**Period:**September 2 - 4, 2008=20**Place:**Shizuoka University B-301 (Common = building)

- (http://www.shizuoka.ac.jp/ippan/shizuoka.html)

836 Ohya, = Suruga-ku,=20 422-8529, Shizuoka

Tel.: 054-238-4722=20 **Workshop Dinner:**18:30 - 20:30, September 3=20**Organizer:**Hideto Asashiba (Shizuoka Univ.)

- (shasash [at] ipc [dot] shizuoka [dot] ac [dot] jp)=20
**Financial support:**Hideto Asashiba; Jun-ichi=20 Miyachi

Grant-in-Aid for Scientific Research (C), JSPS=20

## Speakers, titles of lectures

**Bernhard Keller**(Paris 7):*Cluster algebras, cluster=20 categories and periodicity*=20**Henning Krause**(Paderborn) (joint with D. Benson and S. = Iyengar):=20*Support varities and stratifications for stable module = categories*=20**Mitsuo Hoshino**(Tsukuba.) (joint with Hiroki Abe):=20*Construction of quasi-Frobenius rings*=20**Kiriko Kato**(Osaka Prefecture) (joint with Osamu Iyama = and=20 Jun-ichi Miyachi):*Recollement of homotopy categories and = Cohen-Macaulay=20 modules*=20**Ryo Takahashi**(Shinshu):*Resolving subcategories of = modules=20 over commutative rings*=20**Hiroyku Minamoto**(Kyoto):*A non-commutative = algebro-geometric=20 characterization of representation type of a quiver*

## Program

Tue. Sept. 2 Wed. Sept. 3 Thu. Sept. 4 09:00 - 10:00 Minamoto Minamoto 10:15 - 11:15 Takahashi Takahashi Krause 11:30 - 12:30 Kato Kato Keller --- Lunch --- 14:00 - 15:00 Krause Krause 15:25 - 16:25 Hoshino Hoshino 16:40 - 17:40 Keller Keller

## Abstract

Cluster algebras were invented = by Sergey=20 Fomin and Andrei Zelevinsky at the beginning of this decade. The = initial=20 motivation came from the study of canonical bases in quantum groups = and total=20 positivity in algebraic groups. In recent years, cluster algebras have = become=20 very popular thanks, notably, to the many links to other subjects that = were=20 discovered. Representation theory figures prominently among these = since it=20 allows to categorify the basic constructions of cluster theory. A = central role=20 is played by cluster categories, certain triangulated categories = associated=20 with quivers without oriented cycles. We will present a survey of the = links=20 between cluster algebras and cluster categories and then generalize = the=20 construction of cluster categories to algebras of global dimension two = following recent work by Claire Amiot. As an application, we will = sketch a=20 proof for a conjecture from mathematical physics known as the = periodicity=20 conjecture, which claims that a certain discrete dynamical system = associated=20 with a pair of Dynkin diagrams is periodic.**Bernhard Keller**

Support varieties are used to = classify=20 representations of finite groups, up to some suitable equivalence = relation.=20 The plan is to explain the various ingredients of this joint work with = Benson=20 and Iyengar. We construct support varieties and local cohomology = functors for=20 triangulated categories with respect to the action of a graded = commutative=20 ring. Combining this with some commutative differential graded = algebra, one=20 obtains a stratification of the stable module category of a finite = group.**Henning Krause**

**Mitsuo Hoshino**<= /H3> =20 Recall that a ring $A$ is said to be quasi-Frobenius if it is artinian = and=20 selfinjective on both sides (see e.g. \cite{Na1, Na2}). There have = been given=20 several characterizations of quasi-Frobenius rings. For instance, a = ring $A$=20 is quasi-Frobenius if it is noetherian on either side and = selfinjective on=20 either side (see e.g. \cite{Fa} and its references). Also, many = attempts have=20 been made to generalize the notion of quasi-Frobenius rings (see e.g.=20 \cite{Th}, \cite{Os} and so on). On the other hand, only a few = attempts seem=20 to have been made to construct quasi-Frobenius rings. We provide = several ways=20 to construct quasi-Frobenius rings.

