Maurice Auslander Memorial Workshop – Abstracts

Here you can find and titles and abstracts. The workshop offers two types of talks, namely background talks on Auslander's work (which are delivered by four younger participants) and lectures on advanced topics. The list begins with the background talks; these abstracts also contain some information about literature.

Background talks on Auslander's work

Hanno Becker (Bonn)
The Auslander–Buchsbaum formula

This talk will be about the Auslander–Buchsbaum formula [AB57, Theorem 3.7]:

Theorem 1 (Auslander-Buchsbaum). If R is a commutative local Noetherian ring and M a f.g. R-module of finite projective dimension proj-dim_R(M), then

Firstly, I will recall some basic notions from commutative algebra that go into understanding, proving and using the Auslander-Buchsbaum formula, and present the classical inductive proof of (AB) afterwards. This part will largely follow [BH93, §1.1-§1.3], in which (AB) is stated as Theorem 1.3.3. As an application, I will discuss the equivalence of regularity and finiteness of global dimension [AB57, Theorem 1.10], [BH93, Theorem 2.2.7].

Afterwards, I will present a quick proof of (AB) on the basis of the machinery of derived categories, see [FI03]. These techniques even extend to yield a proof of the Auslander-Bridger formula [AB69, Theorem 4.13], [Chr00, Theorem 1.4.8]:

References

Ryo Kanda (Nagoya)
Auslander-Buchweitz approximations and cotilting modules

In this talk, I will give an introduction to [AB89] and [AR91]. These two papers relate Cohen-Macaulay representations and tilting theory by making an analogy between canonical modules and cotilting modules. I explain these theories from the viewpoint of cotorsion pairs.

More precisely, it is shown in [AB89] that for a commutative noetherian Cohen-Macaulay local ring with a canonical module, every module over the ring has two kinds of approximations to maximal Cohen-Macaulay modules and modules of finite injective dimension. [AB89] proved more general statements using abelian categories, which can be applied to Iwanaga-Gorenstein rings and Cohen-Macaulay orders.

The paper [AR91] illustrates how to construct a cotorsion pair from a cotilting module (in the sense of Miyashita and Happel) over an artin algebra. We will also see a correspondence between basic cotilting modules and contravariant resolving subcategories, which becomes a bijection if the algebra has finite global dimension.

References

Jeremy Russell (Ewing)
Functorial methods in representation theory

According to Auslander himself as stated in [3], many of his results in representation theory were motivated and originally proved using functorial methods; however, he initially presented the results without using this language in order to reach a wider audience. This talk will focus on the functorial techniques Auslander developed in order to study representation theory.

The main idea behind the approach is that one may study an abelian category A by studying the category of finitely presented functors. I will begin by introducing this category and discussing some of its important homological properties. This will be followed by a discussion of how Auslander applied this approach to representation theory of finite dimensional k-algebras. Next I will explain how the approach has been adapted to study A-modules for an arbitrary pre-additive category A. Auslander's duality between the categories of covariant and contravariant finitely presented functors will be discussed. I will finish by establishing deep connections between finitely presented functors and model theory.

References

Johan Steen (Trondheim)
Morphisms determined by objects

In this talk, we will discuss the notion of morphisms determined by objects, originally introduced by Auslander in his Philadelphia notes [Aus78a, Aus78b]. This topic was treated in the last chapter of Auslander-Reiten-Smalø [ARS95], but otherwise left untouched for a long time, before it was reinvigorated in work by Ringel [Rin12, Rin13], Krause [Kra13, Kra14] and Chen-Le [CL13].

Let C be an additive category, and fix an object Y in C. A morphism α:X→Y is right C-determined if for each α':X'→Y such that im Hom_C(C,α') is a subset of im Hom_C(C,α), there is a factorization of α' through α.

The main results of Auslander state:

Theorem 1 ([Aus78a]). The category mod Λ of finite length modules over an artin algebra has right determined morphisms, i.e.,

Following Ringel, this can be interpreted as a lattice isomorphism between equivalence classes of right C-determined morphisms ending in Y (where the order is given by factorization) and the lattice of submodules of Hom_Λ(C,Y).

