The 42nd Symposium on Ring Theory and Representation Theory

ABSTRACT

Osaka Kyoiku University, Osaka October 10 - 12, 2009

The 42nd Symposium onRing Theory and Representation Theory Program

October 10 （Saturday）

09：00 − 09：45 Shuichi Ikehata (Okayama University) George Szeto (Bradley University) Lianyong Xue (Bradley University)

On Galois Extensions with an Inner Galois Group and a Galois Commutator Subring 10：00 − 10：45 Edward Poon （Embry-Riddle University) Hisaya Tsutsui (Embry-Riddle University) Yasuyuki Hirano (Naruto Kyoiku University)

Fully Weakly Prime Rings 11 ： 00 − 12 ： 00 Fred Van Oystaeyen (Antwerp University) Crystalline graded rings 13：30 − 14：15 Kazuho Ozeki (Meiji Institute for Advanced Study of Mathematical Sciences) Shiro Goto (Meiji University) Koji Nishida (Chiba University)

On the structure of Sally modules of rank one 14 ： 30 − 15 ： 15 Futoshi Hayasaka (Meiji University) The Buchsbaum-Rim function of a parameter module 15 ： 30 − 16 ： 15 Noritsugu Kameyama (Shinshu University) Extension of the Matlis duality to a filtered Noetherian ring 16 ： 30 − 17 ： 15 Kohji Yanagawa (Kansai University) Dualizing complex of the Stanley ring associated with a simplicial poset

October 11 (Sunday)

09 ： 00 − 09 ： 45 Ryo Takahashi (Shinshu University) Thick subcategories of the stable category of Cohen-Macaulay modules 10 ： 00 − 10 ： 45 Hiroyuki Minamoto (Kyoto University) Ampleness of two-sided tilting complexes and Fano algebras 11 ： 00 − 12 ： 00 Fred Van Oystaeyen (Antwerp University) The projective scheme of the blow-up ring 13：30 − 14：15 Hiroki Abe (University of Tsukuba) Mitsuo Hoshino (University of Tsukuba)

Derived equivalences for endomorphism rings

14 ： 30 − 15 ： 15 Kota Yamaura (Nagoya University) The classification of tilting modules over Harada algebras

15 ： 30 − 16 ： 15 Kaoru Motose (Hirosaki) The Stickelberger relation and Loewy series of group algebras Map(𝔽_{q}, 𝔽_{q})

16 ： 30 − 17 ： 30 Changchang Xi (Beijing Normal University) Homological conjectures and radical-full extensions, I

October 12 (Monday)

09 ： 00 − 09 ： 45 Mamoru Kutami (Yamaguchi University) Almost comparability and related comparabilities in von Neumann regular rings

10 ： 00 − 10 ： 45 Martin Herschend (Nagoya University) The Clebsch-Gordan problem for quiver representations

11 ： 00 − 12 ： 00 Changchang Xi (Beijing Normal University) Homological conjectures and radical-full extensions, II On Galois Extensions with an Inner Galois Group and a Galois Commutator Subring

Shûichi Ikehata, George Szeto and Lianyong Xue

It was shown ([1], Theorem 3) that B is a central Galois algebra over its center C with an inner Galois group G if and only if it is an Azumaya projective group algebra CG_{f} where f : G × G ⟶ units of C is a factor set. We shall generalize the above theorem to any Galois extension B with an inner Galois group G where G = {g ∈ G | g(x) = U_{g}xU_{g}^{−1} for some U_{g} ∈ B and for all x ∈ B}. It is shown that B contains a projective group algebra CG_{f}. An equivalent condition for a central Galois algebra CG_{f} with Galois group induced by G is given, and characterizations for a Galois extension B with an inner Galois group G generated by {U_{g} | g ∈ G} over B^{G} are obtained. When B is also an Azumaya algebra, some properties are given for a Galois extension B with an inner Galois group G.

References

[1] F.R. DeMeyer, Galois Theory in Separable Algebras over Commutative Rings, Illinois J. Math., 10 (1966), 287-295.

[2] S. Ikehata and G. Szeto, On H-separable polynomials in skew polynomial rings of automorphism type, Math. J. Okayama Univ., 34 (1992), 49-55.

[3] G. Szeto and L. Xue, The Structure of Galois Algebras, Journal of Algebra, 237(1) (2001), 238-246.

[4] G. Szeto and L. Xue, The Galois Algebra with Galois Group which is the Automorphism Group, Journal of Algebra, 293 (2005), no. 1, 312-318.

[5] G. Szeto and L. Xue, On Projective Group Rings with an Inner Automorphism Group, (submitted).

Shûichi Ikehata

Department of Environmental and Mathematical Science Faculty of Environmental Science and Technology Okayama University Tsushima, Okayama 700-8530, JAPAN Email: ikehata@ems.okayama-u.ac.jp

George Szeto

Department of Mathematics

Bradley University

Peoria, Illinois 61625, U.S.A.

Email: szeto@bradley.edu

Lianyong Xue

Department of Mathematics

Bradley University

Peoria, Illinois 61625, U.S.A.

