. Let X −→Y −→ Z be a complex in E. Put M := Coker(Pg ) ∈ Mod E.
Then the following hold.
(1) g is an epimorphism in E if and only if HomE (M, E) = 0.
fg(2) X −→Y −→ Z is a kernel-cokernel pair if and only if the following conditions are satisfied.
PfPg(a) 0 → PX −−→ PY −−→ PZ → M → 0 is exact.
(b) ExtiE (M, E) = 0 for i = 0, 1.
Pg

Proof. (1) We have an exact sequence 0 → HomE (M, E) → P Z −−→ P Y . Thus g is an
epimorphism if and only if P g is a monomorphism if and only if HomE (M, E) = 0.
(2) The condition (a) is equivalent to that f is a kernel of g. Under (a), the condition (b) isPg

Pfequivalent to that 0 → P Z −−→ P Y −−→ P X is exact, which holds precisely when g is a cokernelof f .
We need the following technical lemma later.

34

3. CLASSIFICATIONS OF EXACT STRUCTURES AND CM-FINITE ALGEBRAS

Lemma 3.2.2. Suppose that E is idempotent complete and there exists an exact sequence
0 → PX → PY → PZ → M → 0
in Mod E. For any morphism h : B → C with Coker(Ph ) ∼= M , there exists an object A in E suchthat Ker Ph ∼= PA .
Proof. Schanuel’s lemma shows that Ker Ph ⊕ PY ⊕ PC ∼= PX ⊕ PB ⊕ PZ , which clearly
implies that Ker Ph is finitely generated projective. Since E is idempotent complete, the assertion
holds.

3.2.2. Construction of maps. First we fix some notations which we need to describe ourmain theorem. The following observation follows immediately from Proposition 3.2.1.
Lemma 3.2.3. For an object M in Mod E, the following are equivalent.
g(1) There exists a kernel-cokernel pair X → Y −→ Z in E such that M is isomorphic to
Coker(Pg ).
(2) There exists an exact sequence 0 → PX → PY → PZ → M → 0 in Mod E and
ExtiE (M, E) = 0 for i = 0, 1.
We denote by C2 (E) the subcategory of Mod E consisting of E-modules satisfying the above
equivalent conditions. This class of modules play an indispensable role throughout this chapter.
Lemma 3.2.4. The category C2 (E) is closed under direct summands in Mod E.
Proof. We denote by mod2 E the subcategory of Mod E consisting of all objects M such that
there exists an exact sequence
P2 → P1 → P 0 → M → 0
in Mod E, where each Pi is finitely generated projective.
In [Ki, Lemma 2.5(a)], it was shown that mod2 E is closed under summands in Mod E. Notethat C2 (E) is equal to the intersection of two subcategories of mod2 E:(1) the subcategory consisting of all objects whose projective dimensions are at most 2 and(2) the subcategory consisting of all objects M such that ExtiE (M, E) = 0 for i = 0, 1.
Since these subcategories are closed under direct summands in Mod E, so is C2 (E).

Lemma 3.2.5. The functor Ext2E (−, E) induces a duality of exact categories C2 (E) → C2 (E op ).
Proof. The same proof as in [Iy2, 6.2(1)] applies here.



By definition, the category C2 (E) is closely related to kernel-cokernel pairs of E. Indeed, we
have two maps between them, as we shall construct below.
fg
Suppose that F is a class of kernel-cokernel pairs in E. We say that a complex X −→Y −→Z
is an F -exact if it belongs to F . In this case, we say that f is an F -monomorphism and that g is
an F -epimorphism.
Definition 3.2.6. (1) Let D be a subcategory of C2 (E). We denote by F (D) the class of all
fgcomplexes X −→Y −→ Z which satisfy the following condition:
Pf

Pg

There exists an exact sequence 0 → PX −−→ PY −−→ PZ → M → 0 in Mod E such that M
belongs to D.
(2) Let F be a class of kernel-cokernel pairs in E. We denote by D(F ) the subcategory of Mod E
consisting of all objects M which satisfy the following condition:fg
There exists an F -exact complex X −→Y −→ Z satisfying M ∼= Coker(Pg ).
The category D(F ) can be seen as the category of contravariant defects of F -exact sequences,in the sense of Auslander (see [ARS, IV.4]). Note that by Lemma 3.2.3, the following hold.
fg• Every complex X −→Y −→ Z in F (D) is a kernel-cokernel pair.
• Every object M in D(F ) is in C2 (E).

3.2. CLASSIFYING EXACT STRUCTURES VIA SERRE SUBCATEGORIES

35

3.2.3. Main theorem. In this subsection, we will state our main theorem and give a proof.
Recall that a Serre subcategory of an exact category E is an additive subcategory D of E such thatfor all conflations L  M  N in E, the object M belongs to D if and only if both L and N
belong to D.
Theorem 3.2.7. Let E be an idempotent complete additive category. Then there exist mutually
inverse bijections between the following two classes.
(1) Exact structures F on E.
(2) Subcategories D of C2 (E) satisfying the following conditions.
(a) D is a Serre subcategory of mod E.
(b) Ext2E (D, E) is a Serre subcategory of mod E op .
The map from (1) to (2) is given by F 7→ D(F ) and from (2) to (1) by D 7→ F (D) (see Definition
3.2.6).
To prove this, we first show that the maps in Definition 3.2.6 induces a one-to-one correspondence between wider classes than in Theorem 3.2.7.
Proposition 3.2.8. Let E be an additive category. Then the maps in Definition 3.2.6 induce
mutually inverse bijections between the following two classes.
(1) Classes of kernel-cokernel pairs F in E satisfying the following conditions.
(a) F is closed under homotopy equivalences of complexes.
(b) F is closed under direct sums of complexes.
(c) F is closed under direct summands of complexes.
(d) F is not empty.
(2) Subcategories D of C2 (E) satisfying the following condition.
(a) D is closed under direct sums.
(b) D is closed under direct summands.
(c) D is not empty.
Proof. First we see that the maps in Definition 3.2.6 induces well-defined maps between (1)and (2).
(1) → (2): Suppose that F is a class of kernel-cokernel pairs in E satisfying the conditions of(1). We will see that D(F ) satisfies the conditions of (2). Clearly D(F ) satisfies (a) and (c) by theconditions (1)(b) and (1)(d) respectively. Hence it suffices to show that D is closed under direct
summands.
Suppose that M1 ⊕ M2 is in D(F ). Then M1 ⊕ M2 is an object of C2 (E), which is closed undersummands by Lemma 3.2.4. Therefore both M1 and M2 are in C2 (E). This gives exact sequences
Pf

Pg

i
i
0 → PXi −−→
PYi −−→PZ i → M i → 0

fi

gi

for some kernel-cokernel pairs Xi −→ Yi −→ Zi in E with i = 1, 2. By taking the direct sum ofthese two complexes, we obtain a kernel-cokernel pair
f1 ⊕f2

g1 ⊕g2

X1 ⊕ X2 −−−−→ Y1 ⊕ Y2 −−−−→ Z1 ⊕ Z2 .

(3.2.1)

Thereby we obtain the following exact sequence.
Pf ⊕PfPg ⊕Pg1
2
1
2
0 → PX1 ⊕X2 −−−−−−→
PY1 ⊕Y2 −−−−−−→
PZ1 ⊕Z2 → M1 ⊕ M2 → 0

On the other hand, since M1 ⊕ M2 is in D(F ), there exists an F -exact complex
f

g

X−
→Y −
→Z

(3.2.2)in E such thatPf

Pg

0 → PX −−→ PY −−→ PZ → M1 ⊕ M2 → 0
is exact. It is standard that two projective resolutions are homotopy equivalent to each other.
Thus the Yoneda lemma implies that (3.2.1) and (3.2.2) are homotopy equivalent to each other as
three-term complexes in E. Because F is closed under homotopy equivalences, (3.2.1) is F -exact.

36

3. CLASSIFICATIONS OF EXACT STRUCTURES AND CM-FINITE ALGEBRAS

By the condition (1)(c), we conclude that each gi is an F -epimorphism, which shows that M1 andM2 are in D(F ).
(2) → (1): This is immediate and we leave the details to the reader.
Now we will show that the maps in Definition 3.2.6 are in fact inverse to each other. It is easy
to see that D = D(F (D)) in general. (Note that subcategories are always assumed to be closed
under isomorphisms.)
Suppose F is a class of kernel-cokernel pairs in E satisfying the conditions of (1). Put D :=D(F ). Then clearly we have F (D) ⊃ F . We will prove F (D) ⊂ F .
Suppose that we have an F (D)-exact sequence
f

g

X−
→Y −
→ Z.

(3.2.3)

Put M := Coker(Pg ). Then M is contained in D. Thus there exists an F -exact sequencef0

g0

X 0 −→ Y 0 −→ Z 0(3.2.4)such that
Pf 0

Pg 0

0 → PX 0 −−→ PY 0 −−→ PZ 0 → M → 0
is exact. Since the Yoneda embedding of (3.2.3) also yields a projective resolution for M , it follows
that (3.2.3) and (3.2.4) are homotopy equivalent to each other. Therefore (3.2.3) is an F -exact
sequence, since F is closed under homotopy equivalences.

Now we are in the position to prove Theorem 3.2.7. First we recall the axiom of exact categories
following [Bü, Definition 2.1].
Definition 3.2.9. Let F be a class of kernel-cokernel pairs in E which is closed under isomorphisms. Consider the following conditions.
(Ex0) For all objects X ∈ E, the complex 0 → X = X is F -exact.
(Ex1) The class of F -epimorphisms are closed under compositions.
(Ex2) The pullback of an F -epimorphisms along an arbitrary morphism exists and yields an
F -epimorphism.
We say that (E, F ) is an exact category if F satisfies (Ex0)-(Ex2) and (Ex0)op -(Ex2)op . In thiscase, F is called an exact structure on E, and we just call E an exact category if F is clear fromcontext.
In an exact category (E, F ), we use the terminologies conflations, deflations and inflations
instead of F -exact sequences, F -epimorphisms and F -monomorphisms respectively.
Lemma 3.2.10. Let (E, F ) be an exact category. Then F satisfies all the conditions of Proposition 3.2.8(1).
Proof. The condition (1)(d) is satisfied by definition. The condition (1)(a) can be proved by
the Gabriel-Quillen embedding. We refer the reader to [Bü, Proposition 2.9] and [Bü, Corollary2.18] for the conditions (1)(b) and (1)(c) respectively.
The following proposition is a technical part of the proof of Theorem 3.2.7.
Proposition 3.2.11. Suppose that E is idempotent complete and a class F satisfies the conditions of Proposition 3.2.8 (1). Put D := D(F ). Then the following are equivalent.
(1) F is an exact structure on E.
(2) D is a Serre subcategory of mod E and Ext2E (D, E) is a Serre subcategory of mod E op .
Proof. (1) ⇒ (2): First we show that D is a Serre subcategory of mod E. Letk→ M2 → 0
0 → M1 → M −be a short exact sequence in mod E.

