# 多元環の表現論ワークショップ(2008年)

これまでにも多元環の表現論シンポジウムが，主に最新の重要な成果を紹介・解説する 勉強会として開催されていましたが，今回はさらに勉強会としての性格を強くし，ワークショップとして 研究集会を開催いたします． また，今回は海外からの招待講演者として，この分野の指導的研究者である， B. Keller氏とH. Krause氏を迎えることができました． ワークショップでは，講演者自身の最新の研究成果を中心にして，1人当たり2時間あるいは3時間かけて， ゆっくり，じっくり解説が行われます． 内容は，講演題目にあるように，cluster categories, support varieties, quasi-Frobenius rings, homotopy categories, resolving subcategories, derived categoriesなどの話題からなります． 以上の話題に興味ある方々のご参加を心から歓迎いたします． 特に，関連する分野である，非可換環論，群の表現論，可換環論，非可換代数幾何学 および数理物理学(脚注)の 研究者諸氏にもご参加を呼びかけたいと思います．

なお，懇親会への参加を予定されている方はe-mailで，下記責任者までお知らせ下さい． 準備の都合上8月31日までにご連絡いただけると大変助かります．

——– 記 ——–

(http://www.shizuoka.ac.jp/ippan/shizuoka.html)
〒422-8529 静岡市駿河区大谷836
Tel.: 054-238-4722
バスで来られる場合「静岡大学前」の1つ前のバス停「片山」でおりた方が便利です．

Tel.: 054-253-1151

(shasash [at] ipc [dot] shizuoka [dot] ac [dot] jp)

（脚注：Keller氏の講演では，数理物理学の”priodicity conjecture”に由来する予想の証明も概説されます． 詳しくは講演概要を参照のこと．）

## 講演者，講演題目

• Bernhard Keller (Paris 7): Cluster algebras, cluster categories and periodicity 講演ノート Photo1 Photo2 Photo-slide
• Henning Krause (Paderborn)（共同研究者：D. Benson, S. Iyengar）: Support varities and stratifications for stable module categories 講演ノート Photo1 Photo2
• 星野 光男（筑波大）（共同研究者：阿部 弘樹）: Construction of quasi-Frobenius rings 講演ノート
• 加藤 希理子（大阪府大）（共同研究者：伊山 修，宮地 淳一）: Recollement of homotopy categories and Cohen-Macaulay modules 講演ノート
• 高橋 亮（信州大）: Resolving subcategories of modules over commutative rings 講演ノート
• 源 泰幸（京都大）: A non-commutative algebro-geometric characterization of representation type of a quiver 講演ノート

## プログラム

 9月2日（火） 9月 3日（水） 9月4日（木） 09:00 – 10:00 源 源 10:15 – 11:15 高橋 高橋 Krause 11:30 – 12:30 加藤 加藤 Keller — 昼休み — 14:00 – 15:00 Krause Krause 15:25 – 16:25 星野 星野 16:40 – 17:40 Keller Keller

## 講演概要

### Bernhard Keller

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

### Henning Krause

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

### 星野 光男pdf

Recall that a ring $A$ is said to be quasi-Frobenius if it is artinian and selfinjective on both sides (see e.g. [7,8]). There have been given several characterizations of quasi-Frobenius rings. For instance, a ring $A$ is quasi-Frobenius if it is noetherian on either side and selfinjective on either side (see e.g. [4] and its references). Also, many attempts have been made to generalize the notion of quasi-Frobenius rings (see e.g. [12], [11] and so on). On the other hand, only a few attempts seem to have been made to construct quasi-Frobenius rings. We provide several ways to construct quasi-Frobenius rings.

We modify the notion of Frobenius extensions of rings due to Nakayama-Tsuzuku [9, 10] as follows. Let $A$ be a ring containing a ring $R$ as a subring. Then $A$ is said to be a Frobenius extension of $R$ if the following conditions are satisfied: (1) $A_{R}$ and ${_{R}A}$ are finitely generated projective; and (2) $A_{A} \cong \mathrm{Hom}_{R}(_{A}A,R)$ and ${_{A}A} \cong \mathrm{Hom}_{R^{\mathrm{op}}}(A_{A},R)$ Frobenius extensions preserve various homological properties. For instance, the following hold: $\mathrm{inj \ dim} \ A_{A} \le \mathrm{inj \ dim} \ R_{R}$ and $\mathrm{inj \ dim} \ {_{A}A} \le \mathrm{inj \ dim} \ {_{R}R}$; if $R$ is a noetherian ring satisfying the Auslander condition (see [3]) then so is $A$; and, if $R$ is a quasi-Frobenius ring, i.e., a selfinjective artinian ring then so is $A$.

Also, we provide several ways to construct weakly symmetric rings inductively, starting from a given quasi-Frobenius local ring. To do so, we provide several ways to extend a given quasi-Frobenius ring $R$ to a quasi-Frobenius ring $A$ containing an idempotent $e$ such that $eAe \cong R$. For instance, we provide a way to construct, up to Morita equivalence, every Brauer tree algebra over a field $k$ inductively, starting from $k[t]/(t^{1+m})$, where $m$ is the multiplicity of the exceptional vertex (see [5] for Brauer tree algebras).

References
1. H. Abe and M. Hoshino, Frobenius extensions and tilting complexes, Algebras and Representation Theory (to appear).
2. H. Abe and M. Hoshino, Constructions of quasi-Frobenius rings, preprint.
3. J. -E. Björk and E. K. Ekström, Filtered Auslander-Gorenstein rings, Progress in Math. 92, 425-447, Birkhäuser, Boston-Basel-Berlin, 1990.
4. C. Faith, Rings with ascending condition on annihilators, Nagoya J. Math. 27 (1966), 179-191.
5. P. Gabriel and C. Riedtmann, Group representations without groups, Comment. Math. Helv. 54 (1979), 240-287.
6. M. Hoshino, Strongly quasi-Frobenius rings, Comm. Algebra 28(8)(2000), 3585-3599.
7. T. Nakayama, On Frobeniusean algebras I, Ann. Math. 40 (1939), 611-633.
8. T. Nakayama, On Frobeniusean algebras II, Ann. Math. 42 (1941), 1-21.
9. T. Nakayama and T. Tsuzuku, On Frobenius extensions I, Nagoya Math. J. 17(1960), 89-110.
10. T. Nakayama and T. Tsuzuku, On Frobenius extensions II, Nagoya Math. J. 19(1961), 127-148.
11. B. J. Osofsky, A generalization of quasi-Frobenius rings, J. Algebra 4 (1966), 373-387; Errata, 9 (1968), p. 120.
12. R. M. Thrall, Some generalizations of quasi-Frobenius algebras, Trans. Amer. Math. Soc. 64 (1948), 173-183.

### 加藤 希理子

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

### 高橋 亮

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

### 源 泰幸

The notion of Calabi-Yau algebra is defined by abstracting the property of the derived category of Calabi-Yau varieties. It is know that finite dimensional path algebra of a quiver is Calabi-Yau if the quiver has a finite representation type. In this talk we introduce the notion of Fano algebra by abstracting the property of derived category of Fano varieties, and prove that finite dimensional path algebra of a quiver is Fano if the quiver has a infinite representation type. Therefore we get a characterization of representation type of a quiver.