Email: lxue@bradley.edu

Fully Weakly Prime Rings

Edward Poon, Yasuyuki Hirano, and Hisaya Tsutsui

We define a proper ideal I of a ring R to be weakly prime if 0 ≠ JK ⊆ I implies either J ⊆ I or K ⊆ I for any ideals J, K of R. In this talk, we investigate the structure of rings, not necessarily commutative,in which all ideals are weakly prime. (A ring whose zero ideal is prime is called prime. In this sense, any ring R is weakly prime since the zero ideal is always weakly prime.)

Crystalline Graded Rings

Fred van Oystaeyen

In joint work with V. Bavula the generalized Weyl algebras have been studied, their homological algebra properties like dimensions and regularity have been studied. The main ingredient in this study is the existence of a gradation with very nice properties e.g. the part of degree zero being a Dedekind domain or a Noetherian integrally closed domain of low dimension. The class of crystalline graded rings generalizes the class of generalized Weyl algebras, even in the case of gradings by integers interesting new examples appear. Actions by a noncommutative group of automorphisms of the degree zero part as well as generalized Z-cocycles appear naturally in this theory and a suitable localization at an Ore set yields a crossed product. The general theory then deals with orders and maximal orders in crossed products. In case of crystalline graded rings over Dedekind domains the ideal theory of Dedekind domains may be used to obtain spectrally twisted cocycles where by taking valuations at primes in the Dedekind domains a set of 2-cocycle like conditions appears. We classify all graded maximal orders containing a given crystalline graded ring and provide classification in some special cases. The main ingredient in this theory is the manipulation of spectrally twisted cocycles and orders constructed starting from them.

On the structure of Sally modules of rank one

Shiro Goto, Koji Nishida and Kazuho Ozeki

Let A be a Cohen-Macaulay local ring with the maximal ideal 𝔪 and d = dim A > 0. We assume the residue class field k = A/𝔪 of A is infinite. Let I be an 𝔪-primary ideal in A and choose a minimal reduction Q = (a_{1}, a_{2}, ⋯, a_{d}) of I. Let

R = R(I) := ⨁_{n≥0}I^{n}, T = R(Q) := ⨁_{n≥0} Q^{n} and G = G(I) := ⨁_{n≥0} I^{n}/I^{n+1}

respectively denote the Rees algebras of I and Q and the associated graded ring of I. We then define

S = IR/IT

and call it the Sally module of I with respect to Q. Let B = T/𝔪T which is the polynomial ring with d indeterminates over the field k = A/𝔪.

The main result of my talk is the following, which is a complete structure theorem of the Sally modules of 𝔪-primary ideal I satisfying the equality

e_{1} = e_{0} − ℓ_{A}(A/I) + 1,

where e_{i} = e_{I}^{i}(A) denotes the i-th Hilbert coefficients of I. Theorem 1 ([?]). The following three conditions are equivalent to each other.

(1) e_{1} = e_{0} − ℓ_{A}(A/I) + 1.

(2) 𝔪S = (0) and rank_{B} S = 1.

(3) S ≅ (X_{1}, X_{2}, ⋯, X_{c})B as graded T -modules for some 0 < c ≤ d, where {X_{i}}_{1≤i≤c} are linearly independent linear forms of the polynomial ring B.

When this is the case, c = ℓ_{A}(I^{2}/QI) and I^{3} = QI^{2}, and the following assertions hold.

(a) depth G ≥ d − c and depth_{T} S = d − c + 1.

(b) depth G = d − c, if c ≥ 2.

(c) Suppose c < d. Then

ℓ_{A}(A/I^{n+1}) = e_{0}\binom{n + d}{d} − e_{1}\binom{n + d − 1}{d − 1} + \binom{n + d − (c + 1)}{d − (c + 1)}

for all n ≥ 0. Hence

e_{i} = \{ \begin{matrix}0 & if i ≠ c + 1, \\ (−1)^{c+1} & if i = c + 1 \end{matrix}

for 2 ≤ i ≤ d.

(d) Suppose c = d. Then

ℓ_{A}(A/I^{n+1}) = e_{0}\binom{n + d}{d} − e_{1}\binom{n + d − 1}{d − 1}

for all n ≥ 1. Hence e_{i} = 0 for 2 ≤ i ≤ d.

References

[1] Shiro Goto, Koji Nishida, and Kazuho Ozeki, The structure of Sally modules of rank one, Math. Res. Lett, 15 (2008), no 5, 881–892.