We modify the notion of=20 Frobenius extensions of rings due to Nakayama-Tsuzuku \cite{NT1, NT2} = as=20 follows. Let $A$ be a ring containing a ring $R$ as a subring. Then = $A$ is=20 said to be a Frobenius extension of $R$ if the following conditions = are=20 satisfied: (1) $A_{R}$ and ${_{R}A}$ are finitely generated = projective; and=20 (2) $A_{A} \cong \mathrm{Hom}_{R}(_{A}A,R)$ and ${_{A}A} \cong=20 \mathrm{Hom}_{R^{\mathrm{op}}}(A_{A},R)$. Frobenius extensions = preserve=20 various homological properties. For instance, the following hold: = $\mathrm{inj=20 \ dim} \ A_{A} \le \mathrm{inj \ dim} \ R_{R}$ and $\mathrm{inj \ dim} = \=20 {_{A}A} \le \mathrm{inj \ dim} \ {_{R}R}$; if $R$ is a noetherian ring = satisfying the Auslander condition (see \cite{BE}) then so is $A$; = and, if $R$=20 is a quasi-Frobenius ring, i.e., a selfinjective artinian ring then so = is=20 $A$.

Also, we provide several ways to construct weakly = symmetric=20 rings inductively, starting from a given quasi-Frobenius ring local = ring. To=20 do so, we provide several ways to extend a given quasi-Frobenius ring = $R$ to a=20 quasi-Frobenius ring $A$ containing an idempotent $e$ such that $eAe = \cong R$.=20 For instance, we provide a way to construct, up to Morita equivalence, = every=20 Brauer tree algebra over a field $k$ inductively, starting from=20 $k[t]/(t^{1+m})$, where $m$ is the multiplicity of the exceptional = vertex (see=20 \cite{GR} for Brauer tree algebras).

References- [AH] H. Abe and M. Hoshino, Frobenius extensions and tilting = complexes,=20
- Algebras and Representation Theory (to appear).=20
- [AH] H. Abe and M. Hoshino, Constructions of quasi-Frobenius = rings,=20 preprint.=20
- [BE] J. -E. Bj\"{o}rk and E. K. Ekstr\"{o}m, Filtered=20 Auslander-Gorenstein rings,=20
- Progress in Math. {\bf 92}, 425--447, Birkh\"{a}user,=20 Boston-Basel-Berlin, 1990.=20
- [Fa] C. Faith, Rings with ascending condition on annihilators,=20
- Nagoya J. Math. {\bf 27} (1966), 179--191.=20
- [GR] P. Gabriel and C. Riedtmann, Group representations without = groups,=20
- Comment. Math. Helv. {\bf 54} (1979), 240--287.=20
- [Ho] M. Hoshino, Strongly quasi-Frobenius rings,=20
- Comm. Algebra {\bf 28}(8)(2000), 3585--3599.=20
- [Na1] T. Nakayama, On Frobeniusean algebras I,=20
- Ann. Math. {\bf 40} (1939), 611--633.=20
- [Na2] T. Nakayama, On Frobeniusean algebras II,=20
- Ann. Math. {\bf 42} (1941), 1--21.=20
- [NT1] T. Nakayama and T. Tsuzuku, On Frobenius extensions I,=20
- Nagoya Math. J. {\bf 17}(1960), 89--110.=20
- [NT2] T. Nakayama and T. Tsuzuku, On Frobenius extensions II,=20
- Nagoya Math. J. {\bf 19}(1961), 127--148.=20
- [Os] B. J. Osofsky, A generalization of quasi-Frobenius rings,=20
- J. Algebra {\bf 4} (1966), 373--387; Errata, {\bf 9} (1968), p. = 120.=20
- [Th] R. M. Thrall, Some generalizations of quasi-Frobenius = algebras,=20
- Trans. Amer. Math. Soc. {\bf 64} (1948), 173--183.

This is joint work with O. Iyama and = J.=20 Miyachi. We study the homotopy category of unbounded complexes with = bounded=20 homologies and its quotient category by the homotopy category of = bounded=20 complexes. We show the existence of a recollement of the above = quotient=20 category and it has the homotopy category of acyclic complxes as a=20 triangulated subcategory. In the case of the homotopy category of = finitely=20 generated projective modules over a coherent ring**Kiriko Kato***R*of finite = self-injective dimension both sides, we show that the above quotient = category=20 are triangle equivalent to the stable module category of = Cohen-Macaulay=20 modules over the 2=81~2 upper triangular matrix ring of =*R*.

The notion of a resolving = subcategory was=20 introduced in the 1960s by Auslander and Bridger in the study of = modules of=20 Gorenstein dimension zero, which are now also called totally reflexive = modules. We study resolving subcategories of the category of finitely=20 generated modules over commutative noetherian rings. First, we = consider=20 contravariant finiteness of resolving subcategories, and show a = classification=20 theorem of contravariantly finite resolving subcategories. Second, we = consider=20 constructing a "better" module from each module in a fixed resolving=20 subcategory, and investigate nonfree loci of resolving = subcategories.**Ryo Takahashi**

TBA**Hiroyuki Minamoto**