The 36th Symposium On Ring and Representation Theory (2003)

ABSTRACT

Hirosaki University

Hirosaki-shi Aomori, JAPAN October 11(Sat.) - 13(Mon.), 2003

CONTENTS

p.1:11-1 EXAMPLES OF BROUÉ’S ABELIAN DEFECT GROUP CONJECTURE

IN REPRESENTATION THEORY OF FINITE GROUPS (Koshitani)

p.2:11,2,3-2 Foundation of the representation theory of finite dimensional algebras (Ringel) p.4:11,2-3 Representation dimension and finitistic dimension conjecture (Iyama) p.5:11-4-1 A generalization of Zwara’s theorem on degeneration of modules (Yoshino) p.6:11-4-2 Finite Generations of Rings of Differential Operators of Semigroup Algebras

(Saito)

p.7:11-5-1 On degenerations of modules over general rings (Ohnuki) p.8:11-5-2 Lifting modules over right perfect rings (Kuratomi) p.9:11-6-1 NON-COMMUTATIVE VALUATION RINGS OF K(X; σ, δ) OVER A DI-

VISION RING K (Xi)

p.10:11-6-2 Non-semisimple Hopf algebra of dimension p (using personal computer (Yoko-

gawa)

p.11:11-7-1 Valuations on coproducts of skew fields and free fields (Chiba) p.12:11-8-1 BGG correspondence and Aramova-Herzog’s theory on exterior algebra (Yana-

gawa)

p.13:11-8-2 Factorizations of enveloping algebras of three dimensional Lie algebras and

their applications (Morita)

p.14:12-1 Some applications of Koszul duality (Mori) p.15:12-4-1 On structures and representations of Frobenius algebras by using Galois

coverings (Yamagata)

p.16:12-4-2 On modules of G-dimension zero over non-Gorenstein local rings (Takahashi) p.17:12-5-1 Tilting complexes associated with a sequence of idempotents (Kato) p.18:12-5-2 TLet’s use cyclotomic polynomials in your lectures for your students (Mo-

tose)

p.19:12-6-1 On calculations of modular irreducible characters with the help of computers

(Waki)

p.20:12-6-2 Self-duality of quasi-Harada rings and locally distributive rings (Koike) p.21:12-7-1 On S_{R}(H)-blocks II (Hieda) p.22:12-7-2 pair of rings with common ideals (Tsutsui) p.23:12-8-1 Finitely Cogenerated Distributive Modules (Shindo) p.24:12-8-2 Global Dimension in Serial left Algebras (Uematsu)

p.25:13-1 Derivations of polynomial rings over a field of characteristic zero (Nowicki) p.26:13-3 On Frobenius algebras (Wakamatsu)

• Remark

p.x:d-n-r means that abstract is in page x and this is n-th lecture held in October d at Room 10+r.

第 36 回環論および表現論シンポジウムプログラム

１０月１１日（土） : 8:50–9:50: 越谷重夫 (千葉大学理学部)

EXAMPLES OF BROUÉ’S ABELIAN DEFECT GROUP CONJECTURE IN REPRESENTATION THEORY OF FINITE GROUPS

10:00–11:00: C.M. Ringel （Bielefeld University）

Foundation of the representation theory of finite dimensional algebras I 11:10–12:10: 伊山 修 (姫路工業大学大学院)

Representation dimension and finitistic dimension conjecture I 13:30–14:15:

第 11 番教室: 吉野雄二 (岡山大学理学部) A generalization of Zwara’s theorem on degeneration of modules 第 12 番教室: 齋藤 睦 (北海道大学大学院理学研究科) Finite Generations of Rings of Differential Operators of Semigroup Algebras

14:25–15:10:

第 11 番教室: 大貫洋介・佐藤眞久 (東京農工大学工学部・山梨大学工学部) On degenerations of modules over general rings 第 12 番教室: 倉富要輔・張宰薫 (北九州工業高等専門学校・山口大学理工学研究科) Lifting modules over right perfect rings

15:30–16:15:

第 11 番教室: Guangming Xie ・丸林英俊・小林滋・小松弘明 (鳴門教育大学・岡山県立大学) Non-Commutative Valuation Rings of K(X, σ, δ) over a Division Ring K 第 12 番教室: 横川賢二 (岡山理科大学) Non-semisimple Hopf algebra of dimension p (using personal computer)

16:25–17:10:

第 11 番教室: 千葉克夫 (新居浜工業高等専門学校) Valuations on coproducts of skew fields and free fields 第 12 番教室: 岡竜也 (筑波大学大学院数理物質科学研究科) 遺伝的ホップ代数の特徴づけとスーパーアフィン群への応用 (Characterization of hereditary Hopf algebra and its applications to super affine group)

17:20–18:05:

第 11 番教室: 柳川浩二 (大阪大学大学院理学研究科) BGG correspondence and Aramova-Herzog’s theory on exterior algebra 第 12 番教室: 森田純 (筑波大学数学系) ３次元リー代数の包絡環の分解について、およびその応用 (Factorizations of enveloping algebras of three dimensional Lie algebras and their applications)

１０月１２日（日） : 8:50–9:50: 毛利 出 (University of Toledo)

Some applications of Koszul duality 10:00–11:00: C.M. Ringel （Bielefeld University）

Foundation of the representation theory of finite dimensional algebras II 11:10–12:10: 伊山 修 (姫路工業大学大学院)

Representation dimension and finitistic dimension conjecture II 13:30–14:15:

第 11 番教室: 山形邦夫 (東京農工大学工学部) ガロア被覆を用いたフロベニウス多元環の構造と表現について (On structures and representations of Frobenius algebras by using Galois coverings) 第 12 番教室: 高橋亮 (岡山大学大学院自然科学研究科) On modules of G-dimension zero over non-Gorenstein local rings

14:25-15:10:

第 11 番教室: 星野光男・加藤義明 (筑波大学数学系・筑波大学数学研究科) Tilting complexes associated with a sequence of idempotents 第 12 番教室: 本瀬香 (弘前大学理工学部) Let’s use cyclotomic polynomials in your lectures for your students

15:30–16:15:

第 11 番教室: 脇克志 (弘前大学理工学部) On calculations of modular irreducible characters with the help of computers 第 12 番教室: 小池寿俊 (沖縄工業高等専門学校) Self-duality of quasi-Harada rings and locally distributive rings

16:25–17:10:

第 11 番教室: 稗田吉成 (大阪府立工業高等専門学校) On S_{R}(H)-blocks II 第 12 番教室: Hisaya Tsutsui ・平野康之 (Millersville University ・岡山大学理学部) A pair of rings with common ideals

17:20–17:50:

第 11 番教室: 新堂安孝 () Finitely Cogenerated Distributive Modules 第 12 番教室: 植松盛夫 (上武大学経営情報学部) Global Dimension in Serial left Algebras

18:00–: 懇親会

１０月１３日（月） : 8:50–9:50: Andrzej Nowicki （Nicholus Copernicus University）

Derivations of polynomial rings over a field of characteristic zero 10:00–11:00: C.M. Ringel （Bielefeld University）

Foundation of the representation theory of finite dimensional algebras III 11:10–12:10: 若松隆義 (埼玉大学教育学部)

フロベニウス多元環について (On Frobenius algebras)

The 36th Symposium On Ring and Representation Theory (2003) Program

October 11, (Sat.): 8:50–9:50: Shigeo Koshitani (Chiba University)

EXAMPLES OF BROUÉ’S ABELIAN DEFECT GROUP CONJECTURE IN REPRESENTATION THEORY OF FINITE GROUPS

10:00–11:00: C.M. Ringel （Bielefeld University）

Foundation of the representation theory of finite dimensional algebras I 11:10–12:10: Osamu Iyama (Himeji Institute of Technology )

Representation dimension and finitistic dimension conjecture I 13:30–14:15:

Room 11: Yuji Yoshino (Okayama University ) A generalization of Zwara’s theorem on degeneration of modules Room 12: Mutsumi Saito (Hokkaido University) Finite Generations of Rings of Differential Operators of Semigroup Algebras

14:25–15:10:

Room 11: Yousuke Onuki, Masahisa Sato (Tokyo Univeristy of Agriculture and Techonology, University of Yamanashi) On degenerations of modules over general rings Room 12: Yousuke Kutratomi, Chang chae hoon (Kita Kyushu National College of Technology,Yamaguchi University) Lifting modules over right perfect rings

15:30–16:15:

Room 11: Guangming Xie, Hidetoshi Marubayashi, Shigeru Kobayashi, Hiroaki Komatsu (Naruto Kyouiku University, Okayama Prefectural University) Non-Commutative Valuation Rings of K(X, σ, δ) over a Division Ring K Room 12: Kenji Yokogawa (Okayama Science University) Non-semisimple Hopf algebra of dimension p (using personal computer)

16:25–17:10:

Room 11: Katsuo Chiba (Niihama National College of Technology) Valuations on coproducts of skew fields and free fields Room 12: Tatsuya Oka (Tsukuba University) Characterization of hereditary Hopf algebra and its applications to super affine group

17:20–18:05:

Room 11: Koji Yanagawa (Osaka University) BGG correspondence and Aramova-Herzog’s theory on exterior algebra Room 12: Jun Morita (Tsukuba University) Factorizations of enveloping algebras of three dimensional Lie algebras and their applications October 12, (San.): 8:50–9:50: Izuru Mori (University of Toledo)

Some applications of Koszul duality 10:00–11:10: C.M. Ringel （Bielefeld University）

Foundation of the representation theory of finite dimensional algebras II 11:10–12:10: Osamu Iyama (Himeji Institute of Technology )

Representation dimension and finitistic dimension conjecture II 13:30–14:15:

Room 11: Kunio Yamagata (Tokyo Univeristy of Agriculture and Techonology) On structures and representations of Frobenius algebras by using Galois coverings Room 12: Ryou Takahashi (Okayama University) On modules of G-dimension zero over non-Gorenstein local rings

14:25-15:10:

Room 11: Mitsuo Hoshino, Yoshiaki Kato (Tsukuba University) Tilting complexes associated with a sequence of idempotents Room 12: Kaoru Motose (Hirosaki University) Let’s use cyclotomic polynomials in your lectures for your students

15:30–16:15:

Room 11: Katsushi Waki (Hirosaki University) On calculations of modular irreducible characters with the help of computers Room 12: Kazutoshi Koike (Okinawa National College of Technology) Self-duality of quasi-Harada rings and locally distributive rings

16:25–17:10:

Room 11: Yoshinari Hieda (Osaka prefectural National College of Technology) On S_{R}(H)-blocks II Room 12: Hisaya Tsutsui, Yasuyuki Hirano (Millersville University, Okayama University) A pair of rings with common ideals

17:20–17:50:

Room 11: Yasutaka Shindo () Finitely Cogenerated Distributive Modules Room 12: Morio Uematsu (Joubu University) Global Dimension in Serial left Algebras

18:00–: Banque

October 13, (Mon.): 8:50–9:50: Andrzej Nowicki （Nicholus Copernicus University）

Derivations of polynomial rings over a field of characteristic zero 10:00–11:00: C.M. Ringel （Bielefeld University）

Foundation of the representation theory of finite dimensional algebras III 11:10–12:10: Takayoshi Wakamatsu (Saitama University)

On Frobenius algebras

EXAMPLES OF BROUÉ’S ABELIAN DEFECT GROUP CONJECTURE IN REPRESENTATION THEORY OF FINITE GROUPS

Shigeo KOSHITANI

In representation theory of finite groups, there is a well-known and important conjecture due to M. Broué. He conjectures that, for any prime p, if a p-block A of a finite group G has an abelian defect group P, then A and its Brauer correspondent p-block B of N_{G}(P) are derived equivalent. In this talk we discuss on Broué’s abelian defect group conjecture for several cases where A has elementary abelian defect group P of order 9 .

Department of Mathematics Faculty of Science Chiba University Chiba, 263-8522 JAPAN

Email: koshitan@math.s.chiba-u.ac.jp Foundation of the representation theory of finite dimensional algebras

CLaus Michael Ringel

The aim of the lectures will be to outline a new fundation of the representation theory of artin algebras based on quite old investigations of Roiter and Gabriel concerning the Brauer-Thrall conjectures, and which complement the usual Auslander-Reiten methods. I will focus the attention to the Gabriel-Roiter measure which was introduced (as ”Roiter measure”) by Gabriel in order to make explicit the combinatorial scheme of Roiter’s proof of the first Brauer-Thrall conjecture. The methods which were introduced in this way were intended just for categories of bounded representation type, but in turns out that their proper value unfolds when dealing with artin algebras of infinite representation type!

Let Λ be an artin algebra. The Λ-modules to be considered will usually be finitely generated. Recall that the first Brauer-Thrall conjecture asserts the following: If there is a bound on the length of the indecomposable Λ-modules (Λ is bounded representation type), then there are only finitely many isomorphism classes of indecomposables (Λ is of finite representation type). This conjecture was solved in 1968 by Roiter, and his method was analysed by Gabriel in 1973 by introducing a combinatorial invariant for any Λ-module. Both Roiter and Gabriel have assumed from the beginning that Λ is of bounded representation type, however this assumption is really misleading: A proper reading of the papers of Roiter and Gabriel shows that the methods exhibited by Roiter and the invariant introduced by Gabriel shed light on the structure of the category of Λ-modules for an arbitrary artin algebra Λ.

The Gabriel-Roiter measure μ(M) of a Λ-module M (as we propose to call this invariant) may be considered as a rational number (say between 0 and 1) which only depends on the submodule lattice of M. In case M is indecomposable, then one looks for suitable filtrations M_{1} ⊂ M_{2} ⊂ ⋯ ⊂ M_{t} = M using indecomposable submodules M_{i}; we will call these filtrations Gabriel-Roiter filtrations, and the corresponding embeddings M_{i−1} ⊂ M_{i} Gabriel-Roiter inclusions. Note that any Gabriel-Roiter inclusion X ⊂ Y is mono-irreducible (this means that for any proper submodule Y′ of Y which includes X, the embedding X ⊆ Y′ splits). In particular, it follows that for any Gabriel-Roiter filtration M_{1} ⊂ M_{2} ⊂ ⋯ ⊂ M_{t} = M, all the factors M_{i}/M_{i−1} are indecomposable.

The main property of the Gabriel-Roiter measure is the following: the class of modules which are direct sums of modules M with μ(M) ≤ r for a fixed real number r is closed under submodules. In this way, one obtains an interesting filtration of the category mod Λ of all Λ-modules by subcategories which are closed under submodules.

We say that r is a Gabriel-Roiter measure for Λ provided there are indecomposable Λ-modules M with μ(M) = r. Let us assume from now on that Λ is a connected artin algebra of infinite representation type. We will show that there are Gabriel-Roiter measures r_{t}, r^{t} for Λ with r_{1} < r_{2} < r_{3} < ⋯ < r^{3} < r^{2} < r^{1} such that any other Gabriel-Roiter measure r for Λ satisfies r_{t} < r < r^{t} for all t ∈ ℕ_{1}. Also, for any t, there are only finitely many isomorphism classes of indecomposables with Gabriel-Roiter measure r_{t} or r^{t}. Note that any infinite set of Gabriel-Roiter measures for Λ provides arbitrarily large indecomposable modules, thus, in particular, we encounter two different proofs of the first Brauer-Thrall conjecture.

We will say that the indecomposables with Gabriel-Roiter measure of the form r_{t} form the take-off part of the category mod Λ. The take-off part can be used in order to construct very interesting indecomposable Λ-modules of infinite length. For certain hereditary algebras, the take-off modules are just the preprojective ones, but in general there will be no indecomposable module which is both a take-off module and preprojective. In contrast, all the indecomposable modules with Gabriel-Roiter measure of the form r^{t} will be preinjective. Note that we get in this way a partition of the module category into three parts: the take-off part, the central part and the preinjective part, where both the take-off part and the preinjective part are in some sense of combinatorial nature. The lectures will provide additional information on this partition.

We also will consider the dual partition. Looking at both partitions at the same time, we obtain a rhombic picture for the category mod Λ which sheds a lot of light on the structure of mod Λ and some of its subcategories. For tame hereditary algebras, the rhombic picture corresponds quite well to the commonly used intuitive visualization using the Auslander-Reiten quiver, but in contrast to the old visualization, the new one is based on invariant data.