Department of Mathematics

School of Science and Technology

Meiji University

1-1-1 Higashimita, Tama-ku, Kawasaki 214-8571, Japan Email: goto@math.meiji.ac.jp

Department of Mathematics and Informatics

Graduate School of Science and Technology

Chiba University

1-33 Yatoi-cho, Inage-ku, Chiba-shi, 263 Japan Email: nishida@math.s.chiba-u.ac.jp

Meiji Institute for Advanced Study of Mathematical Sciences Meiji University 1-1-1 Higashi-mita, Tama-ku, Kawasaki 214-8571, Japan

Email: kozeki@math.meiji.ac.jp The Buchsbaum-Rim function of a parameter module

Futoshi Hayasaka

Let (A, 𝔪) be a commutative Noetherian local ring with the maximal ideal 𝔪 and d = dim A > 0 the Krull dimension of A. Let F = A^{r} be a free module of rank r > 0, and let S = S_{A}(F) be the symmetric algebra of F. For a submodule M of F, let R = R(M) be the image of the natural homomorphism S_{A}(M) → S_{A}(F). Let S_{ν} (resp. M^{ν}) be a homogeneous component of degree ν of S (resp. R). Assume that the quotient F/M has finite length and M ⊆ 𝔪F.

In 1964, in order to generalize the multiplicity to the situation of finitely generated modules F/M rather than just cyclic modules A/I, Buchsbaum and Rim introduced and studied the length function λ(ν) = ℓ_{A}(S_{ν}/M^{ν}) , and proved in [?] that the function λ is eventually a polynomial of degree d+r−1. The polynomial P corresponding to λ can then be written in the form P (ν) = ∑_{i=0}^{d+r−1}(−1)^{i}e_{i}\binom{ν+d+r−2−i}{d+r−1−i} with integer coefficients e_{i}. The Buchsbaum-Rim multiplicity of F/M, denoted by e(F/M), is now defined to be the leading coefficient e_{0}. The other coefficient e_{i} is called the Buchsbaum-Rim coefficient of F/M, and is denoted by e_{i}(F/M) for i = 1, 2, …, d + r − 1.

Buchsbaum and Rim also introduced in [?] the notion of a parameter module (matrix), which generalizes the notion of a parameter ideal (system of parameters). The module N in F is said to be a parameter module in F, if the following three conditions are satisfied: (i) F/N has finite length, (ii) N ⊆ 𝔪F, and (iii) the minimal number of generators of N is just d + r − 1.

With this notation, the purpose of this talk is to prove the following:

Theorem 1. Let (A, 𝔪) be a Noetherian local ring of dimension d > 0. Then, for any rank r > 0 and any integer ν ≥ 0, the inequality

ℓ_{A}(S_{ν+1}/N^{ν+1}) ≥ e(F/N)\binom{ν + d + r − 1}{d + r − 1}

holds true for every parameter module N in F = A^{r}. Moreover, the ring A is Cohen-Macaulay, if the equality ℓ_{A}(S_{ν+1}/N^{ν+1}) = e(F/N)\binom{ν+d+r−1}{d+r−1} holds for some integer ν ≥ 0 and some parameter module N in F.

This is a generalization of the result in [?]. Moreover, it seems that this result contains some new information even in the ideal case. Indeed, as a direct consequence of the inequality in Theorem 1, we have that e_{1}(F/N) ≤ 0 hold true for any parameter module N in F. This is a generalization of the recent result of Mandal and Verma that e_{1}(A/Q) ≤ 0 for any parameter ideal Q in A. Our proof based on the inequality in Theorem 1 is completely different from theirs and is considerably more simpler.

References

[1] F. Hayasaka and E. Hyry, A note on the Buchsbaum-Rim multiplicity of a parameter module, to appear in Proc. Amer. Math. Soc.

[2] D. A. Buchsbaum and D. S. Rim, A generalized Koszul complex. II. Depth and multiplicity, Trans. Amer. Math. Soc. 111 (1964), 197–224.

Department of Mathematics

School of Science and Technology

Meiji University

1-1-1 Higashimita, Tama-ku, Kawasaki 214–8571, JAPAN Email: hayasaka@isc.meiji.ac.jp Extension of the Matlis duality to a filtered Noetherian ring

Noritsugu Kameyama

We will give a duality over a certain filtered ring which includes, for example, Iwasawa algebras. We define a filtration as follows [?]. A family F = {F_{p} : p ∈ ℕ} of additive subgroups of a ring Λ is called a filtration, if (i) 1 ∈ F_{0}Λ, (ii) F_{p}Λ ⊂ F_{p+1}Λ, (iii) (F_{p}Λ)(F_{q}Λ) ⊂ F_{p+q}Λ, (iv) Λ = ⋃_{p∈ℕ} F_{p}Λ. Let M be a (left) Λ-module. A family F = {F_{p}M : p ∈ ℤ} of additive subgroups of M is called a filtration of M, if (i) F_{p}M ⊂ F_{p+1}M, (ii) F_{−p}M = 0 for p >> 0, (iii) (F_{p}Λ)(F_{q}M) ⊂ F_{p+q}M, (iv) M = ⋃_{p∈ℤ} F_{p}M.

Our setting is as follows. Let Λ be a left and right Noetherian filtered ring with a Zariskian filtration F Λ = {F_{i}Λ}_{i∈ℤ} such that

(a1) H_{i} = F_{i}Λ is an ideal of Λ for every i ∈ ℤ,

(a2) Λ is complete with respect to F Λ,

(a3) Λ/H_{i} is of finite length as a right and left Λ-module for every i ∈ ℤ.

