Proceedings of the 37th Symposium on Ring Theory and Representation Theory

September 3 (Fri.) – 5 (Sun.), 2004 Shinshu University, Japan

Edited by Hideto Asashiba Osaka City University

January, 2005 Osaka, Japan

第 37 回環論および表現論シンポジウム報告集

２００４年９月３日（金）−５日（日）信州大学理学部

編集: 浅芝秀人 (大阪市立大学)

２００５年１月大阪市立大学

Organizing Committee of The Symposium on Ring Theory and Representation Theory

The Symposium on Ring Theory and Representation Theory has been held annually in Japan and the Proceedings have been published by the organizing committee. The first Symposium was organized in 1968 by H. Tominaga, H. Tachikawa, M. Harada and S. Endo. After their retirement, a new committee was organized in 1997 for managing the Symposium. The present members of the committee are Y. Hirano (Okayama Univ. ), Y. Iwanaga (Shinshu Univ., 1997–2004), S. Koshitani (Chiba Univ.), K. Nishida (Shinshu Univ.) and M. Sato (Yamanashi Univ., 2004–).

The Proceedings of each Symposium is edited by program organizer. Anyone who wants these Proceedings should ask to the program organizer of each Symposium or one of the committee members.

The Symposium in 2005 will be held at Aichi University of Technology in Aichi Prefecture for Sep. 2–4, and the program will be arranged by T. Wakamatsu (Saitama Univ.).

Concerning several information on ring theory group in Japan containing schedules of meetings and symposiums as well as the addresses of members in the group, you should refer the following homepage, which is arranged by M. Sato (Yamanashi Univ.):

http://fuji.cec.yamanashi.ac.jp/ ring/ (in Japanese)

civil2.cec.yamanashi.ac.jp/ ring/japan/ (in English)

Yasuo Iwanaga Nagano, Japan December, 2004

CONTENTS

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vi Program (Japanese) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Program (English, ISO 8601) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Remarks on transitivity of exceptional sequences Araya, Tokuji . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1

Frobenius full matrix algebras and Gorenstein tiled orders Fujita, Naoaki and Sakai, Yosuke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

A construction of Auslander-Gorenstein rings Hoshino, Mitsuo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Higher dimensional Auslander-Reiten theory on maximal orthogonal subcategories Iyama, Osamu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Morphisms represented by monomorphisms Kato, Kiriko . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Morita duality and ring extensions Koike, Kazutoshi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

On indecomposability of a module given by Brauer construction Koshitani, Shigeo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Some topics on derived equivalent blocks of finite groups Kunugi, Naoko . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Gaussian sums are just characters of the multiplicative group of a finite field Motose, Kaoru . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55

Relative invariants of groups acting on Krull domains Nakajima, Haruhisa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Nakayama isomorphisms for the maximal quotient ring of a left Harada ring Nonomura, Kazuaki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 On the holonomic rank for A-hypergeometric system

Okuyama, Go . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

On the decomposability of a syzygy of the residue field Takahashi, Ryo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

On the principal 3-blocks of the Chevalley groups G_{2}(q)

Usami, Yoko . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .98

On the ℤD_{∞} category Van den Bergh, Michel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Some examples of stably equivalent but not derived equivalent symmetric algebras obtained by using generalized tilting modules

Wakamatsu, Takayoshi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Stability of Frobenius algebras with positive Galois coverings Yamagata, Kunio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .116

Frobenius and quasi-Frobenius property for mod \underline{C} Yoshino, Yuji . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

PREFACE

The 37th Symposium on Ring Theory and Representation Theory was held at Matsumoto City M-Wing from September 3 to 5 in 2004. The symposium and the proceedings are financially supported by Kenji Nishida (Shinshu University) Grant-in-Aid for Scientific Research (B)(1), No.14340007, JSPS.

This volume consists of eighteen articles presented at the symposium. We would like to thank all speakeres and coauthors for their contributions.

We would also like to express our thanks to all the members of the organizing committee for their helpful suggestions concerning the symposium, especially to Masahisa Sato for his administration of the web page and for his kind advice and help to publish the proceedings. Finally, we would like to express our gratitude to Kenji Nishida and all the staffs in Shinshu University for their great effort in the organization of the symposium.

Hideto Asashiba Osaka, Japan January, 2005

第 37 回環論および表現論シンポジウム (2004) プログラム

９月３日（金）

09:10/09:40 山形邦夫（東京農工大）正ガロア被覆をもつフロベニウス多元環の安定性について

09:50/10:20 若松隆義（埼玉大）一般傾斜加群から導かれる安定同値だが導来同値でない対称多元環

10:30/11:20 Rickard, Jeremy (Univ. of Bristol)

Derived Categories in Representation Theory, I

11:30/12:00 若松隆義，飛田明彦（埼玉大） Tilting modules and Gorenstein property

12:10/12:40 星野光男（筑波大） A construction of Auslander-Gorenstein rings

14:00/14:30 奥山豪（北大理） On the holonomic rank for A-hypergeometric system

14:40/15:10 浅芝秀人（大阪市大理） Domestic canonical algebras and simple Lie algebras

15:20/16:10 Van den Bergh, Michel (Limburgs Universitair Centrum)

Non-commutative Del Pezzo surfaces, I

16:30/17:00 小池寿俊（沖縄工高専） Morita duality and ring extensions

17:10/17:40 丸林英俊; Brungs, H., Osmanagic, E. （鳴門教育大; Alberta Univ.） Non-commutative valuation rings of crossed product algebras

