Zentralblatt MATH 1931 – 2007

◯^{c} European Mathematical Society, FIZ Karlsruhe & Springer-Verlag Berlin-Heidelberg

Zbl. Math. Zbl 1087.20010

Marcus, Andrei Tilting complexes for group graded algebras. II. (English) Osaka J. Math. 42, No. 2, 453-462 (2005). http://projecteuclid.org/Dienst/UI/1.0/Journal?authority=euclid.ojm

[For part I cf. J. Group Theory 6, No. 2, 175-193 (2003; Zbl 1037.20001).] The paper under review is concerned with equivalences between symmetric algebras (over an algebraically closed field k) graded by a finite group G. The equivalences are between stable, homotopy and derived module categories. Several results are extended from the ungraded to the graded setting. We mention one of these: Let R = ⨁_{g∈G} R_{g} be a symmetric crossed product, let I be a finite G-set, and let X_{i} ∈ D^{b}(R_{1}-mod), i ∈ I, be objects satisfying (a) Hom(X_{i}, X_{i}[m]) = 0 for m < 0, (b) Hom(X_{i}, X_{i}) = k and Hom(X_{i}, X_{j}) = 0 for i ≠ j, (c) X_{i}, i ∈ I, generate D^{b}(R_{1}-mod) as a triangulated category, (d) R_{g} ⊗_{R_{1}} X_{i} ≅ X_{gi} in D^{b}(R_{1}-mod) for g ∈ G. Then there are a symmetric crossed product R′ = ⨁_{g∈G} R′_{g} and a G-graded derived equivalence between R and R′ whose restriction to R_{1} sends X_{i}, i ∈ I, to the simple R′_{1}-modules.

Burkhard Külshammer (Jena) Keywords:

Broué conjecture; stable equivalences; derived equivalences; splendid equivalences; tilting modules; graded algebras; crossed products; symmetric algebras; derived module categories Classification :

∗*20C20 Modular representations and characters of groups 20C05 Group rings of finite groups and their modules (group theory) 16W50 Graded associative rings and modules 18E30 Derived categories, etc. 16S35 Twisted and skew group rings, etc. 16E30 Homological functors on associative modules

Zbl. Math. Zbl 1072.20003

Marcus, Andrei Blocks with cyclic defect groups and Clifford extensions. (English) J. Algebra 287, No. 1, 1-14 (2005). doi:10.1016/j.jalgebra.2004.11.022

The author shows that Dade’s Inductive Conjecture holds for a block b with cyclic defect groups. One of the main tools is a theorem demonstrating that the residual Clifford extension of any non-exceptional character in b coincides with the Clifford extension of any simple module in b. It is also proved that a structure theorem by the reviewer on blocks with normal defect groups extends to twisted group algebras.

Burkhard Külshammer (Jena) Keywords:

blocks; defect groups; stable equivalences; group-graded algebras; Clifford extensions; exceptional characters; Dade conjecture; twisted group algebras Classification :

∗*20C05 Group rings of finite groups and their modules (group theory)

20C20 Modular representations and characters of groups

Zbl. Math. Zbl 1046.20009

Marcus, Andrei Broué’s Abelian defect group conjecture for alternating groups. (English) Proc. Am. Math. Soc. 132, No. 1, 7-14 (2004). http://www.ams.org/journal-getitem?pii=S0002-9939-03-07214-9