Let (R, 𝔪, k) be a commutative local Noetherian ring and Λ an R-algebra which is finitely generated as an R-algebra. We consider that R is a subring of Λ via a structure map R → Λ. Put I_{i} := R ∩ H_{i}(i ∈ ℤ) and F R = {I_{i}}(i ∈ ℤ). Then F R is a filtration of R. We assume further that

(b1) R is complete with respect to F R,

(b2) R/I_{i} is a finite length R-module for every i ∈ ℤ,

(b3) 𝔪^{n} is open for all n > 0, i.e., 𝔪^{n} ⊃ I_{i} for some i ∈ ℤ,

(b4) Λ/H_{i} is a module-finite R/I_{i}-algebra for every i ∈ ℤ, i.e., Λ/H_{i} is a finitely

generated R/I_{i}-module. Let E := E_{R}(k) be an injective hull of k as an R-module. We put (−)^{∨} :=

HOM_{R}(−, E), (−)′ := Hom_{R}(−, E). (−)′ induces usual Matlis Duality. Let M be a filtered Λ-module with a filtration F M = {F_{i}M}_{i∈ℤ}. We call M pseudocompact, if M ≅ \underleftarrow{lim} M/F_{i}M, that is, M is complete and H_{i}M ⊂ F_{i}M for every i ∈ ℤ. Dually, a filtered Λ-module N with a filtration F N = {F_{i}N}_{i∈ℤ} is called copseudocompact, if N ≅ \underrightarrow{lim} F_{i}N and H_{−i}F_{i}N = 0 for every i ∈ ℤ.

Let F_{Λ} be a category such that:

Objects: all filtered Λ-modules,

Morphisms: HOM_{Λ}(M, N) for M, N ∈ F_{Λ}. Here, we put F_{p}HOM_{Λ}(M, N) = {f ∈ Hom_{R}(M, N)} | f(F_{i}M) ⊂ F_{i+p}N for all

i ∈ ℤ} and HOM_{Λ}(M, N) = ⋃_{p∈ℤ} F_{p}HOM_{Λ}(M, N). Let C be a full subcategory of F_{Λ} consisting of all finitely generated pseudocompact Λ-modules, and D a full subcategory of F_{Λ} consisting of all finitely cogenerated copseudocompact Λ-modules. The main result is the following:

Theorem The functor (−)^{∨} gives a duality between the categories C and D.

References

[1] L.Huishi and F.van Oystaeyen, Zariskian Filtrations, K-Monograph in Mathematics, 2, 1996.

Department of Mathmatics

Shinshu University

Matsumoto, Nagano 390-8621 JAPAN Email: s08a102@shinshu-u.ac.jp Dualizing complex of the Stanley ring associated with a simplicial poset

Kohji Yanagawa

A poset (partially ordered set) P is called simplicial, if it has the smallest element ˆ0, and the interval [ˆ0, x] is isomorphic to a boolean algebra (i.e., the power set of a finite set with order given by inclusion) for all x ∈ P. The face poset of a finite simplicial complex is clearly a simplicial poset. Similarly, any simplicial poset P is given by a regular cell complex Γ(P). For example, if two d-simplices are glued along their boundaries, then it is not a simplicial complex, but gives a simplicial poset.

As is well-known, the Stanley-Reisner ring of a finite simplicial complex is a powerful tool for combinatorics. Generalizing this idea, Stanley ([?]) constructed a graded commutative ring A_{P} from a simplicial poset P.

M. Masuda and his coworkers studied A_{P} with a view from toric topology, since the equivariant cohomology ring of a torus manifold is of the form A_{P} (cf. [?]). In this talk, we give the following result by another approach.

(1) We describe a dualizing complex of A_{P} in a concise way.

(2) The theory of squarefree modules (see, for example, [?]) can be developed over A_{P}. To a squarefree module M over A_{P}, we can assign the constructible sheaf M^{+} on Γ(P). The dualizing complex of A_{P} is essentially a complex of squarefree modules, and its sheafification is Verdier’s dualizing complex of (the underlying space of) Γ(P), which give a Poincaré-Verdier duality.

(3) The Cohen-Macaulay (resp. Gorenstein*, Buchsbaum) property of A_{P} is a topological property of the underlying space of Γ(P).

Some part of (3) is known result. But our argument is more systematic and direct (I believe).

References

[1] M. Masuda, h-vectors of Gorenstein* simplicial posets, Adv. Math. 194 (2005), 332–344.

[2] R. Stanley, f-vectors and h-vectors of simplicial posets, J. Pure Appl. Algebra 71 (1991), 319–331.

[3] K. Yanagawa, Stanley-Reisner rings, sheaves, and Poincaré-Verdier duality, Math. Res. Lett. 10 (2003), 635–650.

