The 39th Symposium on Ring Theory and Representation Theory

(2006)

ABSTRACT

CHARACTER THEORY OF SEMISIMPLE BI-FROBENIUS ALGEBRAS

Yukio DOI

Bi-Frobenius algebras were introduced by the author and Takeuchi. They are Frobenius algebras with some coalgebra structures, and generalize both finitedimensional Hopf algebras and scheme rings of association schemes. We study a character theory of bi-Frobenius algebras and its application to association schemes.

We work over a field k; unadorned Hom and ⊗ are assumed to be taken over k. Recall that a Frobenius algebra (A, ϕ) is a finite-dimensional k-algebra A together with a linear function ϕ ∈ A^{∗} = Hom(A, k) such that the map θ : A → A^{∗}, a ↦ (ϕ ↼ a), is a right A-module isomorphism. Here A^{∗} is a two-sided A-module via (a ⇀ f)(b) := f(ba) and (f ↼ a)(b) := f(ab). Let ∑_{i} x_{i}⊗y_{i} ∈ A⊗A be the element

corresponding to id_{A} under the isomorphism: A⊗A ⟶^{id_{A}⊗θ} A⊗A^{∗} = End(A). Then the inverse of θ is given by θ^{−1}(f) = ∑_{i} f(x_{i})y_{i} for all f ∈ A^{∗} and we see

a = ∑ x_{i}ϕ(y_{i}a) = ∑ ϕ(ax_{i})y_{i}, ∑ ax_{i} ⊗ y_{i} = ∑ x_{i} ⊗ y_{i}a

for all a ∈ A. The pair (ϕ, ∑_{i} x_{i} ⊗ y_{i}) is called a Frobenius system for A. The element v(A) := ∑_{i} x_{i}y_{i} is in Z(A), the center of A. If v(A) is an invertible element, then A is a separable algebra, since ∑_{i} v(A)^{−1}x_{i} ⊗ y_{i} is a separable idempotent. The element v(A) is called the volume of A. Note that A×A → k, (a, b) ↦ ϕ(ab), is an associative, non-degenerate bilinear form. If we take {x_{i}} to be linearly independent, then it satisfies ϕ(y_{i}x_{j}) = δ_{ij} (dual basis property).

Dually, let C be a k-coalgebra with comultiplication Δ : C → C ⊗ C and counit ε : C → k. We write Δ(c) = ∑_{(c)} c_{(1)} ⊗ c_{(2)} = ∑ c_{1} ⊗ c_{2} for c ∈ C. The dual C^{∗} is an algebra by convolution, that is, (f ∗ g)(c) = ∑ f(c_{1})g(c_{2}), and C has a two-sided C^{∗}-module structure via f ⇀ c := ∑ c_{1}f(c_{2}) and c ↼ f := ∑ f(c_{1})c_{2} for f ∈ C^{∗}, c ∈ C. A Frobenius coalgebra (C, t) is a finite-dimensional k-coalgebra C together with an element t ∈ C such that the map κ : C^{∗} → C, f ↦ (t ↼ f) is a right C^{∗}-module isomorphism.

Definition. Let H be a finite-dimensional algebra with coalgebra structure Δ : H → H ⊗ H, ε : H → k. Given ϕ ∈ H^{∗} and t ∈ H and define a map S : H → H by S(h) := t ↼ (h ⇀ ϕ) = ∑ ϕ(t_{1}h)t_{2}. Then the 4-tuple (H, ϕ, t, S) is called a bi-Frobenius algebra (or bF algebra) if it satisfies the following (BF1)–(BF6): (BF1) ε ∈ Alg_{k}(H, k); (BF2) 1 ∈ G(H); i.e. Δ(1) = 1 ⊗ 1, ε(1) = 1; (BF3) (H, ϕ) is a Frobenius algebra; (BF4) (H, t) is a Frobenius coalgebra; (BF5) S is an antialgebra map (i.e. S(hh′) = S(h′)S(h), S(1) = 1); (BF6) S is an anti-coalgebra map (i.e. Δ(S(h)) = ∑ S(h_{2}) ⊗ S(h_{1}), ε(S(h)) = ε(h)).

Since H is finite-dimensional, its k-linear dual H^{∗} is also an algebra and coalgebra. One can easily show that (H^{∗}, t, ϕ, S^{∗}) becomes a bF algebra. We call it the dual of (H, ϕ, t, S).

References

[DT] Y. Doi and M. Takeuchi, BiFrobenius algebras, in: Contemporary Mathematics 267 ”New trends in Hopf algebras”, (2000), 67-97.

[D] Y. Doi, Bi-Frobenius Algebras and Group-like Algebras, Lecture notes in pure and applied Mathematics, vol.237 “ Hopf Algebras ” , (2004) 143-155 .