Let n be a positive integer and let G = S_{n}, G^{+} = A_{n} and G̃ = Aut(G^{+}). Let O be a complete discrete valuation ring with an algebraically closed residue field k = O/J(O) of prime characteristic p. The main result of this paper is Theorem 3.1. Assume that p > 2 and that b^{+} is a block of OG^{+} with a nontrivial Abelian defect group D. Set H^{+} = N_{G^{+}}(D) and let c^{+} denote the Brauer correspondent block of b^{+}. Then there exists a splendid tilting complex of G̃/G^{+}-graded (b^{+}OG^{+}, c^{+}OH^{+})-bimodules. The result is proved by constructing a bounded complex X of (b^{+}OG^{+}, c^{+}OH^{+})- bimodules such that the components of X are p-permutation modules whose indecomposable summands have vertices contained in δ(D) = {(u, u) | u ∈ D} and such that X ⊗_{c^{+}OH^{+}} X^{∨} ≅ b^{+}OG^{+} in the homotopy category of complexes of (b^{+}OG^{+}, b^{+}OG^{+})- bimodules and X^{∨} ⊗_{b^{+}OG^{+}} X ≅ c^{+}OH^{+} in the homotopy category of complexes of (c^{+}OH^{+}, c^{+}OH^{+})-bimodules where X^{∨} denotes the O-dual of X. Thus the Broué-Rickard conjecture is verified for A_{n} (with the additional G̃/G^{+}-graded condition that is used in the principal block case to reduce this conjecture for all groups to the case of simple groups). The proof of Theorem 3.1 consists in showing that the splendid tilting complexes constructed by J. Chuang, R. Kessar and R. Rouquier in the S_{n} case are complexes of C_{2}-graded bimodules. Then the desired conclusions are consequences of a general result (Theorem 2.6) on complexes in the Clifford theory context in which G/N ≅ C_{n} with (n, p) = 1. Reviewer’s remark: Page 9, line 5 should read: m_{σ_{j}} = n^{−1}(∑_{i=0}^{n−1}(ϵ^{−j}σ̂)^{i}m).

Morten E. Harris (Chicago) Keywords:

blocks; Abelian defect groups; Rickard equivalences; splendid tilting complexes; group graded algebras Classification :

∗*20C20 Modular representations and characters of groups

20C30 Representations of finite symmetric groups 16W50 Graded associative rings and modules 20C05 Group rings of finite groups and their modules (group theory)

Zbl. Math. Zbl 1037.20001

Marcus, Andrei Tilting complexes for group graded algebras. (English) J. Group Theory 6, No. 2, 175-193 (2003). http://dx.doi.org/10.1515/jgth

Motivated by Broué’s Abelian defect group conjecture, the author investigates equivalences between derived module categories, in the context of algebras graded by a finite group. One of the main results in the paper under review is a graded version of Rickard’s theorem characterizing such equivalences. Moreover, methods by Okuyama for the construction of derived equivalences are adapted to a graded setting. As an application, it is shown that Broué’s conjecture holds for the principal p-block of GL(2, p^{n}), for a prime p and a positive integer n.

Burkhard Külshammer (Jena) Keywords:

derived categories; tilting complexes; stable equivalences; Broué’s conjecture; group graded algebras; triangulated categories; derived equivalences; principal blocks; defect groups Classification :

∗*20C05 Group rings of finite groups and their modules (group theory)

20C20 Modular representations and characters of groups

16W50 Graded associative rings and modules

16E30 Homological functors on associative modules

18E30 Derived categories, etc.

Zbl. Math. Zbl 1035.16037

Marcus, Andrei; Modoi, Ciprian Graded endomorphism rings and equivalences. (English) Commun. Algebra 31, No. 7, 3219-3249 (2003). doi:10.1081/AGB-120022221

Let G be a group, M a G-graded Σ-projective R-module, and E = END_{R}(M) its graded ring of endomorphisms. The aim of this paper is to establish equivalences between categories of modules graded by G-sets, which cover, extend, or unify results of A. Marcus [Mathematica 37, No. 1-2, 131-154 (1995; Zbl 0886.16029)], E. C. Dade [J. Reine Angew. Math. 369, 40-86 (1986; Zbl 0583.16001)], T. Albu and C. Năstăsescu [J. Pure Appl. Algebra 59, No. 2, 125-150 (1989; Zbl 0676.16001)], T. Albu [Commun. Algebra 18, No. 3, 839-862 (1990; Zbl 0709.16021)], etc.