Department of Mathematics, Kansai University Suita 564-8680, Japan

Email: yanagawa@ipcku.kansai-u.ac.jp THICK SUBCATEGORIES OF THE STABLE CATEGORY OF

COHEN-MACAULAY MODULES

Ryo Takahashi

Various classification theorems of thick subcategories (i.e., full triangulated subcategories closed under direct summands) of a triangulated category have been obtained in many areas of mathematics. In this talk, as a higher dimensional version of the classification theorem of thick subcategories of the stable category of finitely generated representations of a finite p-group due to Benson, Carlson and Rickard [?], we consider classifying thick subcategories of the stable category of Cohen-Macaulay modules over a (commutative) Gorenstein local ring.

Let R be a Gorenstein local ring. We denote by mod R the category of finitely generated R-modules, by CM(R) the full subcategory of mod R consisting of all Cohen-Macaulay R-modules, and by \underline{CM}(R) the stable category of CM(R). Then it is well-known that \underline{CM}(R) is a triangulated category. Let Spec R denote the prime ideal spectrum of R, that is, the set of prime ideals of R. Let Sing R denote the singular locus of R, that is, the set of prime ideals 𝔭 of R such that the local ring R_{𝔭} is not regular. A subset Φ of Spec R is called specialization-closed if 𝔭 ∈ Φ and 𝔭 ⊆ 𝔮 ∈ Spec R imply 𝔮 ∈ Φ. A resolving subcategory is by definition a full subcategory closed under direct summands, extensions and syzygies.

Recall that a (local) hypersurface is defined to be a ring isomorphic to S/fS for a regular local ring S and an element f of S. A hypersurface is always a Gorenstein local ring. The main result of this talk is the following theorem.

Theorem 1. Let R be a hypersurface. Then there are one-to-one correspondences among the following three sets:

• the set of nonempty thick subcategories of \underline{CM}(R),

• the set of specialization-closed subsets of Spec R contained in Sing R,

• the set of resolving subcategories of mod R contained in CM(R).

The bijective maps among the three sets are explicitly given in this talk.

References

[1] D. J. Benson; J. F. Carlson; J. Rickard, Thick subcategories of the stable module category, Fund. Math. 153 (1997), no. 1, 59–80.

[2] E. S. Devinatz; M. J. Hopkins; J. H. Smith, Nilpotence and stable homotopy theory, I, Ann. of Math. (2) 128 (1988), no. 2, 207–241.

[3] E. M. Friedlander; J. Pevtsova, Π-supports for modules for finite group schemes, Duke Math. J. 139 (2007), no. 2, 317–368.

[4] M. J. Hopkins, Global methods in homotopy theory, Homotopy theory (Durham, 1985), 73– 96, London Math. Soc. Lecture Note Ser., 117, Cambridge Univ. Press, Cambridge, 1987.

[5] M. J. Hopkins; J. H. Smith, Nilpotence and stable homotopy theory, II, Ann. of Math. (2) 148 (1998), no. 1, 1–49.

[6] A. Neeman, The chromatic tower for D(R), With an appendix by Marcel Bökstedt, Topology 31 (1992), no. 3, 519–532.

[7] R. Takahashi, Classifying thick subcategories of the stable category of Cohen-Macaulay modules, Preprint (2009).

[8] R. W. Thomason, The classification of triangulated subcategories, Compositio Math. 105 (1997), no. 1, 1–27.

Department of Mathematical Sciences Faculty of Science Shinshu University Matsumoto, Nagano 390-8621, JAPAN

Email: takahasi@math.shinshu-u.ac.jp Ampleness of two-sided tilting complexes and Fano algebras

Hiroyuki Minamoto

From the view point of noncommutative algebraic geometry, two-sided tilting complexes are an analogue of line bundles. In algebraic geometry for line bundles, ampleness is an important notion. In this talk we introduce a notion of ampleness for two-sided tilting complexes over finite dimensional algebras of finite global dimension. A two-sided tilting complex σ is called very ample if H^{i}(σ) = 0 for i ≥ 1 and H^{i}(σ^{n}) = 0 for i ≠ 0 and n ≫ 0. We justify this definition by using the theory of noncommutative projective schemes due to Artin-Zhang [?] and Polishchuk [?]. In the theory of noncommutative projective schemes, for a graded coherent ring R over k, we attach an imaginary geometric object projR = (qcohR, \overline{R}, (1)). An abelian category qcohR is considered as the category of coherent sheaves on projR. If σ is a very ample tilting complex over A, then the tensor algebra T := T_{A}(H^{0}(σ)) of H^{0}(σ) over A is a graded connected coherent ring over A and there is a natural equivalence of triangulated categories D^{b} (mod-A) ≃ D^{b} (qcohT) .

From the view point of noncommutative algebraic geometry, Serre functors are considered as shifted canonical bundles. We define Fano algebras by the antiampleness of shifted Serre functor. Fano algebras have remarkable properties. Some classes of algebras studied before are Fano. For example a path algebra of a quiver of infinite representation type is Fano. Moreover we can characterize the representation type of a quiver from the noncommutative algebro-geometric point of view.

Theorem 1. Let Q be a finite quiver without oriented cycles. Then Q has finite representation type if and only if its path algebra kQ is fractionally Calabi-Yau. Q has infinite representation type if and only if kQ is Fano.