Faculty of Education

Okayama University

Okayama 700-8530 JAPAN Email: ydoi@cc.okayama-u.ac.jp Linearity Defect of graded modules over an exterior algebra

Kohji Yanagawa

Let S = K[x_{1}, …, x_{n}] be a polynomial ring over a field K, and E := K⟨y_{1}, …, y_{n}⟩ an exterior algebra. We regard them as graded rings with deg x_{i} = deg y_{i} = 1.

Let A = ⨁_{i≥0} A_{i}, A_{0} = K, be a Koszul algebra (e.g. A = S or E), and gr A the category of finitely generated graded left A-modules. For M ∈ gr A and i ∈ ℤ, M_{⟨i⟩} denotes the submodule of M generated by its degree i part M_{i}. We say M is weakly Koszul, if M_{⟨i⟩} has a graded free resolution of the form

⋯ → A^{⊕β_{j}}(−i−j) → ⋯ → A^{⊕β_{2}}(−i−2) → A^{⊕β_{1}}(−i−1) → A^{⊕β_{0}}(−i) → M_{⟨i⟩} → 0

for all i. If M is weakly Koszul, so is its ith syzygy Ω_{i}(M) for all i ≥ 0. We define the linearity defect ld(M) of M by

ld_{A}(M) := inf{ i | Ω_{i}(M) is weakly Koszul }.

Clearly, ld_{S}(M) ≤ proj. dim_{S}(M) ≤ n. But even if A is commutative, ld_{A}(M) can be infinite. But we have the following.

Theorem 1. (1) (Eisenbud et. al. [1]) We have ld_{E}(N) < ∞ for all M ∈ gr E.

(2) If n ≥ 2, then sup{ ld_{E}(N) | N ∈ gr E } = ∞. But ld_{E}(N) ≤ w^{n!}2^{(n−1)!}, where w := max{ dim_{K} N_{i} | i ∈ ℤ }.

(3) (Herzog-Römer) If J is a monomial ideal of E = K⟨y_{1}, …, y_{n}⟩, then ld_{E}(J) ≤ n − 1.

The parts (1) and (2) of the theorem are closely related to the Bernstein-Gel’fand-Gel’fand correspondence D^{b}(gr S) ≅ D^{b}(gr E). The main results of this talk improve/refine the part (3).

Set [n] := {1, …, n}, and let Δ ⊂ 2^{[n]} be a simplicial complex (i.e., F ⊂ G ∈ Δ implies F ∈ Δ). To Δ, we assign monomial ideals of S and E as follows:

I_{Δ} := ( ∏_{i∈F} x_{i} | F ⊂ [n], F ∉ Δ ) ⊂ S, J_{Δ} := ( ∏_{i∈F} y_{i} | F ⊂ [n], F ∉ Δ ) ⊂ E.

Any monomial ideal of E is of the form J_{Δ} for some Δ. And S/I_{Δ} is called the Stanley-Reisner ring of Δ, and important in combinatorial commutative algebra. The following theorem follows from BGG correspondence for S/I_{Δ} and E/J_{Δ}.

Theorem 2. (Okazaki-Y [2]) Let Δ ≠ ∅ be a simplicial complex on [n]. Then; (1) ld_{S}(I_{Δ}) = ld_{E}(J_{Δ}). (So we denote this value by ld(Δ).)

(2) ld(Δ) is a topological invariant of the geometric realization |Δ^{∨}| of the dual complex Δ^{∨} := { F ⊂ [n] | [n] − F ∉ Δ } of Δ.

(3) For n ≥ 4, ld(Δ) = n − 3 (this is the largest possible value) if and only if Δ is an n-gon.

References

[1] D. Eisenbud, G. Fløystad and F.-O. Schreyer, Sheaf cohomology and free resolutions over exterior algebra, Trans. Amer. Math. Soc. 355 (2003), 4397-4426.

[2] R. Okazaki and K. Yanagawa, Linearity Defects of Face Rings, preprint (math.AC/0607780).

Department of Mathematics

Graduate School of Science

Osaka University

Osaka University, Toyonaka, Osaka 560-0043 JAPAN Email: yanagawa@math.sci.osaka-u.ac.jp Compactly generated triangulated categories, Well generated triangulated categories, The homotopy category of flat R-modules

Amnon Neeman

(Talk 1) Compactly generated triangulated categories

ABSTRACT: Compactly generated triangulated categories are easy enough to define. There is a short list of main theorems about them. We will state the definitions and the theorems, and then illustrate their power by looking at simple examples from algebraic geometry.

(Talk 2) Well generated triangulated categories

ABSTRACT: These are generalizations of compactly generated categories. We will explain the need for looking at the more general picture, and then outline the theorems one has been able to prove so far.

(Talk 3) The homotopy category of flat R-modules

ABSTRACT: This is a conference on ring theory, and in this talk we will explain how the general theory applies to give a string of results on categories of representations. We will begin with a survey of recent papers by Krause, Jorgensen, and by Iyengar and Krause. These three articles explore the relationship between the categories K(InjR), K(P rojR) and K(F latR). It turns out that the relation becomes clearer and more transparent when one uses the techniques of well generated triangulated categories. We will explain what was proved in the three papers mentioned above, and also the recent improvements on them.