In the first part of the talk, largely based on [Rin13], we will introduce the necessary terminology, discuss Auslander's theorem, and see how determined morphisms generalize the almost split morphisms of Auslander--Reiten theory.

In the second part we consider determined morphisms in certain additive categories. In particular, we will see that there is a strong relationship of determined morphisms in triangulated categories [Kra13] and abelian categories [CL13] with Serre duality.

References

Research talks

Claire Amiot (Grenoble)
Derived invariants for surface algebras

Let (S,M) be unpunctured surface with marked points, with genus g and b boundary components. A surface algebra arises from a cut of an ideal triangulation of (S,M). In a joint work with Yvonne Grimeland, we associate to any surface algebra and any generating set of the fundamental group of S, an element in Z^2g+b. We show that this element determines the derived equivalence class of the algebra up to homeomorphism of the surface. In this talk I will explain this result, the main ingredients of the proof, and the information we can deduce on the corresponding derived categories.

Lidia Angeleri-Hügel (Verona)
Minimal approximations

The talk is devoted to the notion of an approximation coined by Auslander and Buchweitz. We will discuss the existence of minimal right approximations in the category of all modules over a ring. Our prototype will be Bass' Theorem P characterizingthe existence of projective covers in terms of closure under direct limitsand of descending chain conditions. Ed Enochs gave an argument for showing that every class A closed under direct limits and providing right A-approximations also provides minimal right A-approximations. Conversely, does the existence of minimal right A-approximations imply that A is closed under direct limits? We will present a positive answer in three special cases.Most results will be based on joint work with Dolors Herberaand on a recent preprint with Jan Saroch and Jan Trlifaj.

Grzegorz Bobiński (Torun)
The Krull--Gabriel dimension of the discrete derived categories

It is well-known that the Krull--Gabriel dimension of the category of modules reflects the representation type of an algebra. More precisely, an algebra is of finite representation type if and only if the Krull--Gabriel dimension of its module category equals 0 (and there is no algebra such that the Krull--Gabriel dimension of its module category equals 1). In the context of derived categories a new phenomenon appears: algebras with discrete derived categories first studied by Vossieck. In a joint work with Henning Krause we have calculated the Krull--Gabriel dimension of the derived categories of the derived discrete algebras. In particular, we have investigated whether this class of algebras can be distinguished via the Krull--Gabriel dimension of the associated derived categories.

Ragnar-Olaf Buchweitz (Toronto)
A McKay Correspondence for Reflection Groups?
(joint work with Eleonore Faber and Colin Ingalls)

Let G be a finite subgroup of GL(n,K) for a field K whose characteristic does not divide the order of G. The group G then acts linearly on the polynomial ring S in n variables over K and one may form the corresponding twisted or skew group algebra A = S*G. With e in A the idempotent corresponding to the trivial representation, one can consider the algebra A/AeA. If G is a finite subgroup of SL(2,K), then A is Morita-equivalent to the preprojective algebra of an extended Dynkin diagram and A/AeA to the preprojective algebra of the Dynkin diagram itself. This can be seen as a formulation of the McKay correspondence for the Kleinian singularities.

We want to establish an analogous result when G is a (nontrivial) group generated by reflections. So far, we can show that A/AeA is Cohen-Macaulay as a module, of finite global dimension as a ring, and of rank (|G|/2)^2 over the discriminant of the group action on S. Thus, it is tempting to expect that A/AeA is the endomorphism ring of a module over the ring of the discriminant and yields a noncommutative resolution of singularities of that discriminant, a hypersurface that is a free divisor, thus, singular in codimension one.

This work in progress relies heavily on Maurice Auslander's work on the McKay correspondence and his joint work on the homological theory of idempotents.