Fakultät für Mathematik Universität Bielefeld D-4800 Bielefeld 1 Germany

Email: ringel@mathematik.uni-bielefeld.de Representation dimension and finitistic dimension conjecture

Osamu Iyama

30 years ago, M. Auslander introduced a concept of representation dimension of artin algebras, which measures homologically how far an artin algebra is from being of finite representation type. His methods have been effectively applied not only for the representation theory of artin algebras, but also for the theory of quasihereditary algebras of Cline-Parshall-Scott by Dlab-Ringel. Recently, the author proved that any artin algebra has a finite representation dimension by showing that any module is a direct summand of some module whose endomorphism ring is quasihereditary. Our method is to construct certain chain of subcategories of mod Λ. It was also applied to solve Solomon’s second conjecture on zeta functions of orders by the author. We will formulate it in terms of rejective subcategories, which was applied to study the representation theory of orders and give a characterization of their finite Auslander-Reiten quivers. Moreover, we will explain the recent result of Igusa-Todorov and that of Rouquier.

A generalization of Zwara’s theorem on degeneration of modules

Yuji Yoshino

We generalize a result of Zwara concerning the degeneration of modules over Artinian algebras to that over general algebras. In fact, let R be any algebra over a field (maybe non-commutative, non-Noetherian, non-Artinian) and let M and N be finitely generated left R-modules. Then, we show that M degenerates to N (along a discrete valuation ring) if and only if there is a short exact sequence of

finitely generated left R-modules 0 → Z →^{(ϕ,ψ)} M ⊕ Z → N → 0 such that the endomorphism ψ on Z is nilpotent. We give several applications of this theorem to commutative ring theory. In particular, we show that the property for a module having G-dimension 0 is an open property.

References

[1] Y.Yoshino, On Degenerations of modules, Preprint (2003).

[2] , Degeneration and G-dimension of modules, Lecture at Lisbon Conference of Commutative Algebra, June, 2003.

[3] , On degenerations of Cohen-Macaulay modules, Journal of Algebra 248, 272–290 (2002).

[4] G.Zwara, Degenerations of finite dimensional modules are given by extensions, Composito Mathematica 121, 205–218 (2000).

Mathematical Institute Faculty of Science Okayama University Okayama 700-8530 JAPAN

Email: yoshino@math.okayama-u.ac.jp

FINITE GENERATIONS OF RINGS

OF DIFFERENTIAL OPERATORS OF SEMIGROUP ALGEBRAS

SAITO, Mutsumi

This talk is based on the papers with William N. Traves [1] and [2]. Let A := { a_{1}, a_{2}, …, a_{n} } ⊂ ℤ^{d} be a finite subset. We denote by ℕA, ℤA, and

ℝ_{≥0}A the monoid, the abelian group, and the cone generated by A, respectively.

Let R_{A} := ℂ[ℕA] denote the semigroup algebra of ℕA. We consider two rings: the ring D(R_{A}) of differential operators of R_{A} and its graded ring Gr(D(R_{A})) with respect to the order filtration.

As a starting point for the study of D(R_{A}), we consider the finite generations of D(R_{A}) and Gr(D(R_{A})).

While considering the finite generation of Gr(D(R_{A})), we encountered the notion of a scored semigroup; a semigroup ℕA is scored if the difference (ℝ_{≥0}A∩ℤA)\ℕA consists of a finite union of hyperlane sections of ℝ_{≥0}A ∩ ℤA parallel to facets of the cone ℝ_{≥0}A.

We have proved the following:

Theorem 1. (1) Gr(D(R_{A})) is finitely generated if and only if R_{A} is a scored

semigroup algebra.

(2) D(R_{A}) is finitely generated for all semigroup algebras R_{A}.

The problem of the Noetherian properties of D(R_{A}) is still open. In the talk, I use some examples to illustrate the idea of the proof of the theorem,

and to explain the scored property. The scored property implies Serre’s condition (S_{2}). However neither the scored property nor the Cohen-Macaulay property implies the other.

References

[1] M. Saito and W.N. Traves, Differential algebras on semigroup algebras, AMS Contemporary Math. 286 (2001), 207–226.

[2] and , Finite generations of rings of differential operators of semigroup algebras, preprint.

Department of Mathematics Graduate School of Science Hokkaido University Sapporo, 060-0810 JAPAN

Email: saito@math.sci.hokudai.ac.jp ON DEGENERATIONS OF MODULES OVER GENERAL RINGS

YOSUKE OHNUKI and MASAHISA SATO

The aim of our talk is to generalize the relationship of degenerations and some tubes, developed by Riedtmann, Zwara and etc. Recall [2], [4] that for a finite dimensional algebra A over an algebraically closed field k and (finite) d-dimensional A-modules M, N, M degenerates to N (i.e., N lies in the Gl_{d}(k)-orbit closure of M) if and only if there is an exact tube (N_{i}, α_{i}, β_{i})

0 ⟵ N_{1} ⟵^{β_{1}} N_{2} ⟵^{β_{2}} N_{3} ⟵^{β_{3}} ⋯
↓  ^{α_{1}}↓  ^{α_{2}}↓  ^{α_{3}}↓
N_{1} ⟵^{β_{1}} N_{2} ⟵^{β_{2}} N_{3} ⟵^{β_{3}} N_{4} ⟵^{β_{4}} · · · .

(each square is exact) such that N_{1} is isomorphic to N and we have some positive integer h with N_{h+m} ≃ N_{h} ⊕ M^{m} for any m ≥ 0.

In order to define a degeneration for modules a general ring, we take an almost exact tube (N_{i}, α_{i}, β_{i}) i.e., all α_{i} are monomorphisms, all β_{i} are epimorphisms and 0 → N_{i} → N_{i−1} ⊕ N_{i+1} → N_{i} → 0 are exact for any i ≥ 2. Then it induces the inverse limit of exact sequences which is the exact sequence of the following form

0 ⟶ lim_{⟵} N_{i} ⟶^{lim_{⟵} α} lim_{⟵} N_{i+1} ⟶ Cokα_{1} ⟶ 0.

We shall define degenerations by the purely ring theoretical notion.

Definition 1. Let R be an associative ring with a unit, and M, N be R-modules. We call N is a degeneration of M if Cokα_{1} is isomorphic to N and lim_{⟵} N_{i+1} is isomorphic to lim_{⟵} N_{i} ⊕ M.

Then N is a degeneration of M if and only if there is an exact sequence 0 → Z → Z ⊕ M → N → 0 for some module Z.

Definition 2. Let 0 → Z \underrightarrow{\binom{f}{g}} Z ⊕ M → N → 0 be an exact sequence which gives a degeneration N of M. A degeneration is called Fitting type if f : Z → Z induces an automorphism Im(f^{n}) → Im(f^{n}) for some n ∈ ℕ.

A degeneration of Fitting type is the notion generalized for a degeneration of finitely generated modules [3]. In this case we can apply the way for construction of an (M, N)-tube, introduced in [1].

We shall give some results and properties with respect to these definitions.

References

[1] R. Aehle, Ch. Riedtmann and G. Zwara Complexity of degenerations of modules, Comment. Math. Helv. 76 (2001), 781–803.

[2] Ch. Riedtmann, Degenarations for Representations of Quivers with Relations, Ann. Sci. École Normal Sup. 19 (1986), 275–301.

[3] Y. Yoshino, On Degenerations of Modules, Preprint.

[4] G. Zwara, Degenerations and Finite-Dimensional Modules are Given by Extensions, Composito Math. 121 (2000), 205–218.

Tokyo University of Aguliculture and Technology Email: ohnuki@cc.tuat.ac.jp

Yamanashi University Email: msato@yamanashi.ac.jp

LIFTING MODULES OVER RIGHT PERFECT RINGS

Chang Chaehoon, Yamaguchi University and Yosuke Kuratomi, Kitakyushu National College of Technology

A right R-module M is said to be an extending module, if it satisfies the following property: For any submodule X of M, there exists a direct summand of M which contains X as an essential submodule, that is, for any submodule X of M, there exists a closure of X in M which is a direct summand of M. Dually, M is said to be a lifting module, if it satisfies the dual property: For any submodule X of M, there exists a direct summand of M which is a co-essential submodule of X, that is, for any submodule X, there exists a co-closure of X in M which is a direct summand of M (cf.[1]). Any submodule X of M has a closure in M, but need not have a co-closure in M. We seem that the difficulty of study of lifting modules is hiding here.

