Higher homological algebra and cluster tilting

日程: 4月29日(日)― 5月1日(火)
4/29, 4/30  A13-323
5/1 A14-321



10:00-11:30  Hiroyuki Nakaoka (Kagoshima University)

On a unification of n-exact categories and n+2-angulated categories


Triangulated categories and exact categories are two major classes of categories in homological algebra. Recently, as their higher-degree analog, n-angulated categories and n-exact categories are introduced by Geiss, Keller, Oppermann and Jasso, respectively. In this talk, I will give a candidate notion to unify n-exact categories and n+2-angulated categories. This talk is based on a joint work with Martin Herschend and Yu Liu.

13:00-14:30 Martin Herschend (Uppsala University)

Categories with higher dimensional Auslander-Reiten theory


This talk is based on work in progress, joint with Steffen Oppermann.

A starting point of higher dimensional Auslander-Reiten theory is to replace the module category of a finite dimensional algebra by a d-cluster tilting subcategory. So far all known examples of such subcategories have only finitely many indecomposable objects. Hence the question of which algebras should be considered as higher analogues of representation infinite algebras does not have a clear answer. In case of algebras of global dimension d an answer could be the d-representation infinite algebras introduced together with Osamu Iyama and Steffen Oppermann. For such algebras there are (as in the classical case) three subcategories of the module category that are considered: the preprojective, preinjective and regular modules. However, in the higher dimensional setting only the preprojective and preinjective modules can be said to have some higher dimensional  Auslander-Reiten theory.

In my talk I will introduce a proposal for what could be considered as subcategories with higher dimensional Auslander-Reiten theory, but which are not necessarily d-cluster tilting. For example the preprojective modules of a d-representation infinite algebra form such a subcategory. For d = 1, any collection of components of the Auslander-Reiten quiver is such a subcategory. I will also present a construction of certain selfinjective algebras that admit subcategories with higher dimensional Auslander-Reiten theory that have infinitely many indecomposable objects. Hence these algebras can be considered as higher analogues of representation infinite selfinjective algebras. In particular, they have infinite global dimension in contrast to the previously known case of global dimension d.

15:00-16:00 Hiroyuki Minamoto (Osaka Prefecture University)

Resolutions of DG-modules

A DG-version of projective resolution and injective resolution of DG-modules over DG-algebra are already known. In this talk, we introduce another DG-version for DG-modules over a connective (=non-positive) DG-algebra and show that they behave nicely with respect to the projective and injective dimensions introduced by Yekutieli. We introduce the global dimension of a connective DG-algebras and show that the finiteness is invariant under derived equivalence.

16:30-18:00 Erik Darpö (Nagoya University)

Things I would like to know about n-representation-finite self-injective algebras

I will review work by Iyama and myself on the construction of self-injective algebras with an n-cluster-tilting module, and discuss some open problems in the field.


10:00-11:30  Izuru Mori (Shizuoka University)

Tilting theory in noncommutative algebraic geometry


This is a survey talk based on the work by Minamoto, Ueyama and myself.  Fano algebras and AS-regular algebras are the most important classes of algebras in noncommutative algebraic geometry.  After showing a strong relationship between Fano algebras and AS-regular algebras, we characterize tilting objects whose endomorphism rings are Fano algebras and whose graded endomorphism rings are AS-regular algebras.  If time permits, we discuss noncommutative smooth quadric surfaces and/or noncommutative projective McKay correspondence in this context.

13:00-14:30 Yusuke Nakajima (IPMU)

Non-commutative crepant resolutions of some toric rings


The notion of non-commutative crepant resolution (= NCCR) was introduced by M. Van den Bergh. This is a non-commutative analogue of the usual crepant resolution, and a module giving an NCCR is related with cluster tilting theory. In this talk, I first review some known results concerning NCCRs (especially I focus on NCCRs of toric rings). Then, I will give some new NCCRs of toric rings using the framework of Hibi rings which are special classes of toric rings arising from partially ordered sets (= poset). Especially, the structure of the associated poset plays an important role for constructing an NCCR. This talk is based on arXiv:1702.07058 and arXiv:1801.05139. 

15:00-16:00 Takahide Adachi (Osaka Prefecture University)

Silting objects and t-structures


In this talk, we study bounded t-structures from the viewpoint of silting theory. We introduce the notion of ST-pairs of triangulated subcategories, a prototypical example of which is the pair of the bounded homotopy category and the bounded derived category of a finite-dimensional algebra. For an ST-pair (C,D), we construct an order-preserving map from silting objects in C to bounded t-structures on D and show that the map is bijective if and only if C is silting-discrete. This is a generalization of results of Koenig--Yang and Keller--Vossieck. This talk is based on a joint work with Yuya Mizuno and Dong Yang.

16:30-18:00 Osamu Iyama (Nagoya University)

Tilting Cohen-Macaulay representations 


There are various triangle equivalences between the singularity categories of Gorenstein rings and the derived or cluster categories of finite dimensional algebras. I will discuss the case of Z-graded commutative Gorenstein rings in dimension one, based on joint works with Buchweitz, Herschend and Yamaura.


10:00-11:30  Yuya Mizuno (Shizuoka University)

Torsion pairs for path algebras and sortable elements


Path algebras are one of the most classical and important classes of algebras. In this talk, we mainly discuss torsion pairs for path algebras. In particular, we review a close relationship with some elements of the Coxeter group, called sortable elements, and explain how to parametrize the torsion pairs by these objects via preprojective algebras. This is a joint work with H.Thomas

13:00-14:30 Aaron Chan (Nagoya University)

Replication and representation-finiteness of hereditary algebras


Take a representation-finite hereditary algebra, then certain orbit algebras of its repetitive algebra become representation-finite; in fact, any representation-finite self-injective algebra arise this way.  Instead of orbit algebras, one can ask whether representation-finiteness determines interesting properties on other types of finite dimensional truncations of the repetitive algebra.  By considering a repetitive algebra as bi-infinite matrix algebra, then we have, for each positive integer, a subfactor algebra given by finite matrices of this given size - such an algebra is called replicated algebra.  It turns out that representation-finiteness of a hereditary algebra is equivalent to having one of its replicated algebras being higher Auslander.  The first aim of the talk is to explain the very elementary reasoning to such connection.  Then we will point out the crucial point in this argument, and show how an analogous connection holds for a larger class of algebras, which includes self-injective algebras, higher hereditary algebras, and higher canonical algebras. This is joint work with Osamu Iyama and Rene Marczinzik.

15:00-16:30 Laurent Demonet (Nagoya University)

Lattice of torsion classes [joint with O. Iyama, N. Reading, I. Reiten, H. Thomas]


We consider the lattice tors A of torsion classes over a finite dimensional algebra. It is in general infinite. However, we show some properties (bialgebraicity, complete semidistributivity, complete congruence uniformity) which are enough to understand tors A from its Hasse quiver. We also give an interpretation of its Hasse quiver in terms of modules. One of our main aim is to understand lattice quotients tors (A/I) of tors A, when I is a two-sided ideal of A. If the time permits it, we will also give some more combinatorial consequences about the Weyl groups of Dynkin type.

世話人: 足立崇英、源泰幸 (大阪府立大学)