3.2. CLASSIFYING EXACT STRUCTURES VIA SERRE SUBCATEGORIES

37

Assume that M1 and M2 are in D. We will see that M ∈ D. By the definition of D, therefigiexists an F -exact sequence Xi −→ Yi −→ Zi such that
Pf

Pg

i
i
0 → PXi −−→
PYi −−→PZ i → M i → 0

is exact for each i = 1, 2. By the horseshoe lemma, we obtain an exact commutative diagram in
Mod E of the following form
0

0

0

0

0


/ PX1

P f1


/ PY1

Pg1


/ PZ 1


/ M1

/0

0


/ PX1 ⊕X2

Pf
/ PY1 ⊕Y2

Pg


/ PZ1 ⊕Z2


/M

/0

0


/ PX2

P f2


/ PY2

Pg2


/ PZ 2


/ M2


0


0


0


0

k

/0

where the columns except the right-most one are split exact. Since P(−) : E → Mod E is fully
faithful by the Yoneda Lemma, we obtain the following commutative diagramX1 /
X1 ⊕ X2

X2 /

/ Y1


/ Y1 ⊕ Y2

f1

f


/ Y2

f2

/ / Z1
/ Z1 ⊕ Z2

g1

g


/ / Z2g2

in E. The top and bottom rows are F -exact, each of the three columns is split exact and gf = 0.
Thus we can apply 3 × 3 lemma in exact categories, see [Bü, Corollary 3.6]. Thus the middle row
is also F -exact, which implies that M is in D.
Next suppose that M is in D. We first see that M1 is in D. We have an F -exact sequence
f

g

Pf

Pg

h

→ M → 0 is exact. Since M1 is in mod E, we
X −→Y −→ Z such that 0 → PX −−→ PY −−→ PZ −have a surjection PW → M1 → 0 for some W in E. This yields the following commutative diagram
/ M1
/0
PW

Pϕ


Pf
Pg
h
/
/
/
/M
/0
P
P
P
0
X
Y
Z
Since F is an exact structure on E, there exists a pullback diagram
a
X /
f
X /

/E

/Y

b

//W

(3.2.5)

ϕ

c
g


//Z

in E where the top row is F -exact. By using the universal property of pullbacks, one can easily
check thatPb
Pa
0 → PX −−→PE −→PW → M 1 → 0
is exact. Thus M1 is in D.
t[b,−c][ϕ,g]

On the other hand, it is well-known that the complex E −−−−→ W ⊕ Y −−−→ Z corresponding
to the pullback square in (3.2.5) is F -exact (see e.g. [Bü, Proposition 2.12]). Moreover, one caneasily check that[Pϕ ,Pg ]kh

PW ⊕ PY −−−−−→ PZ −→ M2 → 0
is exact, where kh is the composition h : PZ  M and k : M  M2 . Therefore N is in D, which
completes the proof that D is a Serre subcategory of mod E.

38

3. CLASSIFICATIONS OF EXACT STRUCTURES AND CM-FINITE ALGEBRAS

Note that an object M in mod E op is contained in Ext2E (D, E) if and only if there exists an
f

Pg

g

Pf

F -exact sequence X −→Y −→ Z in E such that 0 → P Z −−→ P Y −−→ P X → M → 0 is exact.
2
Hence ExtE (D, E) is a Serre subcategory of mod E op by the dual argument.
(2) ⇒ (1): By duality, it suffices to show that F satisfies (Ex0)-(Ex2). Note that (Ex0)automatically holds by the condition (1)(d) in Proposition 3.2.8.
f

g

h

k

(Ex1) Let A −→B−→ C and X −→C−→ D be F -exact sequences. We show that kg is also an
F -epimorphism. We have the following commutative diagram with exact rows
0

PX

PX

Ph

0

/ PA

Pf

/ PB
PB

Pg

Pkg


/ PC


/L

/0

/ PD


/M

/0


N


N


0


0



Pk

where L and N are in D by the definition of D. The right-most column is exact by diagram chasing.
Since D is a Serre subcategory of mod E, we immediately have that M is in D. In particular, M iskg
lcontained in C2 (E). By Lemma 3.2.2, there exists a kernel-cokernel pair Y −→ B −→ D such that
P

Pkg

l
0 → PY −→PB −−→ PD → M → 0 is exact. Thus kg is an F (D)-epimorphism. Since F (D) = F ,the morphism kg is an F -epimorphism.
fg(Ex2) Let X −→Y −→ Z be an F -exact sequence and h : W → Z an arbitrary morphism in E.
Then we have the commutative diagram

0
PW
P

0

/ PX

Pf

/ PY

Pg

 h
/ PZ

a


/L

/0


/M

/0

b


N

0

where all the rows and columns are exact. Since M is in D by the definition, L and N are also
[Ph ,Pg ]in D. In particular, N is contained in C2 (E). On the other hand, we have that PW ⊕ PY −−−−−→ba

PZ −→ N → 0 is exact. Thus by Lemma 3.2.2, there exists an exact sequence
h

0

/ PE

Pk
−Pl

i

/ PW ⊕ PY [Ph ,Pg ] / PZ

/N

ba

/0

in Mod E. It is standard that this exact sequence yields a pullback diagram
E

Y

k

l

/W
h

g


/Z

(3.2.6)

3.2. CLASSIFYING EXACT STRUCTURES VIA SERRE SUBCATEGORIES

39

in E. Thus the existence of the pullback has been proved. By the universal property of the pullbackik(3.2.6), there exists a complex X −→E−→ W in E such that
P

P

i
k
0 → PX −→PE −−→
PW → L → 0ikis exact. Thus the complex X −→ E −→ W belongs to F (D) = F , which implies that k is an
F -epimorphism as desired.
Proof of Theorem 3.2.7. It is immediate from Proposition 3.2.11 and Lemma 3.2.10.

The following description of projective objects in E in terms of D is needed later.
Proposition 3.2.12. Let E be an idempotent complete exact category and D the corresponding
Serre subcategory of mod E given in Theorem 3.2.7. Then for an object Z in E, the following are
equivalent.
(1) W is projective in E.
(2) M (W ) = 0 for all object M ∈ D.
Pg

Proof. The category D consists of all functors M such that PY −−→ PZ → M → 0 is exact
Pg (W )

for some deflation g : Y  Z, so (2) is equivalent to that PY (W ) −−−−→ PZ (W ) is surjective forevery deflation g. This occurs if and only if W is a projective object in E.
Next let us consider the particular case when the simpler classification is available. This
includes the case when E is abelian, or more generally, quasi-abelian (see [Ru1] for the detail).
Lemma 3.2.13. Let E be an idempotent complete category. Then the following are equivalent.
(1) C2 (E) and C2 (E op ) are Serre subcategories of mod E and mod E op respectively.
(2) The class of all kernel-cokernel pairs in E defines an exact structure of E.
In this case, there exists a bijection between the following two classes.
(1) Exact structures on E.
(2) Serre subcategories of C2 (E).
Proof. Let F be the class of all kernel-cokernel pairs in E. Then D(F ) = C2 (E) holds, thus
Proposition 3.2.11 applies. The latter part is clear from Theorem 3.2.7.

Now we will show that a more familiar description for D is available if E has enough projectives.
First we recall the notation of the projectively stable category of an exact category. Let E be anexact category with enough projective objects. Denote by [P](X, Y ) the set of all morphisms fromX to Y which factor through projective objects in E. Then [P] is a two-sided ideal of E and wedenote by E := E/[P], which we call the projectively stable category.
Lemma 3.2.14. Let E be an exact category with enough projectives and D the subcategory ofmod E corresponding to all conflations (see Definition 3.2.6). Then D ' mod1 E holds.
Proof. We have an embedding mod1 E → mod E and denote its essential image by D0 . Weshow that the following conditions are equivalent for an object M in mod E.
(1) M ∈ D.
(2) There exist a deflation g : Y → Z in E and an exact sequence(−)◦gE(−, Y ) −−−−→ E(−, Z) → M → 0.
0(3.2.7)(3) M ∈ D .
(1) ⇔ (2): This is easily shown by diagram chasing.
(2) ⇒ (3): Clear.
(3) ⇒ (2): Suppose that M is in D0 . Since M is finitely presented E-module, we have anexact sequence of the form (3.2.7) for some g : Y → Z. By assumption, there exists a deflation
ϕ : P  Z for some projective object P . Using this, we may replace g by [g, ϕ] : Y ⊕ P → Z,
which is a deflation. This proves the claim.
40

3. CLASSIFICATIONS OF EXACT STRUCTURES AND CM-FINITE ALGEBRAS

Combining Lemma 3.2.13 and 3.2.14, we immediately obtain the following conclusion, which
gives a generalization of Buan’s result [Bua, Proposition 3.3.2], where E was assumed to be mod Λ
for some artin algebra Λ.
Corollary 3.2.15. Let E be an idempotent complete additive category such that the class of
all kernel-cokernel pairs defines an exact structure with enough projectives (e.g. abelian category
with enough projectives). Denote by E the projectively stable category in this exact structure. Then
there exists a bijection between the following two classes.
(1) Exact structures on E.
(2) Serre subcategories of mod1 E.
3.3. Exact categories of finite type
We use our previous results to classify idempotent complete exact categories of finite type.
For an additive category E, an object M is called an additive generator of E if add M = E holds,
where add M is the subcategory of E consisting of all direct summands of finite direct sums of M .
We call that an additive category E is of finite type if it has an additive generator.
We say that an additive category E is a Krull-Schmidt category if every object in E is a finite
direct sum of indecomposable objects whose endomorphism rings are local. For the basics of
Krull-Schmidt categories, we refer the reader to [Kr]. An additive category E is Krull-Schmidt
if and only if E is idempotent complete and EndE (X) is semiperfect for every X ∈ E. If E is
Krull-Schmidt, then E is of finite type precisely when E has finitely many indecomposables up toisomorphism.
Let M be an additive generator of E and Γ := EndE (M ). Then we have a fully faithful functor
E(M, −) : E → Mod Γ to the category of right Γ-modules. Its essential image coincides with thecategory proj Γ of finitely generated projective Γ-modules precisely when E is idempotent complete.
Thus, when we deal with an idempotent complete additive category of finite type, we may assume
that E = proj Γ for some ring Γ. To avoid technical complications, we restrict our attention to thecase when Γ is a noetherian ring. Note that proj Γ is Krull-Schmidt if and only if Γ is semiperfect.
3.3.1. Basic properties. We start with reformulating our result in Section 3 in terms of
the ring Γ, where we use the same notation C2 (Γ):
C2 (Γ) := {M ∈ mod Γ | pd MΓ ≤ 2 and ExtiΓ (M, Γ) = 0 for i = 0, 1}.
Note that every non-zero module M in C2 (Γ) satisfies pd MΓ = 2.
Theorem 3.3.1. Let Γ be a noetherian ring and put E := proj Γ. Then there exists a bijection
between the following two classes.
(1) Exact structures F on E.
(2) Subcategories D of C2 (Γ) satisfying the following condition.
(a) D is a Serre subcategory of mod Γ.
(b) Ext2Γ (D, Γ) is a Serre subcategory of mod Γop .
The correspondence is given as follows.
f