In 1984, Okado [2] has studied the decomposition of extending modules over right noetherian and obtained the following:

Theorem (Okado) A ring R is right noetherian if and only if every extending R-module is expressed as a direct sum of indecomposable modules.

A dual problem is open. Our talk is concerned with this problem and the main result is the following:

Result Any lifting module over right perfect rings is expressed as a direct sum of indecomposable modules.

References

[1] K. Oshiro, Semiperfect modules and quasi-semiperfect modules, Osaka J. Math. 20 (1983), 337-372.

[2] M. Okado, On the decomposition of extending modules, Math. Japonica 29 (1984), 939-941.

NON-COMMUTATIVE VALUATION RINGS OF K(X; σ, δ) OVER A DIVISION RING K

Guangming Xie^{∗}, Hidetoshi Marubayashi^{∗} Shigeru Kobayashi^{∗}, Hiroaki Komatsu^{∗∗}

Let K be a division ring with a σ-derivation δ, where σ is an endomorphism of K and K(X; σ, δ) be the quotient division ring of the Ore extension R = K[X; σ, δ] over K in an indeterminate X. Let V be a total valuation ring of K and (σ, δ) be compatible with V. Then R^{(1)} = V [X; σ, δ]_{J(V)[X;σ,δ]}, the localization of V [X; σ, δ] at J(V)[X; σ, δ], is a total valuation ring of K(X; σ, δ) such that R^{(1)} ∩K = V, X ∈ R^{(1)}. The purpose of this talk is to describe non-commutative valuation rings B of K(X; σ, δ) such that R^{(1)} ⊇ B, B ∩ K = V and X ∈ B. First we shall denote non-commutative valuation rings of K(X; σ, δ) containing R. If δ is not a quasialgebraic σ-derivation, then there are no proper non-commutative valuation rings of K(X; σ, δ) containing R. In the case where δ is a quasi-algebraic σ-derivation and σ ∈ Aut(K), we shall describe all non-commutative valuation rings of K(X; σ, δ) containing R by using the classification of maximal ideals of R. If σ ∉ Aut(K) and δ is a quasi-algebraic σ-derivation, then there is a monic invariant polynomial p(X) of minimal non-zero degree such that K[X; σ, δ]p(X) is a maximal ideal. Set K̂ = ∪_{i=1}^{∞}p(X)^{−i}Kp(X)^{i}, a division ring containing K. We can extend σ and δ to σ̂ and δ̂ with σ̂ ∈ Aut(K̂) and obtain R̂ = ∪_{i=1}^{∞}p(X)^{−i}Rp(X)^{i} = K̂[X; σ̂, δ̂] with a maximal ideal P̂ = R̂p(X). If K̂ is left algebraic over K and Rp(X) is completely prime, then R̂P̂ is the only proper total valuation ring of K(X; σ, δ) such that R̂P̂ ⊇ R.

Next we assume that V is a total valuation ring of K and (σ, δ) is compatible with V. Then σ and δ naturally induce σ̄ and δ̄ of V̄ = V/J(V), a division ring. Let φ : R^{(1)} = V [X; σ, δ]_{J(V)[X;σ,δ]} → R̄^{(1)} = R^{(1)}/J(R^{(1)}) ≅ V̄ (X; σ̄, δ̄) be the natural homomorphism. Then there is a one-to one correspondence between the set of all total valuation rings B of K(X; σ, δ) satisfying B ∩ K = V, X ∈ B, B ⊆ R^{(1)} and the set of all total valuation rings 𝔅 of V̄ (X; σ̄, δ̄) with 𝔅 ⊇ V̄ [X; σ̄, δ̄], which is given by φ(B) = 𝔅 and φ^{−1}(𝔅) = B. By using this correspondence, we shall describe non-commutative valuation rings B of K(X; σ, δ) satisfying B ∩ K = V, X ∈ B, B ⊆ R^{(1)}.

References

[1] G. Xie, H. Marubayashi, S.Kobayashi, H.Komatsu, Non-commutative valuation rings of K(X; σ, δ) over a division ring K, preprint (2003)

∗Department of Mathematics Naruto University of Education Takashima, Naruto, 772-8502, JAPAN

Email: {e0220104,marubaya,skoba}@naruto-u.ac.jp

∗∗Faculty of Computer Science and System Engineering Okayama Prefectural University Kuboki, Soja, Okayama, 719-1197, JAPAN

Email: komatsu@pine.cse.oka-pu.ac.jp Non-semisimple Hopf algebras over a field of characteristic p ≠ 0

Kenji Yokogawa

Let K be a field of characteristic p ≠ 0.

Definition We call a Hopf algebra H of dimension p over K generated by a p-nilpotent element as a p-nilpotent Hopf algebra.

A group ring H = K[σ], σ^{p} = 1, is such a Hopf algebra with a p-nilpotent generator σ−1 (we call Hopf algebra of this type as a Hopf algebra of automorphism type). A Hopf algebra generated by a derivation d, d^{p} = 0, is also such a Hopf algebra (we call as a Hopf algebra of derivation type). Another type of p-nilpotent Hopf algebra is not known. So the question whether these two types of Hopf algebra are the all of p-nilpotent Hopf algebras arises.

これらの algebra は, non-semisimple で、分類するのに適当な tool が無い. そこで、具体的に計算する事から始めた. 計算と言っても基本的には, coassociative な algebra-homo Δ : H → H ⊗ H を決めてやれば良い訳であるが, 計算量が多くて人間業ではない. そこでパソコンと SoftWare 「Mathematica」を使って以下のような結果が得られた. ここでは, 結果よりもパソコンと SoftWare の利用方法を中心に話を進めたいと考えています. 注: SoftWare には「Mathematica」以外に「Maple」がある. 色々な点で「Maple」の方が優れているように思われる.

主な結果

Proposition 1. A non-semisimple Hopf algebra H of dimension p is a pnilpotent Hopf algebra.

Lemma 2. There exists a p-nilpotent generator x whose cocoefficients satisfy the relations; c_{11} = c_{21} = c_{22} = c_{32} = c_{33} = ⋯ = c_{q q−1} = c_{qq} = 0 (where p = 2q + 1, c_{i,j} denotes the coefficient of x^{i} ⊗ x^{j} in Δ(x)).

Definition If a p-nilpotent generator x satisfies the relations of Lemma 2, we call x as a normal generator.

Theorem 3. The cocoefficients of a normal generator x of a p-nilpotent Hopf algebra satisfy the relations; c_{10} = c_{01} = 1, c_{p−1 1} = c, c_{p−k k} = (−1)^{k−1}k^{−1}c, (2 ≤ k ≤ p − 1), c ∈ K, the other terms are equal to 0 and S(x) = −x (S is an antipode).

Conversely, an algebra generated by a p-nilpotent element whose cocoefficients and antipode satisfy the above relations becomes a p-nilpotent Hopf algebra.

Corollary 4. If K is a prime field then there exists exactly p non-isomorphic p-nilpotent Hopf algebras, and if K is an algebraically closed field then there exists exactly 2 non-isomorphic p-nilpotent Hopf algebras.

Faculty of Science

Okayama University of Science Okayama 700, Japan

Email: yokogawa@mdas.ous.ac.jp Valuations on coproducts of skew fields and free fields

Katsuo Chiba

Niihama National College of Technology

The universal field of fractions of free algebras are called free fields. By using free fields P.M.Cohn [1,2] constructed skew field extensions of arbitrary finite right degree and infinite left degree. Free fields are sometimes difficult to manipulate owing to the lack of a convenient normal form for their elements. Valuations form a useful tool in the analysis of free fields. In this lecture we give several properties of valuations on coproducts of skew fields and free fields. Our main results are the following:

Let D be a skew field with a valuation ν and K a bicentral subfield of D. Let {D_{i} | i ∈ I } be a family of subfields of D which contain the common subfield K of D and ν_{i} (i ∈ I) the valuation on D_{i} (i ∈ I) which is a restriction of ν to D_{i}. Then the ring coproduct ∗_{K}D_{i} of D_{i} over K has a valuation ω extending all the valuation ν_{i}. Moreover, if the subfields D_{i} (i ∈ I) satisfies certain conditions, then the field coproduct ∘_{K}D_{i} has a valuation ω extending all the ν_{i}.

