CLASSIFICATIONS OF EXACT STRUCTURES AND
COHEN-MACAULAY-FINITE ALGEBRASHARUHISA ENOMOTO

Abstract. We give a classification of all exact structures on a given additive category. Using this, we investigate the structure of an exact category with finitely many
indecomposables. As an application of this, we give an explicit classification of CohenMacaulay-finite Iwanaga-Gorenstein algebras.

1. Introduction
Exact categories, in the sense of Quillen, have been playing an important role in the representation theory of algebras. In general, an additive category has many exact structures.
Recently, Rump [9] showed that every additive category has the largest exact structures,
but no general description of exact structures was known. We give an explicit descriptionof all exact structures on a given additive category E by using particular modules over E.
Apart from the purely theoretical interest, another motivation for classifying exact
structures comes from the “relative representation theory” of algebras, such as the CohenMacaulay representation theory. First, let k be a field and consider the “absolute” representation theory of finite-dimensional k-algebras, which investigates the category mod Λof finitely generated Λ-modules. An algebra is representation-finite if there exists only
finitely many indecomposable objects in mod Λ up to isomorphism. As for such algebras,
there are bijections between the following classes, called an Auslander correspondence.
(1) Representation-finite finite-dimensional k-algebras Λ.
(2) Abelian Hom-finite k-categories E with finitely many indecomposables.
(3) Finite-dimensional k-algebras Γ satisfying gl.dim Γ ≤ 2 ≤ dom.dim Γ.
The map from (1) to (2) is given by Λ 7→ mod Λ, and from (3) to (2) by Γ 7→ proj Γ.
Actually, the bijection between (2) and (3) is induced from the bijection between:(2)0 Hom-finite Krull-Schmidt k-categories E with finitely many indecomposables.
(3)0 Finite-dimensional k-algebras Γ.
We call the algebra Γ in corresponding to E under the above bijection an Auslander algebra
for E. Under the bijection between (2)0 and (3)0 , it is natural to guess that categoricalproperties of E and the homological property of Γ are related. In this viewpoint, theAuslander correspondence can be interpreted as saying that the abelianness of E can becharacterized by the purely homological behavior of its Auslander algebra.
Now let us move to the relative situation. For a given algebra Λ, in addition to themodule category mod Λ, there are several important subcategories of it which have been
The detailed version [6] of this paper has been submitted for publication elsewhere.

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investigated, e.g. the category CM Λ of Cohen-Macaulay Λ-modules. Usually, these “relative representation categories” are extension-closed in mod Λ, thus naturally has the
structure of an exact category. The main motivating question in our study is whether
there exists a bijection between the following:(1) Relative representation-finite algebras Λ.
(2) Hom-finite Krull-Schmidt k-categories E with finitely many indecomposables which
satisfy some categorical properties.
(3) Finite-dimensional k-algebras satisfying some homological properties.
Here we fix some relative theory, and an algebra is relative representation-finite if its
relative representation category has finitely many indecomposables. The maps are defined(if possible) as follows: For Λ in (1), we define E in (2) to be the relative representation
category of Λ, and the map between (2) and (3) is a restriction of (2)0 and (3)0 above.
Actually several bijections of this type are known, e.g. for one-dimensional orders [1],1-cotilting modules and 1-Iwanaga-Gorenstein algebras [7].
In general, however, 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 suchthat CM Λ and CM Λ0 are equivalent. The problem is that the algebra Λ cannot be recovered from the additive structure of the representation category like CM Λ. Nevertheless,
we can usually recover Λ from the representation category of Λ together with the exact
structure on it. Therefore, by using exact structures, we should modify and divide thequestion into the following.
(A) Construct a bijection between the “exact version” of (1) and (2), that is, characterize exact categories which are exact-equivalent to the representation category.
(B) Construct a bijection between the “exact version ” of (2)0 and (3)0 , that is, classify
the exact structure on a given category in terms of its Auslander algebra.
Problem (A) is something like an Morita-type theory of exact categories, and was tackledwith in [5]. In this article, we focus on Problem (B).
Remark 1. Although we concentrate on categories of finite type in this article, the completely similar argument works for general idempotent complete additive categories byusing functor categorical method, except arguments about 2-regular simples. We refer
the interested reader to [6] for details.
1.1. Convention. For simplicity, throughout this article, we fix a field k. All algebras
are finite-dimensional k-algebras. All categories are skeletally small Hom-finite KrullSchmidt k-categories, and all subcategories are assumed to be full, additive and closed
under direct summands.
For an algebra Λ, we denote by mod Λ (resp. proj Λ) the category of finitely generated
right Λ-modules (resp. finitely generated projective right Λ-modules).
For an object G in a category E, we denote by add G the subcategory of E consistingof direct summands of finite direct sums of G. A category E is called of finite type if E
has only finitely many indecomposable objects up to isomorphism, or equivalently, there
exists G ∈ E such that E = add G. In this case such an object G is called an additive
generator of E.