17:50/18:20 藤田尚昌，酒井洋介 Frobenius full matrix algebras and Gorenstein tiled orders

９月４日（土）

09:10/09:40 野々村和晃 Nakayama isomorphisms for the maximal quotient ring of a left Harada ring 09:50/10:20 本瀬香（弘前大理工）

Gaussian sums are just characters of the multiplicative group of a finite field

10:30/11:20 Rickard, Jeremy (Univ. of Bristol)

Derived Categories in Representation Theory, II

11:30/12:00 巧刀刀直子（愛知教育大） Some topics on derived equivalent blocks of finite groups

12:10/12:40 宇佐美陽子（お茶の水大理） On the principal 3-blocks of the Chevalley groups G_{2}(q)

14:00/14:30 加藤希理子（大阪女子大理） Morphisms represented by monomorphisms

14:40/15:10 荒谷督司（岡山大理） Remarks on transitivity of exceptional sequences

15:20/16:10 Van den Bergh, Michel (Limburgs Universitair Centrum) Non-commutative Del Pezzo surfaces, II

16:30/17:00 中島晴久（城西大理） Relative invariants of groups acting on Krull domains

17:10/17:40 吉野雄二（岡山大理） Frobenius and quasi-Frobenius property for mod \underline{C}

17:50/18:20 高橋亮（岡山大理） On the decomposability of a syzygy of the residue field

９月５日（日）

09:10/09:40 越谷重夫（千葉大理） On indecomposability of a module given by Brauer construction

09:50/10:40 Rickard, Jeremy (Univ. of Bristol)

Derived Categories in Representation Theory, III

10:50/11:40 Van den Bergh, Michel (Limburgs Universitair Centrum) Non-commutative crepant resolutions

11:50/12:30 伊山修（兵庫県立大理） Higher dimensional Auslander-Reiten theory on maximal orthogonal subcategories

Program of the 37th Symposium On Ring Theory and Representation Theory (2004)

2004-09-03

09:10/09:40 Yamagata, Kunio （Tokyo Univ. of Agr. and Tech.） Stability of Frobenius algebras with positive Galois coverings

09:50/10:20 Wakamatsu, Takayoshi (Saitama Univ.)

Some examples of stably equivalent but not derived equivalent symmetric algebras obtained by using generalized tilting modules

10:30/11:20 Rickard, Jeremy (Univ. of Bristol)

Derived Categories in Representation Theory, I

11:30/12:00 Wakamatsu, Takayoshi and Hida, Akihiko (Saitama Univ.)

Tilting modules and Gorenstein property

12:10/12:40 Hoshino, Mitsuo (Univ. of Tsukuba)

A construction of Auslander-Gorenstein rings

14:00/14:30 Okuyama, Go (Univ. of Hokkaido)

On the holonomic rank for A-hypergeometric system

14:40/15:10 Asashiba, Hideto (Osaka City Univ.)

Domestic canonical algebras and simple Lie algebras

15:20/16:10 Van den Bergh, Michel (Limburgs Universitair Centrum)

Non-commutative Del Pezzo surfaces, I

16:30/17:00 Koike, Kazutoshi (Okinawa National College of Tech.)

Morita duality and ring extensions

17:10/17:40 Marubayashi, Hidetoshi; Brungs, H. and Osmanagic, E. (Naruto Univ. of Education; Albereta Univ.) Non-commutative valuation rings of crossed product algebras

17:50/18:20 Fujita, Naoaki and Sakai, Yosuke (Univ. of Tsukuba)

Frobenius full matrix algebras and Gorenstein tiled orders

2004-09-04

09:10/09:40 Nonomura, Kazuaki Nakayama isomorphisms for the maximal quotient ring of a left Harada ring 09:50/10:20 Motose, Kaoru (Hirosaki Univ.)

Gaussian sums are just characters of the multiplicative group of a finite field

10:30/11:20 Rickard, Jeremy (Univ. of Bristol)

Derived Categories in Representation Theory, II

11:30/12:00 Kunugi, Naoko (Aichi Univ. of Education)

Some topics on derived equivalent blocks of finite groups

12:10/12:40 Usami, Yoko (Ochanomizu Univ.)

On the principal 3-blocks of the Chevalley groups G_{2}(q)

14:00/14:30 Kato, Kiriko (Osaka Women’s Univ.)

Morphisms represented by monomorphisms

14:40/15:10 Araya, Tokuji (Univ. of Okayama)

Remarks on transitivity of exceptional sequences

15:20/16:10 Van den Bergh, Michel (Limburgs Universitair Centrum) Non-commutative Del Pezzo surfaces, II

16:30/17:00 Nakajima, Haruhisa (Josai Univ.)

Relative invariants of groups acting on Krull domains

17:10/17:40 Yoshino, Yuji (Univ. of Okayama)

Frobenius and quasi-Frobenius property for mod \underline{C}

17:50/18:20 Takahashi, Ryo (Univ. of Okayama)

On the decomposability of a syzygy of the residue field

2004-09-05

09:10/09:40 Koshitani, Shigeo (Univ. of Chiba)

On indecomposability of a module given by Brauer construction

09:50/10:40 Rickard, Jeremy (Univ. of Bristol)

Derived Categories in Representation Theory, III

10:50/11:40 Van den Bergh, Michel Non-commutative crepant resolutions

11:50/12:30 Iyama, Osamu (Univ. of Hyogo)

Higher dimensional Auslander-Reiten theory on maximal orthogonal subcategories