g

f

g

• For a given D, a complex X −→Y −→ Z in E is in F if and only if 0 → X −→Y −→Z→
M → 0 is an exact sequence in mod Γ with some M in D.
• For a given F , a Γ-module M ∈ mod Γ is in D if and only if there exists a deflation g in
E with M ∼= Coker g.
When we deal with exact structures on proj Γ, the 2-regular condition for simple modules (seeDefinition 3.1.1) are quite essential. For the set S of simple Γ-modules, we denote by Filt S thesubcategory of mod Γ consisting of all modules M such that M has finite length and all composition
factors of M are contained in S. The following observation is useful.
Lemma 3.3.2. Let Γ be a noetherian ring and S a set of simple Γ-modules. Then the following
are equivalent.
(1) Every module in S satisfies the 2-regular condition.

3.3. EXACT CATEGORIES OF FINITE TYPE

41(2) D := Filt S is contained in C2 (Γ) and satisfies the conditions of Theorem 3.3.1(2).
(3) There exists a subcategory D of C2 (Γ) containing S such that D satisfies the conditionsof Theorem 3.3.1(2).
Proof. (1) ⇒ (2): Clearly S ∈ C2 (Γ) holds for every S ∈ S. Since C2 (Γ) is closed under
extensions in mod Γ, we have that Filt S is a Serre subcategory of mod Γ contained in C2 (Γ). Since
Ext2Γ (−, Γ) gives an exact duality C2 (Γ) ' C2 (Γop ) and Ext2Γ (S, Γ) is simple for every S ∈ S, wehave Ext2Γ (D, Γ) = Filt Ext2Γ (S, Γ). Thus this category is a Serre subcategory of mod Γop , whichshows that Filt S satisfies the conditions of Theorem 3.3.1(2).
(2) ⇒ (3): Obvious.
(3) ⇒ (1): Let S be a simple module contained in S. It follows from S ∈ C2 (Γ) that pd SΓ =2 and ExtiΓ (S, Γ) = 0 for i = 0, 1. Notice that Ext2Γ (−, Γ) gives the duality between D and
Ext2Γ (D, Γ), both of which are abelian categories. Thus it is immediate that Ext2Γ (S, Γ) is a simple
object in the abelian category Ext2Γ (D, Γ), which implies that Ext2Γ (S, Γ) is a simple left Γ-module.
Therefore S satisfies the 2-regular condition.
Every additive category admits a trivial exact structure whose conflations are split exact
sequences. We have the following criterion on the existence of non-trivial exact structures onproj Γ.
Proposition 3.3.3. Let Γ be a noetherian ring. Then proj Γ admits a non-trivial exact structure if and only if there exists a simple Γ-module satisfying the 2-regular condition.
Proof. Suppose that there exists a simple Γ-module S satisfying the 2-regular condition.
Then Lemma 3.3.2 and Theorem 3.3.1 imply the existence of a non-trivial exact structure on
proj Γ.
Conversely, suppose that proj Γ has a non-trivial exact structure. By Theorem 3.3.1, we havea non-zero Serre subcategory D of mod Γ. Since any non-zero Γ-module in mod Γ has a surjection
onto simple Γ-module, D contains at least one simple Γ-module S. Then Lemma 3.3.2 implies
that S satisfies the 2-regular condition.

Example 3.3.4. Let R be a commutative noetherian local ring. Then Proposition 3.3.3
implies that there exists a non-trivial exact structure on proj R if and only if R is a regular local
ring of dimension 2. In this case, proj R has exactly one non-trivial exact structures. In this exact
fg
structure, P2 −→ P1 −→ P0 is a conflation if and only if f is a kernel of g and Coker(g) is of finite
length over R.
3.3.2. Admissible exact structures. We introduce a nice class of exact categories whichis completely controlled by the simple modules satisfying the 2-regular condition. For a ring Γ,
we denote by f.l. Γ the subcategory of mod Γ consisting of Γ-modules of finite length. Similarly,
we denote by f.l. E the category consisting of E-modules of finite length.
Definition 3.3.5. Let (E, F ) be an exact category and D the subcategory of mod E corresponding to F under Theorem 3.2.7. We say that F is admissible if D ⊂ f.l. E holds.
Let (proj Γ, F ) be an exact category for a noetherian ring Γ and D the subcategory of mod Γ
corresponding to F under Theorem 3.3.1. Then it is clear that F is admissible if and only if
D ⊂ f.l. Γ holds. Therefore, for an artinian ring Γ, every exact structure on proj Γ is admissible.
We show that the admissibility is left-right symmetric under some assumptions.
Proposition 3.3.6. Let (proj Γ, F ) be an exact category for a noetherian ring Γ. Then it isadmissible if and only if (proj Γop , F op ) is admissible.
Proof. Let D be the subcategory of mod Γ corresponding to F under Theorem 3.3.1. Then
Ext2Γ (D, Γ) is the subcategory corresponding to F op . Since F is admissible, every object in D has
finite length as a Γ-module. Thus D is an abelian category in which every object has finite length.
Because Ext2Γ (D, Γ) is dual to D, every object in Ext2E (D, E) has finite length in the abelian category
Ext2Γ (D, Γ). Since Ext2Γ (D, Γ) is a Serre subcategory of mod Γop , we have Ext2Γ (D, Γ) ⊂ f.l. Γop . 

42

3. CLASSIFICATIONS OF EXACT STRUCTURES AND CM-FINITE ALGEBRAS

Using this notion, we can classify all admissible exact structures on proj Γ.
Theorem 3.3.7. Let Γ be a noetherian ring. For E := proj Γ, there exists a bijection between
the following two classes.
(1) Admissible exact structures F on E.
(2) Sets S of isomorphism classes of simple Γ-modules satisfying the 2-regular condition.
The correspondence is given as follows.
fg• For a given S, a complex X −→Y −→ Z in E is a conflation if and only if there exists anfgexact sequence 0 → X −→Y −→ Z → M → 0 in mod Γ with M in Filt S.
• For a given F , a simple Γ-module S is in S if and only if there exists a deflation g : Y → Z
in E such that Coker(g) ∼= S in mod Γ.
Proof. By Theorem 3.3.1, there exists a bijection between (1) and(3) Subcategories D of C2 (Γ) satisfying the following condition.
(a) D is a Serre subcategory of mod Γ satisfying D ⊂ f.l. Γ.
(b) Ext2Γ (D, Γ) is a Serre subcategory of mod Γop .
We have mutually inverse bijections between (2) and (3) as follows; For a given S in (2), we
put D := Filt S, and for a given D in (3), we denote by S the set of simple modules contained inD. By Lemma 3.3.2, these maps are well-defined. These are mutually inverse to each other bydefinition.

Next we will focus on the case over the ground ring R. In the rest of this subsection, we fix acommutative noetherian complete local ring R. For an additive R-category E for a commutative
noetherian ring R, we say that E is Hom-noetherian (resp. Hom-finite) if the R-module E(X, Y )is finitely generated (resp. of finite length) for every X, Y ∈ E. Every Hom-noetherian idempotent
complete R-category is Krull-Schmidt. For an exact R-category E, we say that E is Ext-noetherian(resp. Ext-finite) if for every object X and Y , the R-module Ext1E (X, Y ) is finitely generated (resp.
of finite length).
The following figure illustrates the relationship between these concepts.
R is artinian

finite typeenough proj.
(or inj.)

Ext-noeth.

finite typeExt-finite

admissible
enough inj.

We will prove these implication in Proposition 3.3.8 and Corollary 3.3.15. We refer the reader toAppendix B.2 for the results and notions we need in the proof.
Proposition 3.3.8. Let E be a Hom-noetherian idempotent complete exact R-category. Thenthe following hold.
(1) If E has either enough projectives or enough injectives, then E is Ext-noetherian.
(2) If E is Ext-noetherian and E is of finite type, then E is Ext-finite.
(3) If E is Ext-finite and E is of finite type, then E is admissible. Conversely, if E is admissible
and has enough injectives, then E is Ext-finite.
Proof. (1) This is clear since extension groups can be computed by projective resolutions or
injective coresolutions.
(2) If E is of finite type, then E has AR conflations by Corollary B.1.4. Since E is Extnoetherian, Proposition B.2.1 implies that E is Ext-finite.
(3) Suppose that E is Ext-finite and of finite type. Take W to be an additive generator of Efgand Γ := EndE W . For any conflation X −→Y −→ Z, the cokernel of g ◦ (−) : E(W, Y ) → E(W, Z)
is a submodule of Ext1E (W, X), thus is of finite length over R, or equivalently, over Γ. Therefore
E is admissible.
Conversely, suppose that the exact structure on E is admissible and E has enough injecf
gtives. For an object X ∈ E, take a conflation X −→ I −→ Z such that I is injective. Then3.3. EXACT CATEGORIES OF FINITE TYPE

43

g◦(−)

E(−, I) −−−−→ E(−, Z) → Ext1E (−, X) → 0 is exact, hence Ext1E (−, X) is of finite length. It
follows that Ext1E (W, X) is of finite length over R for any W in E.
Next we interpret Theorem 3.3.7 in terms of the quiver of Γ. Recall that for a noetherian
R-algebra Γ, the valued quiver Q(Γ) is defined as follows, where J denotes the radical of proj Γ.
(1) The set of vertices is ind(proj Γ), that is, the isomorphism classes of all indecomposableprojective right Γ-modules.
(2) We draw an arrow from P to Q if J (P, Q)/J 2 (P, Q) 6= 0 with a valuation (dP,Q , d0P,Q ),where dP,Q (resp. d0P,Q ) is the dimension of J (P, Q)/J 2 (P, Q) as a kP -vector space (resp.
kQ -vector space). Here kP := EndE (P )/ rad EndE (P ) and kQ := EndE (Q)/ rad EndE (Q).
We introduce a translation on this quiver Q(Γ).
Definition 3.3.9. Let Γ be a noetherian R-algebra. The translation τ of P ∈ Q(Γ) is definedwhen P/ rad P satisfies the 2-regular condition. In this case, τ P is the projective module Q suchthat HomΓ (Q, Γ) is a projective cover of the simple left Γ-module Ext2Γ (P/ rad P, Γ). We draw adotted arrow from P to τ P whenever τ P is defined.
This construction yields a valued translation quiver (see [ARS]). Since we will not use valued
arrows, we omit the proof.
Now admissible exact structures on proj Γ can be visually classified by the dotted arrows.
Corollary 3.3.10. Let Γ be a noetherian R-algebra over a noetherian complete local ring R.
Then there exists a bijection between the following two classes.
(1) Admissible exact structures on proj Γ.
(2) Sets of dotted arrows in Q(Γ).
Moreover, the Auslander-Reiten quiver of the exact category E is given by the quiver Q(Γ) withthe dotted arrows chosen in (2).
Proof. Note that each dotted arrow τ P L99 P in Q(Γ) bijectively corresponds to the simple
Γ-module P/ rad P satisfying the 2-regular condition by the definition. Thus Theorem 3.3.7 implies
the assertion.