–2–2. Preliminaries on exact categories and Auslander algebras
We fix some terminology and recall basic properties of exact categories and auslander
algebras. We refer the reader to [4] for the basics of exact categories.
fg

Let E be an additive category. A complex X −→Y −→ Z in E is a kernel-cokernel pair
if f is a kernel of g and g is a cokernel of f . An exact category is a pair (E, F ) consistingof an additive category E and a class F of kernel-cokernel pairs in E which satisfy someconditions (see [4]). In this case, we say that F is an exact structure for E, and a complex
in F is called a conflation.
Example 2. Let Λ be an algebra and E a subcategory of mod Λ. If E is closed under
extensions, then E has the natural exact structure, whose conflations are short exact
sequences in mod Λ with all terms in E.
As we mentioned in the introduction, categories of finite type are just finite-dimensional
algebras, seen from another perspective:
Proposition 3. There exists a bijection between the following classes.
(1) Equivalence classes of Hom-finite Krull-Schmidt categories E of finite type.
(2) Morita-equivalence classes of finite-dimensional k-algebras Γ.
The map from (1) to (2) is given by Γ := EndE (G), where G is an additive generator ofE, and from (2) to (1) by E := proj Γ.
In what follows, Assumption (∗) means that E is of finite type, G is an additive generator of E and Γ := EndE (G). Our aim is to translate the information of exact structures
on E into the homological behavior of Γ. To this purpose, the following equivalence called
Auslander’s projectivization plays an important role.
Lemma 4. Assume (∗). Then we have the following equivalence and duality:(1) P(−) := E(G, −) : E ' proj Γ.
(2) P (−) := E(−, G) : E ' proj Γop .
Moreover these satisfies HomΓ (P(−) , Γ) ' P (−) .
Since an exact structure on E is a class of kernel-cokernel pairs in E, let us investigate
kernel-cokernel pairs in E in terms of Γ.
fg

Proposition 5. Assume (∗). Let X −→ Y −→ Z be a complex in E and put M :=coker(Pg ) ∈ mod Γ.
(1) f is a kernel of g if and only if the following is exact in mod Γ. Thus pd MΓ ≤ 2.
Pf

Pg

0 → PX −→ PY −→ PZ → M → 0(2) Suppose (1). Then (f, g) is a kernel-cokernel pair if and only if ExtΓ0,1 (M, Γ) = 0.
Proof. We give a sketch of the proof. (1) is immediate from the definition of kernels in anPg

Pfadditive category. By duality, g is a cokernel of f if and only if 0 → P Z −→ P Y −→ P X
is exact. Since P (−) ' HomΓ (P(−) , Γ) and we are assuming (1), this is equivalent toExtΓ0,1 (M, Γ) = 0.
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As can be inferred from this, the following category will play a crucial role.
Definition 6. Let Γ be an algebra. A subcategory C2 (Γ) of mod Γ consists of modules
M satisfying pd MΓ ≤ 2 and ExtΓ0,1 (M, Γ) = 0.
Under assumption (∗), each object in C2 (Γ) corresponds to a kernel-cokernel pair in E.
Therefore, it is natural to guess that exact structures on E correspond to a somewhat nicesubcategories of C2 (Γ).
3. Main Results
Let Γ be an algebra. Recall that a subcategory D of mod Γ is called Serre if it is closed
under extensions, submodules and quotients. Now we state the main result of this article.
Theorem 7. Assume (∗). Then there exists a bijection between the following:(1) Exact structures F on E.
(2) Subcategories D of C2 (Γ) satisfying the following conditions.
(a) D is Serre in mod Γ.
(b) Ext2Γ (D, Γ) is Serre in mod Γop .
More precisely, suppose that F and D correspond to each other.
fg• M ∈ mod Γ belongs to D if and only if there is a conflation X −→Y −→ Z in F∼
such that M = coker Pg .
f