There is good supply of new Fano algebra. We can construct Fano algebras from AS-regular algebras (without finiteness of Gelfand-Kirillov dimension). This is a joint work with I.Mori.

Theorem 2 (M-Mori). Let A be an AS-regular algebra of gldimA = d ≥ 1 and Gorenstein parameter e. Then the finite dimensional algebra

F = (\begin{matrix}A_{0} & & & A_{1} & & & ⋯ & & & A_{e−1} \\ 0 & & & A_{0} & & & ⋯ & & & A_{e−2} \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 0 & & & 0 & & & ⋯ & & & A_{0} \end{matrix})

is an extremely Fano algebra of dimension d − 1.

As a byproduct, we prove the following corollary, which was conjectured by A.Bondal.

Corollary 3. AS-regular algebras are graded coherent.

References

1. M. Artin, and J.J. Zhang, Noncommutative projective schemes, Adv. Math. 109 (1994), pp. 228-287.

2. A. Polishchuk, Nonncommutative proj and coherent algebras, Math. Res. Lett., 12 (2005),1, pp. 63-74.

Department of Mathematics Graduate School of Science Kyoto University Kyoto 606-8502, JAPAN

Email: minamoto@math.kyoto-u.ac.jp.

The Projective Scheme of the Blow-up Ring

Fred van Oystaeyen

Noncommutative Geometry has become a many-headed animal nowadays. In our version of noncommutative algebraic geometry we tried to actually keep the idea that there is a noncommutative space really present and not just virtual while one actually studies noncommutative algebras as the assumed ring of functions on the virtual space. This leads to noncommutative spaces (in fact endowed with noncommutative topologies) allowing a noncommutative version of the Serre global section theorem in scheme theory. The topological space is replaced by the set (lattice) of localizations on some nice Grothendieck category like modules or graded modules in the projective case, where intersection of opens corresponds to compositions of localization functors. The latter are exact pretorsion radicals but not localizations corresponding to torsion theories or Serre quotient categories. Viewing algebras as given by generators and relations they appear as quotients of the free algebra hence equipped with a standard filtration. The Rees ring of the filtration corresponds to a projective noncommutative scheme that acts as the projectivization of the affine noncommutative variety associated to the algebra. The condition for some noncommutative geometry to work for some noncommutative algebra is the so called ”schematic” condition. We study the transfer of this condition from the associated graded algebra of nice (Zariskian) filtered ring to the blow-up or Rees ring of it. We establish the lifting of the projective schematic condition and obtain a statement about noncommutative geometries expressing how Proj of the Rees ring contains Spec of the filtered ring as a closed substructure, the latter being the part at ”infinity” for the affine noncommutative variety

Then we also establish a noncommutative version of Serre’s global section theorem relating noncommutative geometry to the quotient category of finitely generated graded modules modulo those of finite length. All kinds of dimensions and regularity conditions can then be studied algebraically and interpreted geometrically.

A list of relatively recent examples is provided.

Derived equivalences for endomorphism rings

Hiroki Abe and Mitsuo Hoshino

In [?], we have shown the following. Let 0 → Y →^{μ} E →^{ε} X → 0 be an exact sequence in an abelian category 𝒜 and P an object of 𝒜. Assume that E ∈ add(P) and that both Hom_{𝒜} (P, ε) and Hom_{𝒜} (μ, P) are epic. Then End_{𝒜} (X ⊕ P) and End_{𝒜} (Y ⊕ P) are derived equivalent to each other. In this talk, we will provide several applications of this fact.

Let A be a representation-finite artin algerba with M_{1}, ⋯, M_{m} ∈ mod-A a complete set of nonisomorphic indecomposable modules and I = {1, ⋯, m}. We assume that m ≥ 2, i.e., A is not simple. Set

M = ⨁_{i∈I} M_{i}, Λ = End_{A}(M).

For each indecomposable module X ∈ mod-A, since there exists a unique i_{X} ∈ I such that X ≅ M_{i_{X}}, we set I(X) = I \ {i_{X}} and set

M_{X} = ⨁_{i∈I(X)} M_{i}, Λ_{X} = End_{A}(M_{X}).

We will show that if X is projective and injective then Λ_{X} is also an Auslander algebra, that if X is nonprojective then Λ_{X} and Λ_{τX} are derived equivalent to each other, where τ = DTr, and that if X is noninjective then Λ_{X} and Λ_{τ^{−1}X} are derived equivalent to each other, where τ^{−1} = TrD.

Let A be a ring, P ∈ Mod-A and 0 → Y →^{μ} E →^{ε} X → 0 an exact sequence in Mod-A. Assume that E ∈ add(P) and that both Hom_{A}(P, ε) and Hom_{A}(μ, P) are epic. We will show that X ⊕ P is a tilting module if and only if so is Y ⊕ P, and that if X ⊕ P is a classical tilting module, then so is Y ⊕ P.