We end this subsection by giving examples of Corollary 3.3.10.
Example 3.3.11. Let Γ be the algebra defined in Example 3.1.2 the introduction. Then Q(Γ)
coincides with Figure 1 (all valuations are trivial, i.e. (1, 1) and all dotted lines are interpreted asarrows from right to left). It has seven dotted arrows, thus proj Γ has 27 = 128 exact structures.
Example 3.3.12. Let k be a field, R := kJtK the ring of formal power series over a field k andR
R ··· R R
 (t) R · · · R R


 2

..
.
.
(t
)
(t)
R
R
Γ := 



 .
.
.
.
.
..
..
. . ..  ..
(tn ) · · · (t2 ) (t) R
Then Q(Γ) is given by the following quiver.
1

2

3

···

n

n+1

Note that Γ is the Auslander order for an R-order


R R
Λ :=
(tn ) Rhence there is an equivalence proj Γ ' CM Λ, and the above translation quiver gives the AuslanderReiten quiver of CM Λ. Corollary 3.3.10 shows that proj Γ has four admissible exact structures,and the usual exact structure on CM Λ corresponds to the set of two dotted arrows.

44

3. CLASSIFICATIONS OF EXACT STRUCTURES AND CM-FINITE ALGEBRASExample 3.3.13. Let T = kJx, yK be the ring of formal power series in two variables overa field k of characteristic 0, and let R = T (n) be the n-th Veronese subring of T , that is,
R = kJxn , xn−1 y, · · · , y n K. Put Γ := EndR (T ). Then all simple Γ-modules satisfy the 2-regularcondition, and the translation quiver Q(Γ) is the following.

···
5

n

1

···
4

2
3

Here the double arrows are interpreted as the single arrows with valuation (2, 2). Note that
add(TR ) = CM R holds, thus proj Γ ' CM R (see [Yo, Chapter 10] for the detail). Therefore
Corollary 3.3.10 implies that CM R has 2n admissible exact structures.
3.3.3. Enough projectivity and admissibility. We collect some properties about therelation between enough projectivity and admissibility of exact R-categories. Throughout thissubsection, we fix a commutative noetherian complete local ring R.
By Proposition 3.3.8(3), If E is a Hom-noetherian idempotent complete exact R-category offinite type which has enough projectives, then E is admissible. The following gives a criterion
when the converse holds.
Proposition 3.3.14. Let E := (proj Γ, F ) be an admissible exact category for a semiperfectnoetherian ring Γ. Take an idempotent e ∈ Γ such that eΓ is an additive generator of projective
objects in E. Then the following hold.
(1) D = f.l.(Γ/ΓeΓ) holds, where D is the subcategory of mod Γ corresponding to F under
Theorem 3.3.1.
(2) E has enough projectives if and only if Γ/ΓeΓ is in f.l. Γ.
Proof. (1) Since Γ is semiperfect, E is a Krull-Schmidt exact category. By Proposition 3.2.12,
an object P in E is projective if and only if HomΓ (P, D) = 0. Thus an indecomposable object
in E is projective in E if and only if it is the projective cover of a simple Γ-module which is notcontained in D. Therefore, a simple Γ-module S is contained in D if and only if HomE (eΓ, S) = 0,
that is, Se = 0. Thus, for a Γ-module M ∈ mod Γ, it follows that M ∈ D holds if and only if
M e = 0 and M ∈ f.l. Γ, which implies the assertion.
(2) Recall that a morphism g : P → X in E is a deflation if and only if Coker(g) in mod Γ
is in D, and every projective resolution of M ∈ D yields a conflation in E. Thus E has enoughprojectives if and only if there exists an exact sequenceg

P −→Γ→M →0(3.3.1)

in mod Γ for some objects P ∈ add(eΓ) and M ∈ D. Suppose that this holds. Then Im(g) ⊂ ΓeΓholds, so we have a surjection M  Γ/ΓeΓ. Since M is in D, it follows that M is of finite length,
thus so is Γ/ΓeΓ. Conversely, suppose that Γ/ΓeΓ has finite length. Then Γ/ΓeΓ is contained in
D = f.l.(Γ/ΓeΓ). Since Γ is noetherian, ΓeΓ is a finitely generated as a right Γ-module. Therefore
there exists an exact sequence of the form (3.3.1) with M = Γ/ΓeΓ.
Consequently, we have the following interesting consequences. It is remarkable that we use
purely module-theoretical argument to show non-trivial properties of exact categories.
Corollary 3.3.15. Let E be a Hom-noetherian idempotent complete admissible exact Rcategory of finite type. Then the following holds.
(1) If R is artinian, then E has enough projectives and injectives.
(2) E has enough projectives if and only if E has enough injectives.

3.3. EXACT CATEGORIES OF FINITE TYPE

45

Proof. (1) Immediate from Proposition 3.3.14(2).
(2) It suffices to show the “only if” part. We may assume E = (proj Γ, F ) for a noetherian
R-algebra Γ. Denote by D = Filt S the Serre subcategory of mod Γ which corresponds to F under
Theorem 3.3.7, and take idempotents e and f in Γ such that eΓ (resp. f Γ) is an additive generatorof projective objects (resp. injective objects) in E. Put Γ := Γ/ΓeΓ and Γ := Γ/Γf Γ. We know
from Proposition 3.3.14(2) that Γ is of finite length, and it suffices to show that Γ is of finite
length.
Recall that we have a duality Ext2Γ (−, Γ) : D ' Ext2Γ (D, Γ). On the other hand, by Proposition
opop
3.3.14(1), we have D = f.l. Γ and Ext2Γ (D, Γ) = f.l. Γ . Therefore we have a duality F : f.l. Γ 'f.l. Γ between two abelian categories.
Observe that f.l. Γ has an injective cogenerator HomR (Γ, I) by the duality HomR (−, I) :op
f.l. Γ → f.l. Γop , where I is an injective hull of R/ rad R. Thus the abelian category f.l. Γ has a
projective generator, which we denote by P .
Suppose that Γ is not of finite length over R. In particular, we have the following infinite
chain of proper surjections· · ·  Γ/ rad3 Γ  Γ/ rad2 Γ  Γ/ rad Γ.
opin f.l. Γ . Take a surjection f1 : P k  Γ/ rad Γ. Then this lifts to morphisms fi : P k → Γ/ radi Γ.
Since the kernel of Γ/ radi Γ  Γ/ radi−1 Γ is contained in rad Γ/ radi Γ, it follows that fi issurjective for each i, which contradicts the fact that P k has finite length.
Remark 3.3.16. If R is not artinian, E does not necessarily has enough projectives. For
example, consider Example 3.3.13. Then proj Γ has the exact structure corresponding to all thedotted arrows. In this exact structure, there exists no non-zero projective object.
3.3.4. AR conflations and the Grothendieck group. In this subsection, we investigate the Grothendieck group of exact categories. Several papers showed that the relation of theGrothendieck group K0 (E) is generated by AR sequences when E is a particular exact category offinite type, e.g. [AR2, But, Yo]. Our aim in this subsection is to unify these results.
Let E be a Krull-Schmidt category. First we recall the following basic concepts in the AR
theory. A morphism g : Y → Z in E is called right almost split if g is not a retraction and any
non-retraction h : W → Z factors through g. Dually we define left almost split. We say that afg
conflation X −→Y −→ Z in E is an AR conflation if f is left almost split and g is right almost
split. We say that E has AR conflations if for every indecomposable non-projective object Z there
exists an AR conflation ending at Z, and for every indecomposable non-injective object X there
exists an AR conflation starting at X. For further properties of AR conflations, we refer thereader to Appendix B.1. In Corollary B.1.4, we will prove that Krull-Schmidt exact category E
has AR conflations if E is of finite type and the endomorphism ring of an additive generator of E
is noetherian.
Next we introduce some notation concerning theLGrothendieck group. For a Krull-Schmidt
exact category E, let G(E) be the free abelian group [X]∈ind E Z · [X] generated by the set ind E
of isomorphism classes of indecomposable objects in E. We denote by Ex(E) the subgroup of G(E)generated by{[X] − [Y ] + [Z] | there exists a conflation X  Y  Z in E}.
We call the quotient group K0 (E) := G(E)/ 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}.
Now we prove the main result about the relation of the Grothendieck groups.
Theorem 3.3.17. Let E be a Krull-Schmidt exact category of finite type such that the endomorphism ring of an additive generator of E is noetherian. If the exact structure on E is admissible,
then Ex(E) = AR(E) holds.

