Classifications of exact structures and Cohen-Macaulay-finite algebras
Haruhisa Enomoto
In this talk, I will discuss a classification of exact structures on a given additive category and its
application, based on [1]. Exact categories, in the sense of Quillen, have been playing an important rolein the representation theory of algebras. In general, an additive category has many exact structures.
Recently, Rump [3] showed that every additive category has the largest exact structures, but no generaldescription of exact structures was known. We give an explicit description of all exact structures on agiven additive category E by using particular modules over E (equivalently, modules over the Auslander
algebra of E).
Let k be a field. For simplicity, all algebras are assumed to be finite dimensional over k. To this end,
the following condition for simple modules plays an indispensable role.
Definition 1. Let Γ be an algebra and S a simple Γ-module. We say that S satisfies the 2-regularcondition if the following conditions are satisfied.
(1) The projective dimension of S is equal to 2.
(2) ExtiΓ (S, Γ) = 0 for i = 0, 1.
(3) Ext2Γ (S, Γ) is simple Γop -module.
Surprisingly, the following shows that categorical notion (exact structures) is deeply related to homological condition (2-regular conditions). Also this can be seen as a classification of exact categories withfinitely many indecomposables.
Theorem 2. Let E be an idempotent complete Hom-finite additive k-category with finitely many indecomposables, and let Γ be its Auslander algebra. Then there exists a bijection between the following twoclasses.
(1) Exact structures on E.
(2) Sets of simple Γ-modules satisfying the 2-regular condition.
(3) Sets of dotted arrows in the translation quiver Q(Γ) associated with Γ.
As an application, we give the Auslander-type correspondence for Cohen-Macaulay-finite IwanagaGorenstein algebras. We say that an algebra Λ is Iwanaga-Gorenstein if the left and right injective
dimension of Γ itself is finite. For such an algebra Λ, a finitely generated Γ-module X is called CohenMacaulay if ExtiΛ (X, Λ) = 0 for all i > 0. We say that an Iwanaga-Gorenstein algebra is Cohen-Macaulayfinite (CM-finite) if there exist finitely many Cohen-Macaulay modules up to isomorphism. By using theprevious theorem and the results in [2], we proved the following.
Theorem 3. There exists a bijection between the following two classes.
(1) Morita-equivalence classes of CM-finite Iwanaga-Gorenstein algebras.
(2) Equivalence classes of pairs (Γ, X), where Γ is an algebra with finite global dimension and X is aunion of stable τ -orbits in the translation quiver Q(Γ).
Moreover, we give an explicit method to construct a CM-finite algebra from the pair (Γ, X). This
gives a systematic method to construct CM-finite Iwanaga-Gorenstein algebras, and all such algebras are
obtained in this way. Thus our result reduces the classification problem of CM-finite Iwanaga-Gorenstein
algebras to that of algebras with finite global dimension.
References[1] H. Enomoto, Classifications of exact structures and Cohen-Macaulay-finite algebras, arXiv:1705.02163.
[2] H. Enomoto, Classifying exact categories via Wakamatsu tilting, J. Algebra 485 (2017), 1–44.
[3] W. Rump, On the maximal exact structure on an additive category, Fund. Math. 214 (2011), no. 1, 77–87.

Graduate School of Mathematics
Nagoya University
Frocho, Chikusaku, Nagoya 464-8602 Japan
Email: m16009t@math.nagoya-u.ac.jp–1–