Center for Mathematics and its Applications Mathematical Sciences Institute Australian National University Canberra, ACT 0200, Australia

Email: Amnon.Neeman@anu.edu.au On S-Cohn-Jordan Extensions

Jerzy Matczuk

In Memory of P.M. Cohn

Let a monoid S act on a ring R by injective endomorphisms. An over-ring A(R; S) of R is called the S-Cohn-Jordan extension of R if (1) the action of S on R extends to an action of S on A(R; S) by automorphisms and (2) for any a ∈ A(R; S), there exists s ∈ S such that s · a ∈ R. A classical result of P.M. Cohn, which was originally formulated in much more general context of Ω-algebras (instead of rings), says that such an extension always exists provided the monoid S posseses a group S^{−1}S = G of left quotients.

The aim of the talk is to present a series of results relating various algebraic properties of R and that of A(R; S). For example primeness, Goldie conditions and other finiteness conditions will be considered.

Some possible applications to the skew semigroup rings R#S and skew polynomial rings R[x; σ, δ] will be also discussed.

References

[1] P.M. Cohn, Universal Algebra, Harper and Row (1965).

[2] J. Matczuk, S-Cohn-Jordan Extensions, Comm. Algebra, to appear.

Institute of Mathematics Warsaw University 02-097 Warsaw, ul.Banacha 2

Email: jmatczuk@mimuw.edu.pl

Local QF-rings with radical cubed zero

Isao Kikumasa, Kiyoichi Oshiro, Hiroshi Yoshimura

In 1976, in his Book ”Algebra II”, Faith conjectured ”no” for the problem: Are semiprimary right self-injective rings quasi-Frobenius ? Since that time, this conjecture is known by the Faith conjecture. Although many ring theorists have tried to solve this conjecture, the conjecture still remains open. In our talk, first, we translate this problem into a problem on two-sided vector spaces. In some sense, the difficulty of this problem stems from the question: Can we make freely non-commutative local QF-rings ? In connection with my translation of the Faith conjecture, we present our construction of local QF-rings with radical cubed zero using a field of complex numbers, quaternion of Hamilton or number fields etc.

Department of Mathematics

Faculty of Science

Yamaguchi University

Yoshida, Yamaguchi 753-8512, JAPAN Email: kikumasa@yamaguchi-u.ac.jp Email: oshiro@yamaguchi-u.ac.jp Email: yoshi@yamaguchi-u.ac.jp DUAL OJECTIVITY OF QUASI-DISCRETE MODULES AND LIFTING MODULES

Yosuke Kuratomi

In [1], K.Oshiro and his students introduced “ojectivity (generalized injectivity)”, a new concept of relative injectivity, and using this injectivity we obtained some results for direct sums of extending modules. Afterward, S.H.Mohamed and B.J.Müller [2] defined a dual concept of ojectivity as follows :

Definition. M is said to be “N-dual ojective (or generalized N-projective)” if, for any epimorphism g : N → X and any homomorphism f : M → X, there exist decompositions N = N_{1} ⊕ N_{2}, M = M_{1} ⊕ M_{2}, a homomorphism h_{1} : M_{1} → N_{1} and an epimorphism h_{2} : N_{2} → M_{2}, such that gh_{1} = f|_{M_{1}} and fh_{2} = g|_{N_{2}}.

The concept of relative dual ojectivity is generalization of relative projectivity and this projectivity has an important meaning for the study of direct sums of lifting modules (cf.[2], [3]).

In this talk we introduce some results on “dual ojectivity” and apply it to direct sums of quasi-discrete modules.

References

[1] K.Hanada, Y.Kuratomi and K.Oshiro, On direct sums of extending modules and internal exchange property, J.Algebra 250 (2002), 115–133.

[2] S.H.Mohamed and B.J.Müller, Cojective Modules, J.Egyptian Math. 12 (2004), 83–96.

[3] Y.Kuratomi, On direct sums of lifting modules and internal exchange property, Comm.Algebra 33 (2005), 1795–1804.

Kitakyushu National College of Technology 5-20-1 Shii, Kokuraminamiku, Kitakyushu, Fukuoka, 802-0985 JAPAN

Email: kuratomi@kct.ac.jp Perfect isometries and the Alperin-Mckay conjecture

Charles W. Eaton

In the representation theory of finite groups, one of the main themes is the relationship between representations of a group G and those of certain subgroups. When we consider also modular representations, with respect to a field k of characteristic p, we see that local subgroups are particularly strongly related to G (a local subgroup is the normaliser of a p-subgroup, although this definition may be generalised).

Examples of the sort of information about G we may wish to obtain from local subgroups are the number of non-projective simple kG-modules, the number of irreducible characters of degree not divisible by |G|_{p}, and even categorical information, such as the derived equivalence class of the module category of kG. Much is known, including for example, the fact that the derived category is determined by normalisers of p-subgroups when a Sylow p-subgroup is cyclic. However, more is still conjecture, and there are other situations where we do not even know what the conjectures should be.