46

3. CLASSIFICATIONS OF EXACT STRUCTURES AND CM-FINITE ALGEBRASProof. Since AR(E) ⊂ Ex(E) always holds, we will check Ex(E) ⊂ AR(E). Let Γ be theendomorphism ring of an additive generator of E. Then E ' proj Γ holds, so we may assume
that E = proj Γ for a semiperfect noetherian ring Γ. Denote by S the set of simple Γ-modules
corresponding to the exact structure on E = proj Γ under Theorem 3.3.7.
f1
g1
Let X1 −→ Y1 −→ Z1 be a conflation in E and M := Coker(g1 ) in mod Γ. Then M is in
Filt S by Theorem 3.3.7. We show [X1 ] − [Y1 ] + [Z1 ] ∈ AR(E). Suppose that there exists anotherf2
g2
conflation X2 −→ Y2 −→ Z2 in E such that M ∼= Coker(g2 ). Then we have the exact sequencesfi

gi

0 → Xi −→ Yi −→ Zi → M → 0
in mod Γ for each i = 1, 2. Thus Schanuel’s lemma shows that X1 ⊕ Y2 ⊕ Z1 ∼= X2 ⊕ Y1 ⊕ Z2 , which
implies that [X1 ] − [Y1 ] + [Z1 ] = [X2 ] − [Y2 ] + [Z2 ] in G(E). Thus it suffices to show the following
claim to prove our theorem.
Claim: For any M ∈ Filt S, there exists at least one exact sequence in mod Γ
0→X→Y →Z→M →0with X, Y, Z ∈ proj Γ and [X] − [Y ] + [Z] ∈ AR(E).
We will show this claim by induction on l(M ), the length of M as a Γ-module. Suppose thatfgl(M ) = 1, that is, M ∈ S. Take the minimal projective resolution 0 → X −→Y −→Z→M →0fg

of M with X, Y, Z ∈ proj Γ = E. By the proof of Proposition B.1.3, we have that X −→Y −→ Z isindeed an AR-conflation in E. Thus [X] − [Y ] + [Z] ∈ AR(E).
Now suppose that l(M ) > 1. Take a simple Γ-module M1 ∈ S which is a submodule of M .
Then we have an exact sequence 0 → M1 → M → M2 → 0, where M1 and M2 are in Filt S. Sincefigil(M1 ), l(M2 ) < l(M ), we have the corresponding projective resolutions 0 → Xi −→ Yi −→ Zi →Mi → 0 such that [Xi ] − [Yi ] + [Zi ] ∈ AR(E) for i = 1, 2 by induction hypothesis. By the horseshoe
f1 ⊕f2

g1 ⊕g2lemma, we obtain a projective resolution 0 → X1 ⊕ X2 −−−−→ Y1 ⊕ Y2 −−−−→ Z1 ⊕ Z2 → M → 0.
Then we haveX
[X1 ⊕ X2 ] − [Y1 ⊕ Y2 ] + [Z1 ⊕ Z2 ] =([Xi ] − [Yi ] + [Zi ]) ∈ AR(E),i=1,2which completes the proof of the claim.



Corollary 3.3.18. Let R be a noetherian complete local ring and E a Hom-noetherian idempotent complete exact R-category of finite type. Suppose either R is artinian or E has enoughprojectives. Then Ex(E) = AR(E) holds.
Proof. In both cases, E is an admissible exact category by Proposition 3.3.8. Thus Theorem
3.3.17 applies.

3.4. Classifications of CM-finite algebras
In this section, we apply our previous results to the representation theory of Iwanaga-Gorenstein
algebras and orders. More generally, we study the left perpendicular category ⊥ U for a cotilting
module U . Throughout this section, we fix a commutative noetherian complete local ring R.
3.4.1. Cotilting modules. First, we introduce the notion of cotilting module following[AR3]. Let Λ be a noetherian ring and U ∈ mod Λ a Λ-module. We denote by ⊥ U the subcategory⊥of mod Λ consisting of all modules X satisfying Ext>0Λ (X, U ) = 0. Since U is an extension-closed⊥
subcategory of mod Λ, we always regard U as an exact category.
Definition 3.4.1. We say that U is a cotilting module if it satisfies the following conditions.
(C1) id UΛ is finite.
(C2) Ext>0Λ (U, U ) = 0.
(C3) ⊥ U has enough injectives add U , that is, for every X in ⊥ U , there exists an exact sequence
0 → X → U0 → Y → 0
in mod Λ with Y ∈ ⊥ U and U 0 ∈ add U .

3.4. CLASSIFICATIONS OF CM-FINITE ALGEBRAS

47

We shall see in Proposition 3.4.4 that if we restrict to R-orders, then our definition of cotilting
modules coincides with the usual one, e.g. in [Iy4]. From our definition, the following property isimmediate. Here we say that an object X in an exact category E is a projective generator (resp.
injective cogenerator ) if E has enough projectives add X (resp. enough injectives add X).
⊥

Proposition 3.4.2. Let Λ be a noetherian ring and U ∈ mod Λ a cotilting Λ-module. Then
U is an exact category with a projective generator Λ and an injective cogenerator U .

Let R be a Cohen-Macaulay local ring admitting a canonical module ω, for example, complete
Cohen-Macaulay local ring. For a noetherian R-algebra Λ, we denote by CM Λ the subcategoryof mod Λ consisting of modules which are maximal Cohen-Macaulay as R-modules. A noetherian
R-algebra Λ is called an R-order if Λ ∈ CM Λ holds. For an R-order Λ, there exists a duality Dd :=HomR (−, ω) : CM Λ ' CM Λop . It is immediate that CM Λ is an extension-closed subcategory ofmod Λ with a projective generator Λ and an injective cogenerator Dd Λ.
We prepare the following well-known properties of R-orders.
Lemma 3.4.3. Let R be a d-dimensional complete Cohen-Macaulay local ring, Λ an R-orderand M ∈ CM Λ. Then the following holds.
(1) CM Λ = ⊥ (Dd Λ) holds, and id(Dd Λ)Λ = d.
(2) id MΛ ≥ d holds, and id MΛ ≤ m if and only if Ext>m−d(X, M ) = 0 for all X ∈ CM Λ.
ΛProof. Both follow from [GN1, Proposition 1.1].

Proposition 3.4.4. Let R be a d-dimensional complete Cohen-Macaulay local ring, Λ a noetherian R-algebra and U a finitely generated Λ-module.
(1) Suppose that Λ is an R-order and U ∈ CM Λ. Then U is a cotilting module if and only
if U satisfies the condition (C1), (C2) in Definition 3.4.1 and the following.
(C30 ) There exists an exact sequence
0 → Un → · · · → U1 → U0 → Dd Λ → 0
in mod Λ for some n such that Ui ∈ add U for each i.
In particular, U is a cotilting module with id UΛ ≤ d if and only if add U = add Dd Λ.
(2) Suppose that U is a cotilting Λ-module. Then ⊥ U ⊂ CM Λ holds if and only if Λ is an
R-order and U ∈ CM Λ holds.
Proof. (1) Suppose that U satisfies (C1), (C2) and (C30 ). Then the same proof as in [AR3,Theorem 5.4] implies that (C3) holds (see [Iy4, Proposition 3.2.2] for the order case).
Conversely, suppose that U ∈ CM Λ satisfies (C1)-(C3). First we show ⊥ U ⊂ CM Λ. Let X be⊥in U . Then by (C3), we have an exact sequence 0 → X → U 0 → U 1 → · · · → U d−1 with U i in
add U for each i. Since each U i is maximal Cohen-Macaulay as an R-module, it follows from thedepth lemma [BH, Proposition 1.2.9] that X is in CM Λ.
b the subcategory of mod Λ
Next we will see that (C30 ) holds. Put X := ⊥ U and denote by U
consisting of modules Y such that there exists an exact sequence
0 → Un → · · · → U1 → U0 → Y → 0
in mod Λ for some n with Ui ∈ add U for each i. Then the Auslander-Buchweitz theory implies
b if Ext>0 (X , Y ) = 0 (see [AB, Proposition 3.6] or Corollary A.1.3 for the detail). Since
that Y ∈ UΛ0b
X ⊂ CM Λ and Ext>0Λ (CM Λ, Dd Λ) = 0, we obtain Dd Λ ∈ U , which implies (C3 ).
It follows from Lemma 3.4.3(2) that id UΛ ≤ d if and only if U is an injective object in CM Λ.
Thus the remaining assertion easily follows from the definition and (C30 ).
(2) If we have ⊥ U ⊂ CM Λ, then in particular Λ and U are in CM Λ, which in particular implies
that Λ is an R-order. Thus the “only if” part follows. The the “if” part has already shown in the
proof of (1).

A noetherian ring Λ is called Iwanaga-Gorenstein if both id(ΛΛ ) and id(Λ Λ) are finite. By [Za,Lemma A], we have id(ΛΛ ) = id(Λ Λ) in this case. For an Iwanaga-Gorenstein ring Λ, a Λ-module
X ∈ mod Λ is called Gorenstein-projective if X is in GP Λ := ⊥ Λ. Then GP Λ is a Frobenius exact

48

3. CLASSIFICATIONS OF EXACT STRUCTURES AND CM-FINITE ALGEBRAS

category with a projective generator Λ. An Iwanaga-Gorenstein ring Λ is GP-finite if GP Λ isof finite type. These concepts are special cases of cotilting modules, as the following propositionshows. If Λ is an R-order, then this can be easily proved by using Proposition 3.4.4.
Proposition 3.4.5. Let Λ be a noetherian ring.
(1) Λ is Iwanaga-Gorenstein if and only if ΛΛ is a cotilting Λ-module.
(2) Let U ∈ mod Λ be a cotilting Λ-module. Then ⊥ U is Frobenius if and only if Λ isIwanaga-Gorenstein and add U = proj Λ holds.
Proof. (1) First we show the “only if” part. Suppose that Λ is Iwanaga-Gorenstein. ThenΛΛ clearly satisfies (C1) and (C2). Moreover (C3) follows from the fact that ⊥ Λ = GP Λ is the
Frobenius category with an injective cogenerator Λ.
To show the “if” part, we use the main result of [HH]. We refer the reader to [HH] and references therein for the unexplained concepts. In [HH], it was shown that Λ is Iwanaga-Gorenstein
if every module in mod Λ has a finite Gorenstein-projective dimension. Thus it suffices to see that
this is the case if ΛΛ is cotilting, which follows from Proposition 2.4.2.
(2) Clear from (1) and Proposition 3.4.2.
In Chapter 2, it was characterized when a given exact category is exact equivalent to the exact
category of the form ⊥ U for a cotilting module U over a noetherian ring. For an integer n ≥ 1,
recall that E has n-kernels if for every morphism f : X → Y in E, there exists a complex
fn

f2

f1

f

0 → Xn −→ · · · −→ X1 −→ X −→Yin E such that the following diagram is exact.
E(−,fn )E(−,f2 )E(−,f1 )E(−,f )

0 → E(−, Xn ) −−−−−→ · · · −−−−−→ E(−, X1 ) −−−−−→ E(−, X) −−−−→ E(−, Y )Proposition 3.4.6 (= Corollary 2.4.12). Suppose that E := (proj Γ, F ) is an exact category
for a noetherian ring Γ and n ≥ 2 is an integer.
(1) The following are equivalent.
(a) There exist a noetherian ring Λ and a cotilting Λ-module U with id UΛ ≤ n suchthat ⊥ U is exact equivalent to E.
(b) E has projective generators and injective cogenerators, and gl.dim Γ ≤ n holds.
(c) E has projective generators, injective cogenerators, and (n − 1)-kernels.
(2) Suppose that the condition in (1) holds. Take a projective generator P and an injective
cogenerator I, and put Λ := EndE (P ) and U := E(P, I) ∈ mod Λ. Then Λ and U satisfies(1) and E(P, −) gives an exact equivalence E ' ⊥ U .
Proof. Note that the endomorphism ring of every object in E is noetherian, see e.g. [Sa,Proposition 2.3]. Thus Theorem 2.4.11 applies.