Pf

g

Pg

• A complex X −→Y −→ Z in E belongs to F if and only if 0 → PX −→ PY −→ PZ →M → 0 is exact for M ∈ D.
Since Serre subcategories of mod Γ are uniquely determined by sets of simple Γ-modules,
we can describe exact structures more explicitly. Here the following notion is essential,
which is similar to the behavior of the simple over 2-dimensional regular local ring.
Definition 8. Let Γ be an algebra. A simple Γ-module S is called 2-regular if it satisfies
the following two conditions.
(1) S ∈ C2 (Γ), that is, pd SΓ = 2 and Ext0,1
Γ (S, Γ) = 0.
(2) Ext2Γ (S, Γ) is a simple Γop -modules.
Assume (∗). As in the classical (functorial) Auslander-Reiten theory, 2-regular simple
modules correspond to almost split kernel-cokernel pairs in E. In order to help understand2-regular simples visually, let us introduce the translation quiver Q(E). The underlying
quiver of Q(E) is nothing but the usual quiver of E, that is, vertices of Q(E) are isomorphism classes of indecomposable objects in E, and arrows are drawn depending on spacesof irreducible maps between objects in E. Next we draw a dotted arrow X L99 Z if there
exists an almost split kernel-cokernel pairs X → Y → Z in E. This happens precisely
when the simple quotient SZ of PZ is a 2-regular simple Γ-module and Ext2Γ (SZ , Γ) is thesimple quotient of P X .
By investigating simple modules corresponding to a Serre subcategory D of mod Γ in
Theorem 7(2), we obtain the following description.
Theorem 9. Assume (∗). Then we can add the following to Theorem 7.
(3) Sets S of 2-regular simple Γ-modules.

–4–(4) Sets A of dotted arrows in Q(E)(= Q(proj Γ)).
Remark 10. Assume (∗) and suppose that E has an exact structure corresponding to A
in Theorem 9(4). Then a kernel-cokernel pair X → Y → Z in E which corresponds to adotted arrow X L99 Z in A is nothing but an almost split conflation of E. Furthermore,
an indecomposable object X is projective (resp. injective) in the exact category E if andonly if there exists no dotted arrow in A starting at (resp. ending at) X.
Example 11. Let Λ be a representation-finite algebra and E := mod Λ. Then the algebraΓ corresponding to E is nothing but the classical Auslander algebra for Λ. The translation
quiver Q(E) is the same is the usual Auslander-Reiten quiver of mod Λ. Theorem 9amounts to say that exact structures on E are parametrized by (basic) generators, by
taking the direct sum of projective objects. This was proved in [3], and actually it is oneof the motivations of this study to generalize it.
4. Applications
Now we apply our results to the relative representation theory.
Definition 12. Let Λ be an algebra.
(1) Λ is called an Iwanaga-Gorenstein if both id ΛΛ and id Λ Λ are finite.
(2) A module M ∈ mod Λ over an Iwanaga-Gorenstein algebra Λ is called CohenMacaulay if Ext>0
We denote by CM Λ the category of CohenΛ (M, Λ) = 0.
Macaulay Λ-modules.
(3) An Iwanaga-Gorenstein algebra Λ is CM-finite if CM Λ is of finite type.
Example 13. Typical examples of CM-finite Iwanaga-Gorenstein algebras are the following.
• A representation-finite self-injective algebra Λ. In this case, CM Λ = mod Λ holds.
• An algebra Λ with finite global dimension. In this case, CM Λ = proj Λ holds.
Note that CM Λ is closed under extensions in mod Λ, thus has the natural exact structure. Concerning Problem (A) in the introduction, we have the following result.
Theorem 14. Assume (∗) and suppose that E is an exact category. Then the following
are equivalent.
(1) There exists an Iwanaga-Gorenstein algebras Λ(which is automatically CM-finite)such that E is exact-equivalent to CM Λ.
(2) Γ has finite global dimension, and the classes of projective and injective objects in
E coincide.
Combining this with Theorem 9, we immediately obtain the following classification
result.
Corollary 15. There exists a bijection between the following:(1) Morita-equivalence classes of CM-finite Iwanaga-Gorenstein algebras Λ.
(2) Equivalence classes of pairs (Γ, A), where Γ is an algebra with finite global dimension, and A is a set of dotted arrows in Q(proj Γ) which is a union of oriented
cycles.