Existing conjectures vary from the purely numerical, for example those of Alperin, Alperin-McKay, Dade and others, where predictions are made which apply to all groups, to the categorical, such as Broué’s conjecture, which applies only in restricted cases (blocks with abelian defect groups). Perfect isometries may be regarded as lying between numerical and categorical information.

The purpose of these lectures will be to explain the range of conjectures, from the Alperin-McKay conjecture to Broué’s conjecture, via perfect isometries, and the problems associated with generalising Broué’s conjecture. A theme will be how numerical and categorical relationships together have an important part to play in the subject.

In the first lecture, I will introduce the necessary background material. This includes blocks of finite groups; invariants of blocks and groups; perfect isometries and a little about module categories. In the second lecture I will give an overview of the existing results and conjectures and their current status. In the third I will present a suggestion for a generalisation of the idea of a perfect isometry.

Department of Mathematics University of Manchester P.O.Box 88, Sackville Street Manchester, M60 1QD, UK

Email: charles.eaton@manchester.ac.uk Valuation Rings and Maximal Orders in Nonassociative Quaternion Algebras

John S. Kauta

A nonassociative quaternion algebra over a field F has the form A = N ⊕ NJ, where N is a separable quadratic extension field of F with an F -involution x ↦ x̂, Jx = x̂J for x ∈ N, and J^{2} = b ∈ N \ F. We define the notion of a valuation ring for A, and we also define a value function w on A with values from a totally ordered group. We determine the structure of the set B_{w} on which the value function assumes non-negative values and we prove that, given a valuation ring of A, there is a value function w associated to it if and only if the valuation ring is integral and invariant under proper F -automorphisms of A. The restriction w|_{N} is a valuation on N with valuation ring S = N ∩B_{w}. If S is inertial over F, then B_{w} is a valuation ring of A if and only if w(b) = 0; if in addition S is a DVR, then B_{w} is a valuation ring of A if and only if it is a maximal order in A and w(b) = 0. On the other hand if S is tamely and totally ramified over F, then B_{w} is a valuation ring of A if and only if the image of b in S/J(S) is not a square. If in addition V is a DVR, then B_{w} is a valuation ring of A if and only if B_{w} is a maximal order in A and the image of b is not a square in S/J(S); further, B_{w} is the intersection of maximal orders if w(b) = 0.

Keywords: Value functions, valuation rings, maximal orders, nonassociative quaternion algebras.

Department of Mathematics

Faculty of Science

Universiti Brunei Darussalam

Bandar Seri Begawan BE1410, BRUNEI Email: kauta@fos.ubd.edu.bn Cohen-Macaulay modules and holonomic modules over Gorenstein

filtered rings

HIROKI MIYAHARA

Let Λ be a (not necessary commutative) ring. A family of additive subgroups of Λ { F_{p}Λ | p ∈ ℕ } (ℕ ： the set of all non negative integer) is called a filtration of Λ, if

(1) 1 ∈ F_{0}Λ

(2) F_{p}Λ ⊂ F_{p+1}Λ

(3) (F_{p}Λ)(F_{q}Λ) ⊂ F_{p+q}Λ

(4) Λ = ⋃_{p∈ℕ} F_{p}Λ A pair (Λ, F) is called filtered ring if Λ has a filtration. Let σ_{p} : F_{p}Λ ⟶

F_{p}Λ/F_{p−1}Λ (F_{−1}Λ = 0) be a natural homomorphism, then grΛ := ⨁_{p∈ℕ} F_{p}Λ/F_{p−1}Λ is a graded ring with multiplication

σ_{p}(a)σ_{q}(b) = σ_{p+q}(ab), a ∈ F_{p}Λ, b ∈ F_{q}Λ

Let Λ be a filtered ring, and M be a (left) Λ-module. A family of additive subgroups of M { F_{p}Λ | p ∈ ℤ } is called a filtration of M, if

(1) F_{p}M ⊂ F_{p+1}M

(2) F_{p}M = 0 for p << 0

(3) (F_{p}Λ)(F_{q}M) ⊂ F_{p+q}M

(4) M = ⋃_{p∈ℤ} F_{p}M A pair (M F) is called filtered Λ-module if M has a filtration. Let τ_{p} : F_{p}M ⟶

F_{p}M/F_{p−1}M be a natural homomorphism, then grΛ の grM := ⨁_{p∈ℤ} F_{p}M/F_{p−1}M is a graded grΛ-module by

σ_{p}(a)τ_{q}(x) = τ_{p+q}(ax), a ∈ F_{p}Λ, x ∈ F_{q}M

We assume that grΛ is a commutative Gorenstein ^{∗}local ring (with unique ∗maximal ideal M) satisfying the following condition (P):

(P) : There exists an element of positive degree in grΛ − 𝔭 for any graded prime ideal 𝔭 ≠ M

Then, id_{Λ}Λ = id_{Λ^{op}}Λ < ∞. so, We call such a filtered ring Λ a ”Gorenstein filtered ring” (the well known example is Weyl algebra)

Let Λ be a filtered Gorenstein ring, M be a finitely generated filtered (left) Λ-module, and grM be a finite grΛ-module. We put ^{∗}dim grM = htM/I (I = [0 :_{grΛ} grM]), n = ^{∗}dim grΛ, and d = idΛ. then, ^{∗}dim grM ≤ n − d. so (like Weyl algebra) We call M a holonomic Λ-module, if ^{∗}dim grM = n − d. Also, we call M a CM Λ-module, if grM is CMgrΛ-module.