As an application of the result, we have:

Let κ(X) be a free field over a commutative field κ on a set X = {x_{i} | i ∈ I } and ν_{i} a κ-valuation of κ(x_{i}) . T hen there is a κ-valuation ν on κ(X) which extends the all ν_{i}.

References

[1] P.M.Cohn, Progress in free associative algebra, Israel J. Math. 19 (1974), 109–151.

[2] , ”Skew Fields (Theory of General Division Rings),” Cambridge Univ. Press, Cambridge, 1995..

BGG correspondence and Aramova-Herzog’s theory on exterior algebra

Kohji Yanagawa

Recently, some commutative algebraists are studying graded modules over an exterior algebra. One of the motivations is to understand/refine results on Combinatorial Commutative Algebra (e.g. Kalai’s theory on “algebraic shifting”).

Let V be an n-dimensional K-vector space, and E := ⋀ V the exterior algebra. We regard E as a negatively graded ring with E_{−i} = ⋀^{i} V. Let N be a finitely generated graded E-module. Then, for each i ∈ ℤ, there is a polynomial P_{i}(t) ∈ ℚ[t] satisfying P_{i}(j) = dim_{K} Ext_{E}^{j}(K, N)_{j−i} for j ≫ 0. If P_{i}(t) ≠ 0, P_{i}(t) is of the form \frac{e}{m!} · t^{m} + (terms with lower degree) for some positive integer e. In this case, we set d_{i}(N) := m + 1 and e_{i}(N) := e. To extract essential information from an injective resolution, Aramova-Herzog [1] and Römer [4] introduced the following notion.

Definition 1 ([1, 4]). We say (d, i) ∈ ℕ × ℤ is a distinguished pair for a finitely generated graded E-module N if d_{i}(N) = d and d_{j}(N) < d + i − j for all j < i.

Note that N^{∗} := Hom_{E}(N, E) is a graded E-module again. Next result induces Bayer-Charalambous-Popescu [2, Theorem 2.8], which is a well-known result on Stanley-Reisner rings.

Theorem 2 (Römer [4]). (d, i) is distinguished for N if and only if (d, 2n − d − i) is distinguished for N^{∗}. If this is the case, we have e_{i}(N) = e_{2n−d−i}(N^{∗}).

Let W be the dual vector space of V, and S := Sym_{K} W the symmetric algebra. The Bernstein-Gel’fand-Gel’fand correspondence (BGG correspondence) states that the derived category D^{b}(mod_{S}) of finitely generated graded S-modules is equivalent to the similar category D^{b}(mod_{E}) for E (see for example [3]). In this talk, we show that, under the BGG correspondence, Theorem 2 is translated into a statement on D^{b}(mod_{S}) which is a natural consequence of the local duality (Serre duality).

We also refine Theorem 2 in the ℤ^{n}-graded context, while the arguments in [1] and [4] does not work in this context (since they use a generic base change of V). And we give a derived category version of [2, Theorem 2.8].

References

[1] A. Aramova and J. Herzog, Almost regular sequences and Betti numbers, Amer. J. Math. 122 (2000), 689-719.

[2] D. Bayer, H. Charalambous and S. Popescu, Extremal Betti numbers and applications to monomial ideals, J. Algebra 221 (1999), 497–512.

[3] D. Eisenbud, G. Fløystad and F.-O. Schreyer, Sheaf cohomology and free resolutions over exterior algebra, (math.AG/0104203) to appear in Trans. Amer. Math. Soc.

[4] T. Römer, Generalized Alexander duality and applications, Osaka J. Math. 38 (2001), 469–485.

[5] K. Yanagawa, BGG correspondence and Römer’s duality of an exterior algebra, preprint.

Department of Mathematics,

Graduate School of Science

Osaka University

Toyonaka, Osaka 560-0043 JAPAN Email: yanagawa@math.sci.osaka-u.ac.jp Factorizations of enveloping algebras of three dimensional Lie algebras and their applications

Jun Morita

Let F be a field of characteristic zero, and L be a Lie algebra over F. The universal enveloping algebra of L is denoted by U(L). Let V be an L-module. Then we will reach the following fact (cf. [1]).

(Fact) Let L be a three dimensional Lie algebra generated by two elements x, y. Then U(L) = ∑_{i,j,k} F x^{i}y^{j}x^{k}. Furthermore, if both of x and y act on V as locally nilpotent operators, then V is locally finite as an L-module.

For example, we take

e = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) , f = ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} )

as a set of generators for L = sℓ_{2}. Then we have U(sℓ_{2}) = ∑_{i,j,k} F e^{i}f^{j}e^{k}. This immediately shows that an sℓ_{2}-module V is integrable if e and f act on V locally nilpotently.

The key formulas to establish the above (Fact) are as follows:

(A_{k}) yxy^{k} ≡ \frac{k}{k+1}xy^{k+1} + \frac{1}{k+1}y^{k+1}x mod U_{k},
(B_{k}) y^{k}xy ≡ \frac{1}{k+1}xy^{k+1} + \frac{k}{k+1}y^{k+1}x mod U_{k},
(C_{k}) yU_{k} ⊂ U_{k+1}, U_{k}y ⊂ U_{k+1},

where U_{k} = ∑_{0≤m≤k}(F xy^{m} + F y^{m}x + F y^{m}). Since we can also apply these to the three dimensional Heisenberg Lie algebra, we have a new factorization of the universal enveloping algebras for Kac-Moody Lie algebras, more generally for Borcherds Lie algebras. Namely, if 𝔤 is a Kac-Moody Lie algebra (or a Borcherds Lie algebra) with a standard triangular decomposition 𝔤 = 𝔤_{−} ⊕ 𝔥 ⊕ 𝔤_{+} and 𝔤′ = [𝔤, 𝔤] is the derived subalgebra of 𝔤, then we obtain

U(𝔤′) = U(𝔤_{±})U(𝔤_{∓})U(𝔤_{±}),

which is called an additive Gauss decomposition. Similar decompositions exist for many extended affine Lie algebras. There is also a quantum group version, which implies a condition for integrability as an application (cf. [3]).

References

[1] S. Berman, J. Morita and Y. Yoshii, Some factorizations of Universal enveloping algebras of three dimensional Lie algebras and generalizations, Canad. Math. Bull. 45(4) (2002), 525–536.

[2] J. Morita and H. Sakaguchi, Some formulae in U_{q}(sℓ_{2}) and diagonalizability, Kyushu J. Math. 37(1) (2003), 165–173.

Institute of Mathematics University of Tsukuba Ten-no-dai 1-1-1, Tsukuba Ibaraki, 305-8571 JAPAN

Email: morita@math.tsukuba.ac.jp Some Applications of Koszul Duality

Izuru Mori

Let A = ⊕_{i=0}^{∞}A_{i} be a connected graded algebra over a field k and M = ⊕_{i=−∞}^{∞}M_{i} a graded left A-module. For each n ∈ ℤ, we define a graded left A-module M(n) by M(n) = M as an ungraded left A-module but M(n)_{i} = M_{n+i} for all i ∈ ℤ. A linear resolution of M is a resolution of the form

⋯ → ⨁ A(−2) → ⨁ A(−1) → ⨁ A → M → 0,

that is, a free resolution in which each differential is given by right multiplication of a matrix whose entries are all degree 1 elements (linear elements) of A. We say that A is Koszul if k has a linear resolution as a graded left A-module.

If A is a Koszul algebra, then we can define the Koszul dual of A by

A^{!} = ⨁_{i=0}^{∞} Ext_{A}^{i}(k, k),

which has a structure of a connected graded algebra over k by the Yoneda product. An important classical result is that there is a duality, known as the Koszul duality, between the categories of graded left modules having linear resolutions over A and A^{!}, respectively. Recently, it has been shown that the Koszul duality can be extended to a duality between the derived categories of finitely generated graded left modules over A and A^{!}, respectively. Through this extended Koszul daulity, we are more able to translate results on A to those on A^{!}, and vice versa.