Igor Burban (Cologne)
Representations of SL(2,R) and derived Auslander-Reiten theory

SL(2,R) is the most basic non-compact simple Lie group. At the same time, it has very rich representation theory. After a series of intermediate steps, the study of representations of SL(2,R) essentially reduces to a description of the finite dimensional modules over a certain one dimensional order also called Gelfand quiver.

In his ICM talk in 1970 Gelfand raised a question about a classification of all indecomposable representations over this quiver. The tameness of this problem was established by Nazarova and Roiter in 1973 but the exact combinatorics of indecomposable objects remained to be clarified. I shall explain another approach to this classification problem based on the study of the derived category of the Gelfand quiver. We introduce new homological invariants allowing to characterize all classes of indecomposable representations (bands, bispecial, special and usual strings) in purely homological terms, without referring to combinatorics. The derived Auslander-Reiten theory plays an important role in our approach. Generalizations to other non-compact simple Lie groups of rank one will be discussed, too. This talk is based on a joint work with Wassilij Gnedin.

Osamu Iyama (Nagoya)
Higher Auslander correspondence and d-Cohen-Macaulay finiteness

Auslander correspondence between representation-finite algebras and algebras with global and dominant dimensions two is a milestone in representation theory. I will discuss higher Auslander correspondence, which naturally give us the notion of d-Cohen-Macaulay finiteness. Higher preprojective algebras as well as Geigle-Lenzing complete intersections give us a rich source of examples.

Charles Paquette (New Brunswick)
Existence and non-existence of Auslander-Reiten sequences

In this talk, we will consider a k-linear category C that is given by the quotient of the path category of a locally finite (but infinite) quiver by an admissible ideal I (any element in k(Q+) is nilpotent in C and C is Hom-finite). We will consider the category rep(Q,I) of covariant functors from C to mod(k), that is, the locally finite dimensional representations of (Q,I). This category has a nice Auslander-Reiten theory and it follows from a result of Auslander that there are Auslander-Reiten sequences of the form 0 --> L --> M --> N--> 0 where N is finitely presented and L is finitely co-presented. I will start by commenting on this result and talk about the converse direction: what happens if N is not finitely presented or if L is not finitely co-presented? Then, I will consider the case where I=0 and I will describe the AR-quiver of rep(Q).

Idun Reiten (Trondheim)
My work with Maurice

I will discuss some of my joint work with Maurice Auslander, from 1971 to 1994.

Jan Šťovíček (Prague)
Higher triangulations via the calculus of homotopy (co)limits

Not so long ago various new structures in homological algebra were axiomatized: higher triangulations by Maltsiniotis and Künzer, May's axioms for tensor triangulated categories, or N-angulations by Geiß, Keller and Oppermann. We show how to derive these structures from a simple calculus of homotopy (co)limits and, in particular, how basic representation theory is very helpful in not getting lost in this maze. In fact, it turns up that some of these phenomena are extremely general and have counterparts in abstract homotopy theory.

Gordana Todorov (Boston)
Morphisms determined by objects

In this talk we will cover some aspects of Morphisms Determined by Objects, a beautiful and fundamental notion introduced, developed and applied, by Maurice Auslander.

Michael Wemyss (Edinburgh)
Auslander–McKay correspondence in dimension 3

Auslander's version of the classical McKay correspondence is that there is a 1–1 correspondence between indecomposable non-free CM modules on the base singularity and exceptional curves in its minimal resolution. The benefit of this approach is that representations are absent, so it lends itself to generalizations away from the quotient singularity case. In the talk, I will describe one such generalization, namely to (possibly singular) minimal models of 3–dimensional cDV singularities. There are now two such correspondences. First there is a 1–1 correspondence between certain CM modules and minimal models, and then for each such pair there is a further correspondence, along the lines of the classical Auslander–McKay correspondence. This approach sounds algebraic, but the proofs are entirely geometric, and rely on realizing the Bridgeland–Chen flop functor as an appropriate mutation of a quiver with relations, together with an understanding of noncommutative deformations of curves. If there is time, I will discuss some other applications.