References

[1] K.Nishida and H.Miyahara, Cohen-Macaulay modules and holonomic modules over Gorenstein filtered rings, preprint

Department of Mathematical Sciences Graduate School of Science and Technology Shinshu University Matsumoto, Nagano 390-8621 JAPAN

Email: miyahara shinshu@yahoo.co.jp Symmetry in the Vanishing of Ext-groups

Izuru Mori

In this talk, we will find a class of rings R satisfying the following property: for every pair of finitely generated right R-modules M and N, Ext_{R}^{i}(M, N) = 0 for all i ≫ 0 if and only if Ext_{R}^{i}(N, M) = 0 for all i ≫ 0. In particular, we will show that such a class of rings includes a group algebra of a finite group and the exterior algebra of odd degree.

Department of Mathematics Faculty of Science Shizuoka University Shizuoka 422-8529 JAPAN

Email: simouri@ipc.shizuoka.ac.jp On derived equivalences in selfinjective algebras

Hiroki Abe and Mitsuo Hoshino

Let A be an artin algebra. Rickard [5, Proposition 9.3] showed that for any tilting complex P^{•} ∈ K^{b}(P_{A}) the number of nonisomorphic indecomposabele direct summands of P^{•} coincides with the rank of K_{0}(A), the Grothendieck group of A, which generalizes earlier results [2, Proposition 3.2] and [4, Theorem 1.19]. He raised a question whether a complex P^{•} ∈ K^{b}(P_{A}) with Hom_{K(Mod-A)}(P^{•}, P^{•}[i]) = 0 for i ≠ 0 is a tilting complex or not if the number of nonisomorphic indecomposable direct summands of P^{•} coincides with the rank of K_{0}(A) (see also [4]). In case P^{•} is a projective resolution of a module T ∈ mod-A with proj dim T_{A} ≤ 1, Bongartz [1, Lemma of 2.1] has settled the question affirmatively. More precisely, he showed that every T ∈ mod-A with proj dim T_{A} ≤ 1 and Ext_{A}^{1}(T, T) = 0 is a direct summand of a classical tilting module, i.e., a tilting module of projective dimension ≤ 1. Unfortunetely, this is not true in general (see [5, Section 8]). Our first aim of this note is to show that if A is a representation-finite selfinjective artin algebra then every P^{•} ∈ K^{b}(P_{A}) with Hom_{K(Mod-A)}(P^{•}, P^{•}[i]) = 0 for i ≠ 0 and add(P^{•}) = add(νP^{•}), where ν is the Nakayama functor, is a direct summand of a tilting complex.

Rickard [6, Theorem 4.2] showed that the Brauer tree algebras over a field with the same numerical invariants are derived equivalent to each other. Subsequently, Okuyama pointed out that for any Brauer tree algebras A, B with the same numerical invariants there exists a sequence of Brauer tree algebras A = B_{0}, B_{1}, ⋯, B_{m} = B such that, for any 0 ≤ i < m, B_{i+1} is the endomorphism algebra of a tilting complex for B_{i} of length ≤ 1. These facts can be formulated as follows. For any tilting complex P^{•} ∈ K^{b}(P_{A}) associated with a certain sequence of idempotents in a ring A, there exists a sequence of rings A = B_{0}, B_{1}, ⋯, B_{m} = End_{K(Mod-A)}(P^{•}) such that, for any 0 ≤ i < m, B_{i+1} is the endomorphism ring of a tilting complex for B_{i} of length ≤ 1 determined by an idempotent (see [3, Proposition 3.2]). Our second aim of this note is to show that for any derived equivalent representation-finite selfinjective artin algebras A, B there exists a sequence of selfinjective artin algebras A = B_{0}, B_{1}, ⋯, B_{m} = B such that, for any 0 ≤ i < m, B_{i+1} is the endomorphism algebra of a tilting complex for B_{i} of length ≤ 1.

References

[1] K. Bongartz, Tilted algebras, in: Representations of algebras (Puebla, 1980), 26–38, Lecture Notes in Math., 903, Springer, Berlin.

[2] D. Happel and C. M. Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), no. 2, 399–443.

[3] M. Hoshino and Y. Kato, Tilting complexes associated with a sequence of idempotents, J. Pure Appl. Algebra 183 (2003), no. 1-3, 105–124.

[4] Y. Miyashita, Tilting modules of finite projective dimension, Math. Z. 193 (1986), no. 1, 113–146.