It is known that A is an Artin-Schelter regular Koszul algebra, which is an important class of algebras in Noncommutative Algebraic Geometry, if and only if A^{!} is a Frobenius Koszul algebra. Further, the extended Koszul duality induces an equivalence between the derived category of a quantum projective space, which is a projective scheme associated to an Artin-Schelter regular Koszul algebra, and the stable category of the correnponding Frobenius Koszul algebra. So there is potential interaction between two research areas, Noncommutative Algebraic Geometry and the study of Frobenius algebras.

In this talk, I will give some examples of such interaction, namely, rationality of the Poincare series, Gorenstein condition, Grothedieck group, Serre duality, e.t.c.

References

1. R. Martinez-Villa, A. Martsinkovsky, Cohomology of Tails, Tate-Vogel cohomology, and noncommutative Serre duality over Koszul quiver algebras, preprint.

2. I. Mori, Rationality of the Poincare series for Koszul algebras, preprint.

3. I. Mori, S. P. Smith, The Grothendieck group of a quantum projective space bundle, preprint.

Department of Mathematics The University of Toledo Toledo, OH 43606-3390 U.S.A Email: imori@utnet.utoledo.edu ガロア被覆を用いたフロベニウス多元環の構造と表現について

Kunio Yamagata

体上の有限次元フロベニウス多元環が，ある多元環の反復多元環（repetitive algebra）と，ある無限次元巡回群をガロア群にもつガロア被覆をもつための判定法を紹介する．また有限次元加群の圏への応用も紹介の予定．

Faculty of Engineering

Tokyo Univeristy of Agriculture and Techonology Koganei, Tokyo 184-8588 JAPAN

Email: yamagata@cc.tuat.ac.jp ON MODULES OF G-DIMENSION ZERO OVER NON-GORENSTEIN LOCAL RINGS

Ryo Takahashi

Let R be a commutative noetherian local ring with residue class field k. Put (−)^{∗} = Hom_{R}(−, R), and denote by Ω_{R}^{n}M the nth syzygy module of a finitely generated R-module M.

Definition 1. Let M be a finitely generated R-module.

(1) If the natural homomorphism M → M^{∗∗} is isomorphic and Ext_{R}^{i}(M, R) = Ext_{R}^{i}(M^{∗}, R) = 0 for every i > 0, then we say that M has G-dimension zero, and write G-dim_{R}M = 0.

(2) If n is the smallest integer such that G-dim_{R}(Ω_{R}^{n}M) = 0, then we say that M has G-dimension n, and write G-dim_{R}M = n. If such an integer n does not exist, then we say that M has infinite G-dimension, and write G-dim_{R}M = ∞.

Conjecture 2. Let R be a non-Gorenstein local ring. Suppose that there exists a non-free R-module of G-dimension zero. Then there exist infinitely many isomorphism classes of indecomposable R-modules of G-dimension zero.

Denote by modR the category of finitely generated R-modules.

Definition 3. Let X be a full subcategory of modR.

(1) Let ϕ : X → M be a homomorphism from X ∈ X to M ∈ modR. We call ϕ an X -precover (or a right X -approximation) of M if for any homomorphism ϕ′ : X′ → M with X′ ∈ X there exists a homomorphism f : X′ → X such that ϕ′ = ϕf. We call ϕ an X -cover (or a right minimal X -approximation) of M if ϕ is an X -precover and any endomorphism f of X with ϕ = ϕf is an automorphism.

(2) The category X is said to be contravariantly finite in modR if every M ∈ modR has an X -precover.

Let G denote the full subcategory of modR consisting of all R-modules of Gdimension zero.

Theorem 4. Let R be a henselian non-Gorenstein local ring of depth zero. Suppose that there exists a non-free R-module in G. Then the residue class field k of R does not admit a G-precover as an R-module. In particular, the category G is not contravariantly finite in modR.

Theorem 5. Conjecture 2 is true if R is a henselian local ring of depth zero, especially if R is an artinian local ring.

References

1. M. Auslander and M. Bridger, Stable module theory, Memoirs of the American Mathematical Society, No. 94, 1969.

2. R. Takahashi, On the category of modules of Gorenstein dimension zero, preprint (2003).

3. , On the category of modules of Gorenstein dimension zero II, preprint (2003).

4. J. Xu, Flat covers of modules, Lecture Notes in Mathematics 1634, Springer-Verlag, Berlin, 1996.

5. Y. Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, London Mathematical Society Lecture Note Series, 146, Cambridge University Press, Cambridge, 1990.

6. , Modules of G-dimension zero over local rings with the cube of maximal ideal being zero, to appear in NATO Sci. Ser. I Math. Phys, Chem.

Graduate School of Natural Science and Technology Okayama University Okayama 700-8530, JAPAN

Email: takahasi@math.okayama-u.ac.jp Tilting complexes associated with a sequence of idempotents Mitsuo Hoshino, Yoshiaki Kato

Rickard [3] showed that the Brauer tree algebras with the same numerical invariants are derived equivalent to each other. Let A be a Brauer tree algebra corresponding to a Brauer tree whose edges are labelled 1, 2, …, n. Note that there exists a partition of the edges {1, …, n} = E_{0} ∪ ⋯ ∪ E_{l}, where E_{s} consists of the edges i for which there exists a sequence of edges i_{0}, i_{1}, …, i_{s} = i such that i_{0} is adjacent to the exceptional vertex and for any 0 ≤ r < l, i_{r} ≠ i_{r+1} and i_{r}, i_{r+1} have a vertex in common. He constructed a tilting complex P^{•} ∈ K^{b}(P_{A}) such that P^{j} = 0 for j > 0 and j < − l, P^{−j} ∈ add(⊕_{i∈E_{l−j}}e_{i}A), where e_{i} ∈ A is a local idempotent corresponding to the edge i, for 0 ≤ j ≤ l and End_{K(Mod-A)}(P^{•}) is a Brauer “star” algebra with the same numerical invariants as A. On the other hand, Okuyama [2] pointed out recently that for Brauer tree algebras A, B with the same numerical invariants there exists a sequence of Brauer tree algebras B_{0} = A, B_{1}, …, B_{l} = B such that B_{r+1} is the endomorphism algebra of a tilting complex for B_{r} of term length two defined by an idempotent. See König and Zimmermann [1] for another example of derived equivalences which are iterations of derived equivalences induced by tilting complexes of term length two. We will formulate these results.

Let A be a noetherian ring and e_{0}, e_{1}, …, e_{l} ∈ A a sequence of idempotents such that add(e_{0}A_{A}) = P_{A}, e_{i+1} ∈ e_{i}Ae_{i} for 0 ≤ i < l and Ext_{A}^{j}(A/Ae_{i}A, e_{i}A) = 0 for 0 ≤ j < i ≤ l. First, we will show that there exists a tilting complex P^{•} ∈ K^{b}(P_{A}) such that P^{i} = 0 for i > 0 and i < − l, P^{−i} ∈ add(e_{i}A) for 0 ≤ i ≤ l and H^{−j}(P^{•}) ∈ Mod-(A/Ae_{i}A) for 0 ≤ j < i ≤ l, and that such a tilting complex P^{•} is essentially unique. Next, we will show that there exists a sequence of rings B_{0} = A, B_{1}, …, B_{l} = End_{K(Mod-A)}(P^{•}) such that for any 0 ≤ i < l, B_{i+1} is the endomorphism ring of a tilting complex for B_{i} of term length two defined by an idempotent. Furthermore, in case A is a selfinjective artin algebra over a commutative artin ring R and add(e_{i}A_{A}) = add(D(_{A}Ae_{i})) for 1 ≤ i ≤ l, where D = Hom_{R}(−, E(R/rad R)), we will show that End_{K(Mod-A)}(P^{•}^{•}) is a selfinjective artin R-algebra whose Nakayama permutation coincides with that of A. Finally, we deal with the case where A is a finite dimensional algebra over a field k and add(e_{i}A_{A}) = add(D(_{A}Ae_{i})) for 1 ≤ i ≤ l, where D = Hom_{k}(−, k). We will construct a two-sided tilting complex which corresponds to P^{•}. Simultaneously, we will provide a sufficient condition for an algebra B containing A as a subalgebra to be derived equivalent to A.