3.4.2. Classifications for noetherian R-algebras. To state our classifications, it is convenient to introduce the following terminology.
Definition 3.4.7. Let R be a noetherian complete local ring.
(1) We say that a pair (Λ, U ) is an n-cotilting pair if Λ is a noetherian R-algebra andU ∈ mod Λ is a cotilting Λ-module with id UΛ ≤ n. Cotilting pairs (Λ1 , U1 ) and (Λ2 , U2 )are said to be equivalent if there exists an equivalence mod Λ1 → mod Λ2 which induces
an equivalence add U1 → add U2 .
(2) We say that a pair (Γ, X) is an algebra with dotted arrows if Γ is a noetherian R-algebra
and X is a set of dotted arrows of Q(Γ) (see Definition 3.3.9). Two such pairs (Γ1 , X1 )and (Γ2 , X2 ) are said to be equivalent if there exists an equivalence proj Γ1 → proj Γ2 suchthat X1 corresponds to X2 under the isomorphism Q(Γ1 ) → Q(Γ2 ). For an algebra withdotted arrows (Γ, X), let PX be the direct sum of indecomposable projective Γ-modules
which are not sources of dotted arrows in X. We fix an idempotent eX ∈ Γ satisfyingeX Γ = PX .

3.4. CLASSIFICATIONS OF CM-FINITE ALGEBRAS

49

The following main theorem classifies an n-cotilting pair of finite type by algebras with finite
global dimension and set of dotted arrows.
Theorem 3.4.8. Let R be a noetherian complete local ring. There exists a bijection between
the following for n ≥ 2.
(1) Equivalence classes of n-cotilting pairs (Λ, U ) such that ⊥ U is of finite type.
(2) Equivalence classes of algebras with dotted arrows (Γ, X) such that gl.dim Γ ≤ n andΓ/ΓeX Γ is of finite length over R.
(3) Exact equivalence classes of Hom-noetherian idempotent complete exact R-category E of
finite type such that E has enough projectives, enough injectives and (n − 1)-kernels.
Proof. By Corollary 3.3.10, we have the bijection between (2) and the following.
(4) Exact equivalence classes of (proj Γ, F ), where Γ is a noetherian R-algebra with gl.dim Γ ≤n and F is an exact structure on proj Γ with enough projectives and injectives.
In fact, Proposition 3.3.8 shows that F in (4) is admissible, and Proposition 3.3.14 and Corollary
3.3.15 implies that F has enough projectives and injectives if and only if Γ/ΓeX Γ is of finite length.
By Proposition 3.4.6, we have the bijections between (2), (3) and (4). Finally, we have maps
between (1) and (3) as follows. For an n-cotilting pair (Λ, U ) in (1), we put E := ⊥ U . For an
exact category E in (3), there exists an n-cotilting pair (Λ, U ) such that ⊥ U is exact equivalent to
E by Proposition 3.4.6, which satisfies the condition of (1). The straightforward argument showsthat these maps are mutually inverse to each other.

Next we shall apply Theorem 3.4.8 to GP-finite Iwanaga-Gorenstein algebras. For a translation
quiver Q, we consider a τ -orbit (that is, a connected component of the graph Q0 consisting of thesame vertices as Q and all the dotted translation arrows of Q). We say that a τ -orbit is stable ifevery vertex on it is both a source and a target of dotted arrows.
Corollary 3.4.9. Let R be a noetherian complete local ring. There exists a bijection between
the following for n ≥ 2.
(1) Morita equivalence classes of GP-finite Iwanaga-Gorenstein noetherian R-algebras Λ withid Λ ≤ n.
(2) Equivalence classes of algebras with dotted arrows (Γ, X) satisfying the following.
(a) gl.dim Γ ≤ n.
(b) Γ/ΓeX Γ is of finite length over R.
(c) X is a union of stable τ -orbits in Q(Γ).
Proof. By Proposition 3.4.5, it suffices to show that the pair (Γ, F ) in Theorem 3.4.8(2) gives
the Frobenius exact structure on proj Γ if and only if X in Theorem 3.4.8(2) is a union of stableτ -orbits. Since the exact category proj Γ has enough projectives and injectives by Proposition
3.3.14(2) and Corollary 3.3.15, the exact structure F is Frobenius if and only if the class ofprojectives and that of injectives coincide. For a set of dotted arrows X, an indecomposable object
M in proj Γ is not projective (resp. not injective) in F if and only if there exists a dotted arrow in
X starting at (resp. ending at) M . Therefore the class of indecomposable non-projective objects
and that of indecomposable non-injective objects coincide if and only if X is a union of some stableτ -orbits, which clearly implies the assertion.

We refer the reader to Example 3.1.2 in Section 1 for the detailed example of this classification.
3.4.3. Classifications for R-orders. Restricting Theorem 3.4.8 to the case of R-orders, we
obtain the corresponding result as follows.
Corollary 3.4.10. Let R be a complete Cohen-Macaulay local ring. Then there exists abijection between the following for n ≥ 2.
(1) Equivalence classes of n-cotilting pairs (Λ, U ) such that Λ is an R-order, U is in CM Λand ⊥ U is of finite type.
(2) Equivalence classes of algebras with dotted arrows (Γ, X) satisfying the following.
(a) gl.dim Γ ≤ n.

50

3. CLASSIFICATIONS OF EXACT STRUCTURES AND CM-FINITE ALGEBRAS(b) Γ/ΓeX Γ is of finite length over R.
(c) ΓeX is maximal Cohen-Macaulay as an R-module.
Proof. We check that the bijections in Theorem 3.4.8 restricts to this case. By Proposition
3.4.4(2), we may replace the equivalence class (1) with(10 ) Equivalence classes of n-cotilting pairs (Λ, U ) such that ⊥ U is of finite type and ⊥ U ⊂CM Λ holds.
Suppose that (Λ, U ) and (Γ, X) correspond to each other under Theorem 3.4.8. It suffices toshow that ⊥ U ⊂ CM Λ if and only if ΓeX is maximal Cohen-Macaulay as an R-module. Noticethat add eX Γ is the category of all projective objects in the exact category (proj Γ, F ), thus eX Γ
is a projective generator of it. Hence we may assume that Λ = EndΓ (eX Γ) = eX ΓeX . In thissituation, we have an embedding HomΓ (eX Γ, −) : proj Γ → mod Λ whose essential image is ⊥ U . It
follows that ⊥ U = add HomΓ (eX Γ, Γ) = add ΓeX . Therefore ΓeX is maximal Cohen-Macaulay as
an R-module if and only if ⊥ U ⊂ CM Λ holds.
This gives the following Auslander-type correspondence for Cohen-Macaulay-finite R-orders
with dim R ≥ 2. It largely improves [Iy4, Theorem 4.2.3].
Corollary 3.4.11. Let R be a complete Cohen-Macaulay local ring with dim R = d ≥ 2.
Then there exists a bijection between the following.
(1) Morita equivalence classes of R-orders Λ such that CM Λ is of finite type.
(2) Equivalence classes of algebras with dotted arrows (Γ, X) satisfying the following.
(a) gl.dim Γ = d.
(b) Γ/ΓeX Γ is of finite length over R.
(c) ΓeX is maximal Cohen-Macaulay as an R-module.
Proof. Let us apply Corollary 3.4.10 to the case n = d. By Proposition 3.4.4, it suffices toshow that for a noetherian R-algebra Γ satisfying the conditions of (2)(c), we have gl.dim Γ ≥ d.
Every noetherian R-algebra Γ satisfies dimR Γ ≤ gl.dim Γ , see e.g. [GN2, Corollary 3.5(4)]. Since
ΓeX is a direct summand of Γ and ΓeX is maximal Cohen-Macaulay, we have d = dimR ΓeX ≤dimR Γ ≤ gl.dim Γ.
APPENDIX A

Some results needed in Chapter 2
In this chapter, we collect some results which we used in Chapter 2.
A.1. The Auslander-Buchweitz theory for exact categories
In this section, we shall study the Auslander-Buchweitz approximation theory, developed in[AB], in the context of exact categories. This is a useful tool to investigate cotilting subcategories.
Our results in this appendix are used in Section 4.
First let us recall the following important notions.
Definition A.1.1. Let C be an additive category and D an additive subcategory of C.
(1) A morphism f : DX → X in C is said to be a right D-approximation if DX is in D andevery morphism D → X with D ∈ D factors through f .
(2) D is said to be contravariantly finite if every object in C has a right D-approximation.
Dually we define a left D-approximation and a covariantly finite subcategories.
(3) D is said to be functorially finite if C is both contravariantly finite and covariantly finite.
The Auslander-Buchweitz theory gives a systematic method to provide right and left approximations by a certain subcategory. The following is an exact category version of [AB, Theorem
1.1] and we present a proof for the convenience of the reader.
Proposition A.1.2. Let E be an exact category and X an extension-closed subcategory of E.
Suppose that X has enough injectives W. Then the following hold.
(1) For any C ∈ Xbn , there exist conflations
YC /

/ XC

g
/ YC
C /

f

/ / C,(A.1.1)/ / XC,(A.1.2)cn−1 and Y C ∈ W
cn .
with XC , X C ∈ X , YC ∈ W(2) If E has enough projectives and X is a preresolving subcategory of E, then f is a rightc
X -approximation for C and g is a left W-approximation
for C.
Proof. (1): The proof is by induction on n. Suppose that C ∈ Xb0 = X . Then 0  C  C
gives (A.1.1). Since X has enough injective objects, we have a conflation C  W  C 0 with W
in W and C 0 in X , which is the desired conflation (A.1.2).
Now let n ≥ 0 be an integer and C in Xbn+1 . By the definition of Xbn+1 , there exists a
conflation D  X  C such that D is in Xbn and X is in X . By the induction hypothesis, we
cn−1 andhave conflations YD  XD  D and D  Y D  X D with XD , X D ∈ X , YD ∈ W
Dnc
Y ∈ W . Then we have the following pushout diagram.
D /


YD /

/X


/E


XD


XD
51

//C
//C

52

A. SOME RESULTS NEEDED IN CHAPTER 2

Since X is closed under extensions, E is in X by the middle column, hence the middle row gives(A.1.1). Because X has enough injective objects, we obtain a conflation E  W  F in X withW ∈ W and F ∈ X , which induces the following diagram.
/E
//C
YD /




//G
/W
YD /

F


F

cn by the middle row, and the right column gives (A.1.2).
Thus G is in W(2): Since E has enough projectives and X is a preresolving subcategory of E, it follows that
>0
c
Ext>0
E (X , W) vanishes. Then it is easy to check ExtE (X , W) = 0, and by using the long exact
sequences of Ext one can easily show that f and g are approximations.