[5] J. Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), no. 3, 436–456.

[6] J. Rickard, Derived categories and stable equivalence, J. Pure Appl. Algebra 61 (1989), 303–317.

Institute of Mathematics, University of Tsukuba, Ibaraki, 305-8571, Japan

Email: abeh@math.tsukuba.ac.jp hoshino@math.tsukuba.ac.jp Azumaya’s conjecture and Harada rings

Kazutoshi Koike

The aim of this talk is to present several results concerning Azumaya’s conjecture as an application of study of Harada and quasi-Harada rings. An artinian ring R is said to be exact in case R has a composition series of two-sided ideals such that each composition factor is simple and balanced as an (R, R)-bimodule. Azumaya [1] introduced the concept of exact rings and conjectured

Azumaya’s conjecture. Every exact ring has a self-duality.

Here an artinian ring R is said to have a self-duality in case R has a self Morita duality, i.e., there exists a duality (contravariant equivalence) between the category of finitely generated left R-modules and the category of finitely generated right Rmodules. We call an artinian ring serial (resp. locally distributive) in case the lattice of submodules of each left and right indecomposable projective module is linearly ordered (resp. distributive). As is well-known, every serial ring has a self-duality. Clearly serial rings are locally distributive. It is shown that locally distributive rings are exact. Therefore if Azumaya’s conjecture is true, the following problem is affirmative.

Problem. Does every locally distributive ring have a self-duality?

However the problem is still open. In this talk, we shall present several partial results (e.g., the self-duality for locally distributive right serial rings) about the problem. These results are obtained from study of quasi-Harada rings. Oshiro call certain QF-3 QF-2 artinian rings Harada rings in [5] and investigate the structure of Harada rings deeply. Baba and Iwase [2] generalize the concept of Harada rings as quasi-Harada rings, which are certain QF-2 rings. Locally distributive right QF-2 rings are left quasi-Harada rings. Thus we can use results for quasi-Harada rings to study self-duality of locally distributive (right QF-2) rings.

We shall also apply the observations of Kado and Oshiro [3] about the self-duality for Harada rings to exact and locally distributive rings and reduce Azumaya’s conjecture and Problem above to other problems. For example, we show the equivalence of Azumaya’s conjecture and the statement that every exact ring has a weakly symmetric self-duality.

References

[1] G. Azumaya. Exact and serial rings. J. Algebra, 85(2):477–489, 1983.

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

[3] J. Kado and K. Oshiro. Self-duality and Harada rings. J. Algebra, 211(2):384–408, 1999.

[4] K. Koike. Self-duality of quasi-Harada rings and locally distributive rings. J. Algebra, 302(2):613–645, 2006.

[5] K. Oshiro. Lifting modules, extending modules and their applications to QF-rings. Hokkaido Math. J., 13(3):310–338, 1984.

[6] W. Xue. Rings with Morita duality. Springer-Verlag, Berlin, 1992.

Okinawa National College of Technology 905 Henoko, Nago City Okinawa 905-2192, JAPAN

Email: koike@okinawa-ct.ac.jp Derived Equivalences of Symmetric Algebras defined by Generalized Tilting Modules

Takayoshi Wakamatsu

Let _{B}T_{A} be a generalized tilting module with Cok(C(T_{A}), PC(T_{A})) = modA and Ker(ID(DT_{B}), D(DT_{B})) = modB. Then, for a symmetric algebra Λ = A⊕M ⊕DA with multiplication

(a, m, f) · (a′, m′, f′) = (aa′, am′ + ma′ + φ(m ⊗ m′), af′ + fa′ + ψ(m ⊗ m′))

where φ : _{A}M ⊗_{A} M_{A} → _{A}M_{A} is nilpotent and ψ : _{A}M ⊗_{A} M_{A} → _{A}DA_{A} is nondegenerate, the corresponding symmetric algebra Λ^{T} = B ⊕ M^{T} ⊕ DB is defined and there is an equivalence of stable module categories Q : \underline{mod}Λ → \underline{mod}Λ^{T} if the modules M_{A} and T ⊗_{A} M_{A} are in the class C(T_{A}) and the homomorphism θ : T ⊗_{A} Hom_{A}(T, M) → Hom_{A}(T, T ⊗_{A} M), θ(t⊗h)(t′) = t⊗h(t′) is bijective. On the other hand, it is known by D. Happel that the stable module category \underline{mod}Λ of any self-injective algebra Λ has a structure of triangulated category. In this talk, we prove that the functor Q is in fact an equivalence of triangulated categories.