References

1. S. König and A. Zimmermann, Tilting selfinjective algebras and Gorenstein orders, Quart. J. Math. Oxford (2) 48 (1997), 351-361.

2. T. Okuyama, Some examples of derived equivalent blocks of finite groups, preprint.

3. J. Rickard, Derived categories and stable equivalence, J. Pure Appl. Algebra 61 (1989), 303-317.

Mitsuo Hoshino, Yoshiaki Kato Institute of Mathematics University of Tsukuba Tsukuba, Ibaraki 305-8571 JAPAN Email: hoshino@math.tsukuba.ac.jp, ykato@math.tsukuba.ac.jp Let’s use cyclotomic polynomials in your lecture for your students

The detailed version of this note has been published in the Journal of Pure and Applied Algebra.

Kaoru Motose

We will present some results on cyclotomic polynomials. We will talk about a factorization of a number, some fundamental theorems on algebra, and applications to a code and a cipher (cryptography). Moreover, we will present some relations between cyclotomic polynomials and Gauss sums.

References

(1) K. Motose, On value of cyclotomic polynomials, Math. J. Okayama Univ. 35(1993), 35-40.

(2) R. Lidl and H. Niederwriter, Finite fields, Encyclopedia of Mathematics and Applications, 20, Cambridge University Press, London, 1984.

(3) G. Stein, Factoring cyclotomic polynomials over large finite fields, Finite fields and applications, London Math. Soc. Lecture Note Ser., 233 (Glasgow, 1995), Cambridge Univ. Press, Cambridge, 1996, 349-354.

Department of Mathematical System Science Faculty of Science and Technology Hirosaki University, Hirosaki 036-8561, Japan

E-mail address: skm@cc.hirosaki-u.ac.jp On calculations of modular irreducible characters with the help of computers

Katsushi Waki

Let p be a prime. We denote p-modular system by (K, R, F) where F is an algebraically closed field of characteristic p. Let G be a finite group. It is not so easy to calculate of irreducible modular characters correspondint to simple F G-module from ordinary characters correspondint to simple KG-module. I will introduce a standard way for the calculation of modular irreducible characters without a concrete construction of simple F G-modules.

I also show some short demonstrations of the calculations of irreducibel modular characters by GAP[1]

References

1. M. Schönert et al., GAP 3.4 Manual (Groups, Algorithms and Programming), RWTH Aachen, Aachen, Germany, 1994.

Dept. of Mathematical System Science Faculty of Science and Technology Hirosaki University Hirosaki, Aomori 036-8561 JAPAN

Email: waki@cc.hirosaki-u.ac.jp Self-duality of quasi-Harada rings and locally distributive rings

Kazutoshi Koike

Recently Y. Baba [1] proved that for a QH ring (quasi-Harada ring) R, if gRg is a local serial ring with some condition, then R has a self-duality, where g is an idempotent of R with gR a minimal faithful right module. (Note that (two-sided) QH rings are QF-3.) Motivated by this result, we also investigate self-duality of QH rings and obtain several results including an improvement of the result of Y. Baba and applications to self-duality of locally distributive rings.

Y. Baba and K. Iwase [2] called a left artinian ring R left QH in case every indecomposable projective right R-module is quasi-injective. In our study, the following easy lemma is very important.

Lemma 1. Let R be a left QH ring and let S = End_{R}(E(R_{R})). Then there exist a diagonally complete subring S′ of S and an ideal I of S′ such that R ≅ S′/I.

Here we say that a subring R′ of a ring R is diagonally complete in case there exists a complete set {e_{1}, …, e_{n}} of orthogonal idempotents for R such that e_{i}Re_{i} ⊂ R′ (1 ≤ i ≤ n). Diagonally complete subrings R′ inherit many properties from R. Particularly we have

Theorem 2. Diagonally complete subrings inherit Morita duality and good selfduality.

Good self-duality is a special type of self-duality. Every serial ring has a good self-duality and factor rings inherit good self-duality. From the results above, we can obtain an improvement of the result of Baba. We can also prove the following. (Compare with properties of left H-rings.)

Theorem 3. (1) Every left QH ring R has a right Morita duality. In particular R is right artinian.

(2) Every left QH ring R can be constructed from a QF ring.

An artinian ring R is said to be locally distributive in case each of its left and right indecomposable projective R-module is distributive. Clearly serial rings are locally distributive. Azumaya conjectured that every exact artinian ring has a selfduality (see [4]). While locally distributive rings are exact, even the self-duality for locally distributive rings is still open. Since locally distributive right QF-2 rings are left QH, we can apply results of left QH rings to these rings and obtain the following partial results. Here almost self-duality is a generalization of self-duality (see [3]).

Theorem 4. Let R be a locally distributive right QF-2 ring with basic set {e_{1}, …, e_{n}} of orthogonal primitive idempotents and let m = #{e_{i} | soc(R_{R})e_{i} ≠ 0}. Then

(1) R has an almost self-duality.

(2) If R is right serial or m ≤ 2, then R has a good self-duality.

(3) If n ≤ 3, then R has a self-duality.

References

1. Y. Baba. On self-duality of Auslander rings of local serial rings. Comm. Algebra, 30(6):2583– 2592, 2002.

2. Y. Baba and K. Iwase. On quasi-Harada rings. J. Algebra, 185(2):544–570, 1996.

3. K. Koike. Almost self-duality and Harada rings. J. Algebra, 254(2):336–361, 2002.

4. W. Xue. Rings with Morita duality. Springer-Verlag, Berlin, 1992.

Okinawa National College of Technology 2-19-2 Ohigashi, Nago City Okinawa 905-0016, JAPAN On S_{R}(H)-blocks II

Hieda Yoshimasa

Let G be a finite group, p a prime divisor of the order of G and (K, R, k) a p-modular system, i.e., R is a complete discrete valuation ring with maximal ideal (π), K is the quotient field of R of characteristic 0 and k(:= R/(π)) is the residue class field of R of characteristic p. Moreover, we assume that K contains the |G|th roots of unity.

For a subset X of G, X̂ denotes the sum of all elements of X in the group algebra 𝔬G, where 𝔬 is R, K or k. We consider the Hecke algebra S_{𝔬}(H) := End_{𝔬G}(Ĥ𝔬G) for a subgroup H of G.

In [7] G.R.Robinson has proved that Z(S_{R}(H)) ≃ A_{R}(H) as R-algebras, where A_{R}(H) denotes the endomorpism ring End_{R[G×G]}(RGĤRG). So he has defined A_{R}(H)-blocks (we call them S_{R}(H)-blocks) and their defect groups in G × G for the (central) primitive idempotents of A_{R}(H) (S_{R}(H)). (We mention that the set of S_{R}(H)-blocks corresponds bijectively to the set of S_{k}(H)-blocks as ĤRG is a permutation module. )

In this talk we show some results on S_{R}(H)-blocks, which are almost proved in [4] and [7] and some new results on the principal S_{R}(H)-blocks and their defect groups.

References

1. C. W. Curtis and I. Reiner : Methods of Representation Theory with Application to Finite Groups and Orders-Volume I, John Wiley and Sons, New York, 1981.

2. Y. Hieda : Some examples of S_{R}(H)-blocks, Bul. of Osaka Pref. of Tech., vol.34,73-76, 2000.

3. Y. Hieda : Some examples of S_{R}(H)-blocks, Proc. of the 33th Symposium On Ring Theory and Representation Theory, 29-35, 2001.

4. Y. Hieda and Y. Tsushima : On S_{R}(H)-Blocks for Finite Groups, J. of Algebra, vol.202, 583-588, 1998.

5. G. D. James : The Representation Theory of Symmetric Groups, Lecture notes in Mathematics 682, 1978) .

6. H. Nagao and Y. Tsushima : Representations of Finite Groups, Academic Press, 1989.

7. G. R. Robinson : Some Remarks on Hecke Algebras, J. of Algebra, vol.163, 806-812, 1994.

Department of Liberal Arts

Osaka Prefectural College of Technology Neyagawa, Osaka 572-8572 JAPAN

Email: hieda@las.osaka-pct.ac.jp A pair of rings with common ideals Yasuyuki Hirano Department of Mathematics, Okayama University Okayama 700-8530, Japan and Hisaya Tsutsui Department of Mathematics, Millersville University Millersville, PA, 17551, USA

Abstract

A study of pairs of commutative rings with the same set of prime ideals appears in the literature. In this talk, we consider some generalizations of the study and investigate pairs of rings, not necessarily commutative, with certain ideals in common. Several examples of such pairs will be given and conditions that pass through such rings will be studied.