Let A be a Noether algebra and X ∈ mod-A. Assume that there exists T ∈ mod-A such that X ⊕ T ∈ mod-A a tilting module. We will show the following.

(1) If there exists an epimorphism of the form f : T^{(l)} → X, then there exists an epimorphism ε : T^{(r)} → X such that Ker ε ⊕ T is a tilting module. In particular, if X ⊕ T is a classical tilting module, then so is Ker ε ⊕ T.

(2) If there exists a monomorphism of the form g : X → T^{(l)}, then there exists a monomorphism μ : X → T^{(r)} such that Cok μ ⊕ T is a tilting module.

References

1. H. Abe and M. Hoshino, Gorenstein orders associated with modules, Comm. Algebra, to appear.

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

Email: abeh@math.tsukuba.ac.jp hoshino@math.tsukuba.ac.jp The classification of tilting modules over Harada algebras

Kota Yamaura

In the 1980s, Harada [?] introduced a new class of algebras now called Harada algebras. We recall left Harada algebras from a structural point of view as follows.

Definition. Let R be a basic finite dimensional algebra over a field K and Pi(R) be a complete set of orthogonal primitive idempotents of R. We call R a left Harada algebra if Pi(R) can be arranged such that Pi(R) = {e_{ij}}_{i=1}^{m},_{j=1}^{n_{i}} where

(1) e_{i1}R is an injective R-module for any i = 1, ⋯, m,

(2) e_{ij}R ≃ e_{i,j−1}J(R) for any i = 1, ⋯, m, j = 2, ⋯, n_{i}.

Here J(R) is the Jacobson radical of R.

In my talk, we give the classification of tilting modules over left Harada algebras. Tilting modules provide us a powerful tool in the representation theory of algebras (we refer to [?, ChapterVI]). We denote by modR the category of finitely generated module over an algebra R.

Definition. Let R be a finite dimensional algebra over a field K. We call T ∈ modR a tilting module if T satisfies the following conditions.

(1) proj.dimT ≤ 1.

(2) Ext_{R}^{1}(T, T) = 0.

(3) There exists an exact sequence 0 ⟶ R_{R} ⟶ T_{0} ⟶ T_{1} ⟶ 0 where T_{0} and T_{1} are direct summands of a direct sum of some copies of T.

A tilting module T over an algebra R induces two category equivalences between certain full subcategories of modR and of mod(End_{R}(T)). As a consequence of these category equivalences, some problems about R can be shifted to those of End_{R}(T) (e.g. finiteness of global dimension). By this reason, finding a classification of tilting modules over a given algebra is an important problem in representation theory.

Now we present the main theorem of my talk. Let R be a left Harada algebra over an algebraically closed field K as in the above definition. We denote by tilt(R) the set of isomorphism classes of basic tilting R-modules and by T_{n}(K) the n × n upper triangular matrix algebra over K.

The following is our main theorem which asserts that tilting R-modules are described by tilting modules over a direct product of algebras of the form T_{n}(K).

Theorem [?]. There exists a bijection

tilt(R) ⟶ tilt(T_{n_{1}}(K)) × tilt(T_{n_{2}}(K)) × ⋯ × tilt(T_{n_{m}}(K)).

It is known that tilting T_{n}(K)-modules are described combinatorially by using non-crossing partitions of a regular (n + 2)-polygon. This fact and our theorem allow us to classify tilting modules over a given Harada algebra.

References

1. I. Assem, D. Simson, A. Skowronski, Elements of the Representation Theory of Associative Algebras, London Mathematical Society Student Texts 65, Cambridge university press (2006)

2. Y. Baba and K. Oshiro, Classical Artinian Rings and Related Topics, preprint

3. M. Harada, Non-small modules and non-cosmall modules, Ring Theory. Proceedings of 1978 Antwerp Conference, New York (1979), 669–690

4. K. Yamaura, The classification of tilting modules over Harada algebras, arXiv:0905.2245

Graduate School of Mathematics

Nagoya University

Furocho, Chikusaku, Nagoya 464-8602 JAPAN Email: m07052d@math.nagoya-u.ac.jp The Stickelberger relation and Loewy series of group algebras Map(𝔽_{q}, 𝔽_{q})

Kaoru Motose

In this talk, I present a proof of the Stickelberger relation (see [1]) using Loewy series of a group algebra Map(𝔽_{q}, 𝔽_{q}) of the additive group of a finite field 𝔽_{q} (see [2], [3] and [4]). This relation is essential in a proof of the Eisenstein reciprocity law (see [1]).

I also present the next result by a special case in this law, namely, the law of cubic reciprocity.

If q^{2} + q + 1 divides 3^{q} − 1 for a prime q > 3, then 9 divides q + 1. This is a slight contribution to the Feit-Thompson conjecture for a prime 3

(see [5]).