Fucalty of Education

Saitama University

Shimo-ohkubo 255, Sakura-ku, Saitama 338-8570, JAPAN Email: wakataka@post.saitama-u.ac.jp On a tensor product of square matrices in Jordan canonical form

Ryo Iwamatsu

To construct graded local Frobenius algebras over an algebraically closed field K, it is importatnt to find out a Joradan canonical form of tensor product of square matrices. In fact, it is known that any local graded Frobenius algebra is of the form of Λ(φ, γ) = T (V)/R(φ, γ), where V is a finite dimensional vector space, γ is an element of GL(V) and φ : V^{⊗n} → K is a linear map satisfying several conditions. Further, if we decompose as (V, γ) = ⨁_{i}(V_{i}, γ_{i}), then the conditions of φ can be described in terms of each φ_{i_{1}···i_{r}} : V_{i_{1}} ⊗ ⋯ ⊗ V_{i_{r}} → K. And then, we have to consider a Jordan canonical form of γ_{i_{1}} ⊗⋯⊗γ_{i_{r}} as an element GL(V_{i_{1}} ⊗⋯⊗V_{i_{r}}). (For detail, see Wakamatsu [3]).

Let K be an algebraically closed field of characteristic p ⩾ 0. And let A = J(a, s) and B = J(b, t) be Jordan blocks, where ab ≠ 0 and 0 < s ⩽ t. Our problem is to find out a Jordan canonical form A ⊗ B. In the case of p = 0, Martsinkovsky and Vlassov [2] gave the solution completely. In this talk, we shall introduce an algorithm for finding out a Jordan canonical form of A ⊗ B. This problem is reduced to the problem of finding the indecomposable decomposition R = K[X, Y]/(X^{s}, Y^{t}) as a K[θ] = K[X + Y]-module. Our algorithm gives the generators of indecomposable summands of R concretely. And we show that there exists some homogeneous elements ω_{0}, ω_{1}, …, ω_{s−1} such that

R ≅ ⨁_{i=0}^{s−1} K[θ]ω_{i}

as K[θ]-modules, where the degree of each ω_{i} is i. Independently, Iima and Yoshino gave a similary result of us. That is announced in [1].

References

1. K. Iima and Y. Yoshino, On the decomposition of tensor product of Jordan canonical form, (http://eprints.math.sci.hokudai.ac.jp/archive/00001589/01/Iima_pr.pdf) (in Japanese)

2. A. Martsinkovsky amd A. Vlassov, The representation rings of k[X], preprint. (http://www.math.neu.edu/ martsinkovsky/GreenExcerpt.pdf)

3. T. Wakamatsu, On graded Frobenius algebras, Journal of Algebra. 267 (2003), 377–395.

Faculty of Education

Saitama University

Shimo-Okubo 255, Sakura, Saitama, Saitama 338-8570, JAPAN Email: wakataka@post.saitama-u.ac.jp Finite groups having exactly one non-linear irreducible character

Kaoru MOTOSE

I talk about finite groups with exactly one non-linear irreducible character, in detail, characterizations, real representations of these groups and relations on Hurwitz theorem.

References

1. W. Feit, Characters of finite groups , Benjamin, 1967

2. D. Grenstein, Finite groups, 1980, Chelsea

3. I. N. Herstein, Noncommutative rings, The Curaus Mathematical Monographs, M.A.A., 1965 4. A.S. Muktibodh, Con-Cos groups, Review Bulletin of Calcutta Mathematical Society, 11(2003),

53-56.

5. A.S. Muktibodh, Characterization of Frobenius Groups of Special type, to appear.

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

Email: mail@address skm@cc.hirosaki-u.ac.jp Integrality of eigenvalues of Cartan matrices in finite groups

Tomoyuki Wada

Let G be a finite group and F be an algebraically closed field of characteristic p > 0. Let B be a block of the group algebra F G with defect group D and C_{B} be the Catan matrix of B. We are concerned with eigenvalues of C_{B}, especially with when all eigenvalues are rational integers and at that time with what each eigenvalue and its eigenvector represent. In this talk, we show, in some blocks, that if the Fobenius-Perron eigenvalue ρ(B) of C_{B} is an integer, then all eigenvalues are integers and those coincide with elementary divisors of C_{B}. Furthermore, we can take an eigenvector matrix for them as a Brauer character table Φ_{β} for some block β which is related to B.

We also show that when G is a fnite group with an abelian 2-Sylow subgroup D, or when G is a finite group with an elementary abelian Sylow 3-subgroup D of order 9, ρ(B) is an integer for the principal 2-block B in the former case, or the principal 3-block B in the latter case of F G if and only if B and its Brauer correspondent block b of F N_{G}(D) are Morita equivalent (even stronger Puig equivalent).

Department of Mathematics

Institute of Symbiotic Science and Technology Tokyo University of Agriculture and Technology Nakacho 2-24-16, Koganei, Tokyo 184-8588 JAPAN

Email: wada@cc.tuat.ac.jp Mackey functor and cohomology of finite groups

Akihiko Hida

Let G be a finite group and k a field of characteristic p > 0. In [1], G. Mislin proved the following.

Theorem 1 ([1, Theorem]). Let H be a subgroup of G containing a Sylow psubgroup of G. Then the following are equivalent. (1)The restriction

res_{G,H} : H^{∗}(G, k) ⟶ H^{∗}(H, k)

is an isomorphism. (2)If Q is a p-subgroup of H, x ∈ G and ^{x}Q ⊆ H, then x ∈ HC_{G}(Q).