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For (Γ, A) in (2), the corresponding algebra Λ is obtained by taking the endomorphismring of the direct sum of modules not lying in A. In this case, CM Λ ' proj Γ holds.
In this theorem, roughly speaking, Γ parametrizes all possible CM categories E, andfor each Γ, A parametrizes all Iwanaga-Gorenstein algebras whose CM category is E.
Remark 16. Actually, Corollary 15 is a special version of the result about cotilting Λmodules U such that its Ext-perpendicular category ⊥ U is of finite type, see [6]. In thiscase, we consider the pair (Λ, U ) in (1) and the restriction of A in (2) is dropped.
Example 17. Put Λ := k[X]/(X 4 ) and E := mod Λ. It is well-known that Λ is representationfinite, thus E is of finite type and the corresponding Γ is the classical Auslander algebraof Λ. Then Q(E) looks like1234
together with three dotted loops starting at 1, 2 and 3. Then there exists 23 = 8 choices
of A satisfying Corollary 15 since every set of dotted arrows are allowed. The resulting
eight Iwanaga-Gorenstein algebras are all the Iwanaga-Gorenstein algebras such that their
CM category is equivalent to E. All of this are 2-Iwanaga-Gorenstein. More generally,
starting from representation-finite algebra Λ, this procedure yields CM-finite 2-IwanagaGorenstein algebras. This process is the same as τ -selfinjective algebras in [2], or 0precluster-tilted algebras in [8].
Corollary 15 actually gives somewhat computable algorithm to produce CM-finite
Iwanaga-Gorenstein algebras. More precisely, all CM-finite Iwanaga-Gorenstein algebras
Λ are obtained by the following steps.
(1) Take an algebra Γ with finite global dimension.
(2) Compute Q(proj Γ)(3) For each cyclic orbit A of Q(proj Γ), compute the endomorphism ring Λ of vertices
not lying on A.
Moreover, Corollary 15 seems to be the best possible classification of CM-finite IwanagaGorenstein algebras by the following reason: If an algebra Λ has finite global dimension,then Λ is Iwanaga-Gorenstein with CM Λ = proj Λ, hence is CM-finite. Therefore thecomplete classification of CM-finite Iwanaga-Gorenstein algebras, if exists, should containthat of algebras with finite global dimension, which is unlikely to be settled without
any further restriction. Corollary 15 decomposes the classification problem of CM-finite
algebras into (1) that of algebras with finite global dimension and (2) the computation of
the translation quiver associated with the algebra.
However, there are several problems on Corollary 15. One of those is that, for agiven algebra Γ, there seems to be no systematic algorithm to draw Q(proj Γ), without
calculating explicitly by hand. For actual computation, we first should chose algebras
with finite global dimension, but there are too many and it is not clear which classes ofsuch algebras are suitable for computation.
One of the candidates is the class of strict τ -algebra introduced in [7], which is an
Auslander algebra of torsion classes of finite type. Purely combinatorial conditions on
the translation quiver Q are known such that its mesh algebra Γ := k(Q) yields a strictτ -algebra, and in this case Q(proj Γ) = Q holds. Thus we do not have to compute thetranslation quiver Q(proj Γ) if we start from Q.

–6–References[1] M. Auslander, K. W. Roggenkamp, A characterization of orders of finite lattice type, Invent. Math.
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Graduate School of Mathematics
Nagoya University
Chikusa-ku, Nagoya, 464-8602, Japan
E-mail address: m16009t@math.nagoya-u.ac.jp–7–