Corollary A.1.3. Let E be an exact category with enough projectives and X a preresolving
subcategory of E with enough injectives W. Suppose that Xb = E holds. Then X is contravariantly
c is covariantly finite. Moreover we have X ∩X ⊥ = W and add X = ⊥ W = ⊥ W.
cfinite and X ⊥ = W
n
⊥
n
c .
If Xb = E, then X = W
Proof. We first show X ∩ X ⊥ = W. Since X is a preresolving subcategory of E, it follows
that W ⊂ X ⊥ holds. Let X ∈ X ∩ X ⊥ . Since X has enough injectives W, there exists a conflation
X  W  X 0 with W in W and X 0 in X . Then this sequence splits because Ext1E (X 0 , X) = 0.
Thus X is contained in add W , which implies that X is injective in X . Therefore X is in W.
c Note that W
c ⊂ X ⊥ holds, so it suffices to prove X ⊥ ⊂ W.
c Let C
Next we show X ⊥ = W.
⊥
be in X . By Proposition A.1.2, we have a conflation YC  XC  C with XC in X and YC inc Since X ⊥ is clearly closed under extensions, we have XC ∈ X ∩ X ⊥ = W. It follows from theW.
c that C is in W.
c Note that in case Xbn = E, we may assume that YC is in W
cn−1 ,
definition of W
cn .
thus C is actually in W
c Clearly add X ⊂ ⊥ W = ⊥ W
c holds. Let C be in
Finally we shall show add X = ⊥ W = ⊥ W.
⊥cc and XC in
W. Then by Proposition A.1.2, we have a conflation YC  XC  C with YC in W⊥c
X . Then C ∈ W implies that this sequence splits, which shows that C is a summand of XC .
Consequently, C is in add X , which completes the proof.

Immediately we obtain the following criterion for two preresolving subcategories to be thesame up to summands.
Corollary A.1.4. Let E be an exact category with enough projectives, and let X and X 0 be
c0 = E andpreresolving subcategories of E with enough injectives W and W 0 respectively. If Xb = X
add W = add W 0 hold, then add X = add X 0 .
Proof. The assertion is clear since in this situation add X = ⊥ W holds by Corollary A.1.3.
For an application to cotilting subcategories we studied in Section 4, we have the following
result.
Proposition A.1.5. Let W be an n-cotilting subcategory of E. Then XW is contravariantly
⊥
⊥c
c cn
finite. Furthermore we have X⊥
W = W = W and XW = W = W.
Proof. The assertions follow from Corollary A.1.3 (apply to the case X := XW ), Proposition
2.3.2 and 2.4.2.

A.2. Constructions of exact structures
In this section, we collect two methods to construct new exact structures from a given one.
One is to change exact structures on exact categories, and the other is to give a natural exact
structures to quotient categories of exact categories.

A.2. CONSTRUCTIONS OF EXACT STRUCTURES

53

In what follows, we denote by E an exact category and by C an additive subcategory of E. Our
aim is to construct a new exact structure in which objects in C behave as projective or injective
objects.
We remark that similar results were given in [DRSS] in the case of artin R-algebras, based on
the theory of relative homological algebra developed by Auslander-Solberg [ASo1, ASo2, ASo3].
Definition A.2.1. Let L  M  N be a conflation in E.
(1) It is called a (C, −)-conflation if E(C, M ) → E(C, N ) is surjective for all C in C. In thiscase L  M is called a (C, −)-inflation and M  N is called a (C, −)-deflation.
(2) It is called a (−, C)-conflation if E(M, C) → E(L, C) is surjective for all C ∈ C.
(3) It is called a C-conflation if it is both a (C, −)-conflation and a (−, C)-conflation.
In the obvious way we define the terms (−, C)-inflation, (−, C)-deflation, C-inflation and Cdeflation.
Theorem A.2.2. Let E be an exact category and C an additive subcategory of E. Then the classof all (C, −)-conflations (resp. (−, C)-conflations, C-conflations) defines a new exact structure on
E.
Proof. We first show that all (C, −)-conflations defines a new exact structure on E. It suffices
to check Keller’s axiom (Ex0), (Ex1), (Ex2) and (Ex2)op in [Ke, Appendix A]. Note that the classof all (C, −)-conflations are clearly closed under isomorphisms and (Ex0) “the identity map of zero
object is (C, −)-deflations” is trivial.
(Ex1) The composition of two (C, −)-deflations is a (C, −)-deflation.
Suppose that X  Y and Y  Z be (C, −)-deflations. From the definition, X  Y andY  Z are deflations in E. Thus the composition X  Y  Z is also a deflation in E. Then theclaim follows since the composition E(C, X)  E(C, Y )  E(C, Z) is surjective.
(Ex2) The class of (C, −)-deflations is stable under pullbacks.
Suppose that L  M  N be a (C, −)-conflation and that X → N is an arbitrary morphism.
Since E is an exact category, there exists a pullback diagram
//X
/E
L /
L /


/M


//N

where two rows are conflations. We should check the above row is also a (C, −)-conflation. It isequivalent to say that any morphism C → X factors through E  X for any C ∈ C. Let C → X
be a morphism with C in C. Since M  N is a (C, −)-deflation, the composition C → X → N
factors through M  N . Since the right square is a pullback diagram, there exists a morphism
C → E such that C → E → X is equal to C → X.
(Ex2)op The class of (C, −)-inflations is stable under pushouts.
Suppose that L  M  N is a (C, −)-conflation, and L → X is an arbitrary morphism.
Since E is an exact category, There exists a pullback diagram
/M
//N
L /

X /


/E

//N

where two rows are conflations. We should show that any morphism C → N factors throughE  N for C ∈ C. This is easy because C → N factors through M  N and M  N factors
through E  N .
Thus we have proved that the class of (C, −)-conflations defines an exact structure on E.
Dually the class of (−, C)-conflations also defines another exact structure on E. It is easy tocheck that the intersection of two exact structures gives another exact structure, which implies
that the class of C-conflations also defines an exact structure on E.
Next we consider the ideal quotients of an exact category. The following observation gives anatural way to introduce an exact structure to the ideal quotient of E.

54

A. SOME RESULTS NEEDED IN CHAPTER 2

Proposition A.2.3. [DI, Theorem 3.6] Let E be an exact category and C an additive subcategory of E. Denote by π : E  E/[C] the natural functor. Suppose that every object in C is bothprojective and injective. Then the following are equivalent.
(1) E/[C] is an exact category whose conflations are precisely the essential images of conflations in E under π.
(2) The images of inflations in E under π are monomorphisms and the images of deflationsin E under π are epimorphisms.
In this case, if moreover E has enough projectives P, then E/[C] has enough projectives add π(P).
Proof. We only prove the last assertion. It is clear from the definition of the exact structure
on E/[C] that the images of objects in P under π are projective in E/[C]. For any object X in E,
we have a conflation ΩX  P  X in E with P being projective. Sending it by π, we obtaina conflation ΩX  P  X. From this it follows that E/[C] has enough projectives, and thatprojective objects are precisely objects in add π(P).
APPENDIX B

The AR theory for exact categories over a noetherian ringThis chapter contains a brief discussion of the Auslander-Reiten theory for exact categories
which are Hom-noetherian over a noetherian complete local ring R. We show that the existenceof AR conflations is closely related to the AR duality and dualizing R-varieties (Theorem B.2.5).
These observations shed new light on the result on isolated singularities by Auslander [Au4], see
Remark B.2.6.
B.1. Existence of AR conflations
Let E be a Krull-Schmidt category. We denote by J the Jacobson radical of E. We need later
the following easy lemma about kernel-cokernel pairs in Krull-Schmidt categories.
Lemma B.1.1. Suppose that X → Y → Z is a kernel-cokernel pair in E. Then this complex
is isomorphic to the direct sum of kernel-cokernel pairs of the following forms.
(1) A = A → 0 for some A in E.
(2) 0 → B = B for some B in E.
(3) X 0 → Y 0 → Z 0 with all morphisms in J .
fg

Let E be a Krull-Schmidt exact category. Recall that a conflation X −→Y −→ Z in E is an
AR conflation if f is left almost split and g is right almost split. It immediately follows from thefgdefinition that if X −→Y −→ Z is an AR conflation, then f is left minimal, that is, any ϕ : Y → Y
satisfying ϕf = f is an automorphism. Dually g is right minimal in this case. Also one can provethe uniqueness of the AR conflation as in the classical case.
The following classical observation is useful for us. For an object X in E, we put SX :=PX /J (−, X) ∈ Mod E and S X := P X /J (X, −) ∈ Mod E op .
Proposition B.1.2. Let E be a Krull-Schmidt category. Then the following hold.
(1) The map X 7→ SX gives a bijection between ind E and the set of isomorphism classes ofsimple object in Mod E.
Pgg(2) A morphism Y −→ Z is right almost split if and only if Z is indecomposable and PY −−→
PZ → SZ → 0 is exact.
(3) There exists a right almost split morphism ending at Z if and only if SZ is finitely
presented.
Proof. It is well-known, see [Au3] for example.



The following proposition says that the existence of the right almost split map ensures theexistence of the AR conflation.
Proposition B.1.3. Let Z be an indecomposable non-projective object in E. Then the following are equivalent.
(1) There exists a right almost split morphism to Z.
(2) There exists an AR conflation X  Y  Z in E ending at Z.
Proof. (1) ⇒ (2): Since Z is not projective, there exists some object M in D such that
M (Z) 6= 0 by Proposition 3.2.12. This means that there exists some non-zero morphism a : PZ →M by the Yoneda lemma. Since Im a is finitely generated and D is a Serre subcategory of mod E,
we have that Im a is also in D. Since PZ has a unique maximal subobject J (−, Z), it is easy to see
55

56

B. THE AR THEORY FOR EXACT CATEGORIES OVER A NOETHERIAN RING

that the cokernel of the composition J (−, Z) → PZ → Im a is isomorphic to SZ . Since J (−, Z)fg

is finitely generated by (1), we have SZ ∈ D. Therefore, we obtain a conflation X −→Y −→ Z suchthat g is right almost split. Now we may assume that f is in J by Lemma B.1.1. This clearly
implies that g is right minimal and the classical argument shows that f is left almost split.
(2) ⇒ (1): Obvious from the definition.