References

1. K. Ireland and M. Rosen, A classical introduction to modern number theory, 2nd ed., GTM 84, 1990, Springer.

2. K. Motose, On Loewy series of group algebras of some solvable groups, J. Algebra, 130 (1990) 261-272

3. K. Motose, On commutative group algebras, Sci. Rep. Hirosaki Univ., 40 (1993), 127-131.

4. K. Motose, On commutative group algebras. II, Math. J. Okayama Univ., 36 (1994), 23-27.

5. K. Motose, Notes to the Feit-Thompsons conjecture , Proc. Japan Acad., 85 (2009), 16-17.

5-13-5 Toriage, Hirosaki, Aomori pref., 036-8171, JAPAN Email: moka.mocha no kaori@snow.ocn.ne.jp Homological conjectures and radical-full extensions

Changchang Xi

This is a series of two lectures. In these lectures, we shall consider the finitistic dimension conjecture and the strong no loop conjecture (and related other homological conjectures). We approach these conjectures by the so-called radical-full extensions, and reduce the verification of the conjectures to the following question: Suppose that B ⊆ A is a radical-full extension such that the radical of B is a left ideal in A, and that these conjectures are true for A, is it possible to prove the conjectures for B ? We shall provide basic definitions and examples, and report results in this direction. Among them are the following two theorems.

Definition 1. An extension of two Artin algebras is a pair of Artin algebras B and A such that B is a subalgebra of A and that A and B have the same identity. In this case, the extension is denoted by B ⊆ A. If, in addition, the Jacobson radical of A is generated as a right ideal by the Jacobson radical of B, we say that the extension B ⊆ A is radical-full.

Theorem 2. Let B ⊆ A be a radical-full extension of Artin algebras. Suppose that the radical of B is a left ideal in A. If the global dimension of A is at most 4, then the finitistic dimension conjecture is true for B.

Theorem 3. Let B ⊆ A be a radical-full extension of Artin algebras. Suppose that the radical of B is a left ideal in A. If the global dimension of A is at most 2, then the strong no loop conjecture is true for B.

References

1. C.C. Xi, On the finitistic dimension conjecture, I. Related to representation-finite algebras, J. Pure Appl. Algbera 193 (2004), 1287–305. Erratum to ”On the finitistic dimension conjecture, I. ”, J. Pure Appl. Algbera 202(1-3) (2005), 325–328.

2. , On the finitistic dimension conjecture, II. Related to finie global dimension, Adv. Math. 201 (2006), 116–142.

3. , On the finitistic dimension conjecture, III. Related to the pair eAe ⊆ A, J. Algebra. 319 (2008), 3666–3688.

4. and D.M. Xu, The finitistic dimension conjecture and relatively projective modules, Preprint, 2007, available at: http://math.bnu.edu.cn/^{∼}ccxi/.

School of Mathematical Sciences Beijing Normal University 100875 Beijing, CHINA

Email: xicc@bnu.edu.cn Almost comparability and related comparabilities in von Neumann regular rings

Mamoru Kutami

Abstract. There are many comparabilities in (von Neumann) regular rings: general comparability, the comparability axiom, s-comparability, weak comparability, almost comparability etc.. We mainly treat almost comparability in the talk. Here, we recall the definition of almost comparability for regular rings. A regular ring R satisfies almost comparability if for each x, y ∈ R either xR ≲_{a} yR or yR ≲_{a} xR, where for any principal right ideals A and B of R, A ≲_{a} B means that A is subisomorphic to B ⊕C for all nonzero principal right ideals C of R. The notion of almost comparability for regular rings was first introduced by Ara and Goodearl [1], for giving an alternative proof of the epoch-making O’Meara’s Theorem [4] that “directly finite simple regular rings with weak comparability are unit-regular”. After that the study of almost comparability for simple regular rings was continued by Ara et al. [2], and moreover Ara et al. [3] touched the relation between s-comparability and almost comparability.

In the talk, we investigate the cancellation property and the unperforation property for regular rings with almost comparability or related comparabilities.

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, K.C. O’Meara and D.V. Tyukavkin, Cancellation of projective modules over regular rings with comparability, J. Pure Appl. Algebra 107 (1996), 19–38.

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

5. M. Kutami, Von Neumann regular rings satisfying weak comparability, Applied Categorical Structures 16 (2008), 183-194.

Acknowledgment. This work was supported by JSPS KAKENHI (21540041).

Department of Mathematics Faculty of Science Yamaguchi University Yamaguchi 753-8512, JAPAN

Email: kutami@yamaguchi-u.ac.jp The Clebsch-Gordan problem for quiver representations

Martin Herschend

Given a quiver Q (i.e. an oriented graph) one can consider its path algebra A. The finite dimensional modules over A are identified with representations of Q. Such a representation consist of vector spaces, one for each vertex in Q and linear maps corresponding to the arrows of Q.

Given two representations of Q one can construct a new representation by, at each vertex taking the tensor product of the corresponding vector spaces and similarly taking the tensor product of the linear maps corresponding to the arrows.

This tensor product of quiver representations gives rise to the following Clebsch-Gordan problem: For each two indecomposable representations of Q find the decomposition of their tensor product into indecomposables.

In my talk I will survey the known solutions to this problem with focus on string algebras, which are a fairly large class of algebras given by quivers with relations.