Finitely Cogenerated Distributive Modules

Yasutaka SHINDOH

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Throughout this talk, “A” will always denote an associative ring with a non-zero identity and all modules will be unitary left A-modules. A module is said to be distributive if the lattice of its submodules satisfies the distributive law. Two modules M_{1} and M_{2} are said to form an unrelated pair when the following condition holds:

If L_{1}/N_{1} is isomorphic to L_{2}/N_{2} for some submodules

N_{i} ≤ L_{i} ≤ M_{i} (i = 1, 2), then L_{1} = N_{1} and L_{2} = N_{2}.

A module M said to be X-cyclic for some module X, when the following condition holds:

If ∑_{α∈Hom_{A}(X,M)} Imα = M, then there exists an epimor-

phism α ∈ Hom_{A}(X, M).

A module M said to be X-cocyclic for some module X, when the following condition holds:

If ⋂_{β∈Hom_{A}(M,X)} Kerβ = 0, then there exists a monomor-

phism β ∈ Hom_{A}(M, X).

Using the above concepts, we obtain the following theorem as a generalization of the main result of [Vam78] and its duality.

Theorem 1. Let M be a module. Then the following conditions hold.

(1) If M is finitely generated semilocal and distributive, then M is P -cyclic for any projective module P.

(2) If M is finitely cogenerated and distributive, then M is Qcocyclic for any injective module Q.

The aim of this talk is to prove and establish the above theorem.

References

Shi03. Y. Shindoh, A submodule unrelated to the corresponding factor module over a commuta-

tive ring, J. Algebra Appl. 2 (2003), no. 1, 101–104.

ShiPR. Y. Shindoh, Finitely cogenerated distributive modules are Q-cocyclic, preprint. Ste74. W. Stephenson, Modules whose lattice of submodules is distributive, Proc. London Math.

Soc. 28 (1974), no. 3, 291–310.

Vam78. P. Vámos, Finitely generated artinian and distributive modules are cyclic, Bull. London Math. Soc. 10 (1978), no. 3, 287–288.

Karita 5-8-21-302, Sumiyoshi-ku, Osaka 558-0011, JAPAN

Email: ring-pub@fan.gr.jp Global dimension in left serial algebras

Morio Uematsu

Let A be a finite dimentional basic conected algebra over an algebraically closed field k, and n is the number of the non isomorphic simple left modules of A. If A is quasi-hereditary, then the global dimension gl.dim.A is less than or equal to 2n − 2, and the Loewy length l(A) of A is less than or equal to 2^{n} − 1 [3]. If A is serial and gl.dim.A is finite, then gl.dim.A ≤ 2n − 2 and l(A) ≤ 2n − 1 [2]. On the other hand, Yamagata constructed the algebras of large global dimension with few simple modules [4], and Deng constructed the algebras of arbitrary finite global dimension with n ≥ 2 arbitrary [1]. So, if we consider the relationship between gl.dim.A and n, we need some conditions on algebras. In this talk we show that left serial quasi-hereditary algebras have global dimension less than or equal to n. Next, we show that left serial algebras have global dimension less than or qual to 2n − 2 when they have finite global dimension.

Lemma 1. If A is an algebra whose quiver has no oriented cycles then the global dimension of A is less than or equal to n − 1.

Example 2. In the above lemma, the bound of global dimension is sharp. Let A be an algebra defined by the following quiver with relations {α_{i+1}α_{i} = 0(1 ≤ i ≤ n − 2)}.

1 ⟶^{α_{1}} 2 ⟶^{α_{2}} ⋯ ⟶^{α_{n−1}} n

Then the global dimension of A is n − 1.

Now we recall the definition of heredity idempotents and quasi-hereditary algebras. Let N be the Jacobson radical of A. An idempotent e is said to be a heredity idempotent of A if eNe = 0 and AeA is projective as a left A-module. An algebra A is said to be quasi-hereditary if there is a sequence {e_{1}, e_{2}, ⋯, e_{m}} of (not necessarily primitive) idempotents of A such that, for any 1 ≤ t ≤ m, \overline{ε_{t}} is a heredity idempotent of A/Aε_{t+1}A and ε_{1} = 1, where ε_{j} = e_{j} +e_{j+1} +⋯+e_{m} for 1 ≤ j ≤ m and ε_{n+1} = 0. Such a sequence called a heredity sequence.

Theorem 3. Let A be a serial quasi-hereditary algebra. Then gl.dim.A ≤ n and l(A) ≤ 2n − 1, and these bounds are sharp.

We can show the same result when A is a right or left (not necessarily both) serial algebra, and more over when A has unique oriented cycle in its quiver.

Theorem 4. Let A be a left serial quasi-hereditary algebra. Then gl.dim.A ≤ n and l(A) ≤ 2n − 1, and these bounds are sharp.

References

1. Deng, Bangming, A construction of algebras with arbitrary finite global dimensions, preprint.

2. Gustafson, William H., Global Dimension in Serial Rings, J. Algebra 97(1985), 14-16.

3. Dlab,V. and Ringel,C.M., Quasi-hereditary algebras, Illinois J. Math. 33(1989),280-291.

4. Yamagata,K., A construction of algebras with large global dimensions, J. Algebra 163(1994), 57-67.

Faculty of Management Information Science Jobu University Shin-machi Tano, Gunma 370-1393 JAPAN

Email: uematsu@jobu.ac.jp Derivations of polynomial rings over a field of characteristic zero

Andrzej Nowicki

We present some old and new results concerning several famous algebraic problems formulated using derivations of polynomial algebras. The main role in this lecture play rings of constants and locally nilpotent derivations. We present also some observations and remarks for simple and semisimple derivations.

Institute of Mathematics

Nicholas Copernicus University

ul chopina 12/18, 87-100 Torun, POLAND Email: anow@mat.uni.torun.pl

フロベニュース多元環について

埼玉大学 教育学部 若松隆義

QF 多元環は数学のいろいろな場面に現れる重要な研究対象である。最近の話題としては Sklyanin A1gebra や AS-Regu1ar Ring の Koszul dual として現れることが知られている。講演ではフロベニウス多元環がかなり具体的に構成され分類されるものであることを説明したい。具体的には、体 K 上の分離多元環 A、 A 上の両側加群 AM_{A}、根基の巾零指数 n ≥ 2 が与えられたとき、 Λ/radΛ ≅ A、 radΛ/rad^{2}Λ ≅ M、 rad^{n}Λ = 0 となるようなフロベニウス多元環 Λ の K 同型類全体の記述について考える。更に graded 多元環の場合には、一般の場合に比べて話が単純になるので、可能な中山同型や多元環の分類例を与えることが出来る。 QF 多元環全体の様子はそれで一応分かったことになるが、他の問題に関してその記述がそのまま使えるわけではない。そのため、幾つかの別の構成法も考える。このうちの一つとして、多元環

A 上の巾零対称多元環 (φ, ψ) : _{A}M ⊗_{A} M_{A} ⟶^{φ} _{A}M_{A}, _{A}M ⊗_{A} M_{A} ⟶^{ψ} D(A), から

対称多元環 A(φ, ψ) = A ⊕ M ⊕ D(A) が構成されること、傾斜加群_{B}T_{A} が与えられたとき、 T_{A} と_{A}M_{A} の間で一定の条件が満たされるとき、 B 上の巾零対称多元環 (φ^{T} , ψ^{T}) が自然に得られ、 A(φ, ψ) と A(φ^{T} , ψ^{T}) が安定同値になることを示す。この二つの対称多元環は、 pd(_{B}T), pd(T_{A}) < ∞ の場合には自明拡大多元環の場合と同様の方法で、導来同値になるということが示される。