We say that E has AR conflations if for every indecomposable non-projective object Z there
exists an AR conflation ending at Z, and for every indecomposable non-injective object X there
exists an AR conflation starting at X.
Corollary B.1.4. Suppose that E := proj Λ is an exact category for a semiperfect noetherianring Λ. Then E has AR conflations.
Proof. We have Mod E ' Mod Λ, so every simple object in this category is finitely presented
because Λ is noetherian. Thus the assertion holds by Propositions B.1.2 and B.1.3.

B.2. AR conflations, the AR duality and dualizing varieties
Next we investigate the relationship between AR conflations and the AR duality in the general
setting. In what follows, we denote by R a noetherian complete local ring.
We denote by I the injective envelope of the simple R-module R/ rad R, and by D : Mod R →Mod R the Matlis dual D = HomR (−, I). It is well-known that this induces a duality D :noeth R ↔ art R and D : f.l. R ↔ f.l. R, where noeth R, art R and f.l. R denotes the subcategory of
Mod R consisting of R-modules which are noetherian, artinian and of finite length respectively.
From now on, we assume that E is a Hom-noetherian idempotent complete exact R-category.
Recall that a morphism f : W → Z in E is projectively trivial if for each object X, the induced
map Ext1E (Z, X) → Ext1E (W, X) is zero. It is easy to see that f is projectively trivial if and only
if f factors through every deflation Y  Z. If E has enough projectives, then f is projectively
trivial if and only if f factors through some projective object. Denote by P(X, Y ) the set of
all projectively trivial morphisms from X to Y . Then P is a two-sided ideal of E and we put
E := E/P, which we call the projectively stable category. Dually we define the notion of injectively
trivial morphisms and the injectively stable category E := E/I.
The following theorem is an exact category version of [RVdB1, Proposition I.2.3], and it
generalizes [LNP, Theorem 3.6] and [Ji, Proposition 2.4]. Also see [LNP, IN] for related work.
Note that we do not assume that R is artinian or E is Hom-finite. This enables us to give anotherproof of the one implication of Auslander’s theorem about isolated singularities, see Remark B.2.6
Proposition B.2.1. Suppose that E is Hom-noetherian Krull-Schmidt exact R-category suchthat either (i) E is Ext-noetherian or (ii) E is Hom-finite. Let Z be an indecomposable nonprojective object in E. Then the following are equivalent.
(1) There exists a right almost split morphism to Z.
(2) There exists an AR conflation X  Y  Z ending at Z.
(3) The functor D Ext1E (Z, −) : E → Mod R is representable.
Moreover, if X  Y  Z is an AR conflation, then D Ext1E (Z, −) ' E(−, X) holds, and thus
Ext1E (Z, W ) and E(W, X) are of finite length over R for every W ∈ E.
Proof. (1) ⇔ (2): This is Proposition B.1.3.
(2) ⇒ (3): Suppose that η : X  Y  Z is an AR conflation. We regard 0 6= η ∈ Ext1E (Z, X).
Since I is a cogenerator of Mod R, there exists a non-zero morphism γ : Ext1E (Z, X) → I in Mod R
such that γ(η) 6= 0. Then we have a morphismh−, −iW : E(W, X) ⊗R Ext1E (Z, W ) → I,(h, µ) 7→ γ(hµ)

in Mod R, where hµ is the image of µ by the induced morphism Ext1E (Z, f ) : Ext1E (Z, W ) →Ext1E (Z, X). Then the same proof as in [Ji, Lemma 2.1] applies here to show that h−, −iW is
non-degenerate in both variables. Therefore, we have two injections E(W, X) ,→ D Ext1E (Z, W )and Ext1E (Z, W ) ,→ DE(W, X) in Mod R. They are obviously isomorphisms if (ii) E is Hom-finite.

B.2. AR CONFLATIONS, THE AR DUALITY AND DUALIZING VARIETIES

57

Suppose that (i) E is Ext-noetherian. Then D Ext1E (Z, W ) is artinian, thus E(W, X) is of finite
length. Therefore two injections are isomorphisms and both of E(W, X) and Ext1E (Z, W ) are offinite length over R. The naturality in W is clear from a direct calculation, so we obtain an
isomorphism of functors D Ext1E (Z, −) ' E(−, X).
(3) ⇒ (2): The proof of [Ji, Lemma 2.2] applies here.
In the rest of the appendix, we show that the existence of AR conflations are closely related
to the notion of dualizing R-varieties, introduced in [AR1] and [AR4].
Definition B.2.2. Let C be a Hom-finite idempotent complete R-category. We say that C isa dualizing R-variety if it satisfies both of the following.
(1) For any finitely presented R-functor F : C op → f.l. R, the composition functor DF :C op → f.l. R → f.l. R is finitely presented.
(2) For any finitely presented R-functor G : C → f.l. R, the composition functor DF : C →f.l. R → f.l. R is finitely presented.
The following properties are immediate, which we state without proofs.
Lemma B.2.3. Let C be a dualizing R-variety. Then the following hold.
(1) The category of finitely presented functors mod1 C and mod1 C op are abelian categories.
(2) D induces a duality D : mod1 C ↔ mod1 C op .
The following technical lemma is elementary.
Lemma B.2.4. Let E be an exact category with enough projectives and injectives. Then mod1 E
is an abelian category with enough projectives and injectives. The functor E → mod1 E defined by
X 7→ E(−, X) gives the equivalence between E and the category of projective objects in mod1 E,
and the functor E → mod1 E defined by X 7→ Ext1E (−, X) gives the equivalence between E and thecategory of injective objects in mod1 E.
Proof. To show that mod1 E is abelian, it suffices to show that E has weak kernels, see [Au1].
Let g : Y → Z be any morphism in E. Since E has enough projectives, we have the following
pullback diagram with P being projective.
Z0 /
Z0 /

/E

/P

a

//Y

(B.2.1)
g

b
c


//Z

It is straightforward to check that a gives the weak kernel of ϕ ∈ E(X, Y ).
The remaining assertion follows from the same argument as in [Iy4, Theorem 2.2.2(1)].



Now we are in position to prove our main results in this appendix, which is an exact category
version of [AR4, Proposition 2.2].
Theorem B.2.5. Let E be a Hom-noetherian idempotent complete exact R-category withenough projectives and injectives. The following are equivalent.
(1) The category E has AR conflations.
(2) E is a dualizing R-variety.
(3) E is a dualizing R-variety.
(4) There exist mutually inverse equivalences τ : E  E : τ − such that we have natural
isomorphisms DE(τ − X, Z) ∼= Ext1E (Z, X) ∼= DE(X, τ Z).
In this case, E is Ext-finite and E and E are Hom-finite.
Proof. (1) ⇒ (2): First note that E is Hom-finite by Proposition B.2.1. Let F ∈ mod1 E
be a finitely presented functor. We have an exact sequence E(−, Y ) → E(−, Z) → F → 0 in
Mod E, which is induced from g : Y → Z in E by the Yoneda lemma. Consider the diagramt[b,−a][c,g](B.2.1). It is standard that the right square gives a conflation E −−−−→ P ⊕ Y −−→ Z (see e.g.

58

B. THE AR THEORY FOR EXACT CATEGORIES OVER A NOETHERIAN RINGfg

[Bü, Proposition 2.12]). Hence by replacing g by [c, g], we may assume that X −→Y −→ Z is a(−)◦g

conflation with E(−, Y ) −−−−→ E(−, Z) → F → 0 being exact.
By the long exact sequence of Ext,
Ext1 (−,f )

(−)◦g

E
E(−, Y ) −−−−→ E(−, Z) → Ext1E (−, X) −−−−−−−→ Ext1E (−, Y )is exact in Mod E. Thus we have an exact sequence
Ext1 (−,f )E
0 → F → Ext1E (−, X) −−−−−−−→ Ext1E (−, Y ).

Since D : Mod E → Mod E op is a contravariant exact functor, we obtain an exact sequence
D Ext1E (−, Y ) → D Ext1E (−, X) → DF → 0.
By (1) and Proposition B.2.1, the functors D Ext1E (−, Y ) and D Ext1E (−, X) are representable,
which shows that DF is finitely presented.
Next suppose that G ∈ mod1 E op is a finitely presented functor. We have an exact sequenceE(B, −) → E(A, −) → G → 0 in Mod E op . Applying D, we obtain an exact sequence 0 → DG →DE(A, −) → DE(B, −) in Mod E. We will show that DE(A, −) ' Ext1E (−, τ A) for some τ A in E.
Without loss of generality, we may assume that A has no non-zero projective summands. Then
we have a direct sum of AR conflations τ A  E  A in E. By Proposition B.2.1, DE(A, −) 'Ext1E (−, τ A) holds. Thus we obtain the following exact sequence in Mod E.
0 → DG → Ext1E (−, τ A) → Ext1E (−, τ B)
Since E has weak kernels by Lemma B.2.4, the subcategory mod1 E is closed under kernels in
Mod E (see [Au1]). Also Lemma B.2.4 shows that Ext1E (−, τ A) is finitely presented. Thus DG isalso finitely presented.
(2) ⇒ (4): Since E is a dualizing R-variety, we have a duality D : mod1 E op ' mod1 E between
abelian categories. It induces a duality D : P ' I, where P (resp. I) denotes the category of
projective objects in mod1 E (resp. injective objects in mod1 E op ). By Lemma B.2.4, we have aequivalence I ' E and the contravariant Yoneda embedding E ' P. Define the equivalence τ bythe composition E ' P ' I ' E and denote by τ − its quasi-inverse. Then it follows from Lemma
B.2.4 that (4) holds.
(4) ⇒ (1): Obvious from Proposition B.2.1.
By duality, (3) is also equivalent to all the other conditions.
This theorem gives an application about the relation between AR conflations and isolated
singularities shown in [Au4]. For a complete Cohen-Macaulay local ring R, we say that an Rorder Λ has at most an isolated singularity if gl.dim Λ ⊗R Rp = ht p holds for every non-maximal
prime ideal p of R. It is well-known that an R-order Λ has at most an isolated singularity if andonly if CM Λ is Hom-finite.
Remark B.2.6. Suppose that CM Λ has AR conflations. Then CM Λ is a dualizing R-variety
by Theorem B.2.5, hence in particular CM Λ is Hom-finite, and therefore Λ has at most an isolated
singularity. This gives a simple conceptual proof of the “only if” part of the main theorem of[Au4]; CM Λ has AR conflations if and only if Λ has at most an isolated singularity.

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