That the conditon (2) implies (1) is well known. The remarkable part is the implication (1) ⇒ (2). The proof of Theorem by Mislin uses deep results from topology. We will give an algebraic proof. We follow the approach of P. Symonds using Mackey functor for G. We consider the structure of the cohomology H^{∗}(−, k) as a Mackey functor for G.

References

1. G. Mislin, On group homomorphisms inducing mod-p cohomology isomorphisms, Comment. Math. Helvetici 65 (1990) 454-461.

Faculty of Education

Saitama University

Shimo-Okubo 255, Sakura-ku, Saitama-city, 338-8570 JAPAN Email: ahida@math.edu.saitama-u.ac.jp Invariants of reductive algebraic groups with simple commutator subgroups

Haruhisa Nakajima

Let G be a reductive algebraic group over the complex number field C whose semisimple part is denoted to G′. We will use any of the notation ϱ, (ϱ, G) or (V, G) to denote a finite dimensional representation ϱ : G → GL(V) over C. An algebraic action of G on an affine variety X (abbr. (X, G)) is said to be coregular (resp. cofree, equidimensional), if C[X]^{G} is a polynomial ring (resp. if C[X] is a free C[X]^{G}-module, if the quotient morphism X → X/G = m Spec(C[X]^{G}) is equidimensional).

For simple connected G’s, their coregular, cofree and equidimensional representations are, respectively, classified by [2, 3], [4], and [1].

In this short communication, we will give some remarks on coregular representations of G with certain simple G′ which are based on relative invariants of algebraic tori on factorial rings and some equidimensional or cofree actions. Our remarks seem to be partially related to the results on quasi-coregular representations of simple algebraic groups and equidimensional toric extensions of their representatitions (e.g., [5]).

For a convenience sake, we omit the detailed statements in this abstract.

References

1. O.M. Adamovich, Equidimensional representations of simple algebraic groups, Transl. II. Ser., Amer. Math. Soc. 128 (1986), 25–29.

2. O.M. Adamovich, E.O. Golovina, Simple linear Lie groups having a free algebra of invariants, Selecta Math. Soviet 3 (1983/1984), 183–220

3. G.W. Schwarz, Representations of simple Lie groups with regular rings of invariants, Invent. Math. 49 (1978), 167-191.

4. , Representations of simple Lie groups with a free module of covariants, Invent. Math. 50 (1978), 1–12.

5. H. Nakajima, Equidimensional toric extensions of symplectic groups, Proc. Japan Acad. 74 (1994), 74–79.

Department of Mathematics Faculty of Science Josai University Sakado 350-0295, JAPAN Von Neumann regular rings with comparability

Mamoru Kutami

A ring R is said to be (von Neumann) regular if for each x ∈ R there exists an element y of R such that xyx = x, and R is said to be unit-regular if for each x ∈ R there exists a unit (i.e., an invertible) element u of R such that xux = x. A ring R is directly finite provided that xy = 1 (x, y ∈ R) implies yx = 1.

Unit-regularity and direct finiteness are important finiteness conditions for regular rings. For these relations, M. Henriksen (1973) first showed that unit-regular rings are always directly finite. But, unfortunately directly finite regular rings are not unit-regular in general, as G. Bergman (1977) showed. Therefore, the following interesting problems between unit-regularity and direct finiteness for regular rings arised:

Problem 1. Is every directly finite simple regular ring always unit-regular?

Problem 2. Which directly finite regular rings are unit-regular? For Problem 1, O’Meara [4] introduced the notion of weak comparability and showed that directly finite simple regular rings with weak comparability must be unit-regular. Here we recall that a regular ring R is said to satisfy weak comparability if, for each nonzero x ∈ R, there exists a positive integer n = n(x) such that n(yR) ≲ R_{R} implies yR ≲ xR for all y ∈ R. Thereafter, simple regular rings with weak comparability were studied in [1], [2], [3] etc..

In this talk, we mainly treat regular rings with weak comparability, and give some results on finitely generated projective modules over these rings, concerning with Problem 2.

References

1. P. Ara and K.R. Goodearl, The almost isomorphism relation for simple regular rings, Publ. Mat. UAB 36 (1992), 369–388.

2. P. Ara, K.R. Goodearl, E. Pardo and D.V. Tyukavkin, K-theoretically simple von Neumann regular rings, J. Algebra 174 (1995), 659–677.

3. P. Ara and E. Pardo: Refinement monoids with weak comparability and applications to regular rings and C^{∗}-algebras, Proc. Amer. Math. Soc. 124(3) (1996), 715–720.

4. K.C. O’Meara, Simple regular rings satisfying weak comparability, J. Algebra 141 (1991), 162–186.

Department of Mathematics

Faculty of Science

Yamaguchi University

Yoshida, Yamaguchi 753-8512, JAPAN Email: kutami@yamaguchi-u.ac.jp