The main result of the paper establishes that for any subgroup H of G certain full subcategories of G/H-graded R-modules are equivalent to a certain category (G/H, E, G^{G/H})- Gr. The paper is very technical and uses quite many notations. Though the authors explain in full detail even some standard notations, the basic notation (G/H, E, G^{G/H})- Gr, which appears in the statement of the main results of the paper (Theorem 3.9 and Corollary 3.10) could not be located by the reviewer in the whole paper.

Toma Albu (Istanbul) Keywords:

modules graded by G-sets; rigid localizing subcategories; graded Gabriel topologies; graded endomorphism rings; equivalences; quasi-projective modules; Grothendieck categories Classification :

∗*16W50 Graded associative rings and modules 16S50 Endomorphism rings: matrix rings 16D90 Module categories (assoc. rings and algebras) 18E15 Grothendieck categories 18E40 Torsion theories 16S90 Maximal ring of quotients, torsion theories

Zbl. Math. Zbl 0999.18004

Madar, Adrian; Marcus, Andrei Frobenius functors and transfer. (English) Publ. Math. 59, No.3-4, 407-427 (2001). http://www.math.klte.hu/publi/contents.php?szam=40

Summary: If the functor F is both a left and a right adjoint of G, then one may define transfer maps associated to F and G. We use these natural transformations to generalize Higman’s criterion for relative projectivity. We also give several applications to situations involving ring extensions and Hopf algebra actions.

Keywords:

adjoint functors; Frobenius functors; relative projectivity; transfer; Frobenius extensions; group graded algebras; Hopf algebra actions Classification :

∗*18A40 Adjoint functors

16W30 Hopf algebras (assoc. rings and algebras) 16W50 Graded associative rings and modules

Zbl. Math. Zbl 0985.20002

Marcus, Andrei Twisted group algebras, normal subgroups and derived equivalences. (English) Algebr. Represent. Theory 4, No.1, 25-54 (2001). http://springerlink.metapress.com/openurl.asp?genre=journalissn=1386-923X

The paper under review deals with graded derived equivalences between algebras graded by a finite group G. Its main results are generalizations and refinements of corresponding results on algebras without gradings. The author shows that graded derived equivalences preserve Clifford extensions of blocks and of simple modules. Then he defines splendid graded derived equivalences and investigates their behaviour with respect to local subgroups. He presents a graded version of Broué’s Abelian defect group conjecture and its consequences on Dade’s inductive conjecture. Also, the author verifies Dade’s inductive conjecture for p′-extensions of p-blocks with cyclic defect groups.

Burkhard Külshammer (Jena) Keywords:

twisted group algebras; crossed products; Rickard equivalences; Clifford extensions; linear source modules; blocks; cyclic defect groups; graded derived equivalences; group graded algebras; simple modules Classification :

∗*20C05 Group rings of finite groups and their modules (group theory)

20C20 Modular representations and characters of groups

16W50 Graded associative rings and modules 16S35 Twisted and skew group rings, etc. 20C25 Projective representations and multipliers of groups 16D90 Module categories (assoc. rings and algebras)

Zbl. Math. Zbl 1027.18001

Madar, Adrian; Marcus, Andrei Frobenius functors and functor categories. (English) Stud. Univ. Babeş-Bolyai, Math. 45, No.4, 75-85 (2000).

Summary: If F is a functor having the functor G both as a left and right adjoint, then one can associate to F a transfer map. The pair (F, G) is called a Frobenius pair. In this paper we discuss Frobenius functors between functor categories and their transfer maps.

Keywords:

Frobenius pair; transfer maps Classification :

∗*18A25 Comma categories 18A40 Adjoint functors

Zbl. Math. Zbl 1013.16033

Marcus, Andrei; Modoi, Ciprian Groups of homomorphisms graded by G-sets. (English) Ital. J. Pure Appl. Math. 2000, No.8, 115-126 (2000).

Let G be a group and R = ⨁_{g∈G} R_{g} a G-graded ring. If M = ⨁_{g∈G} M_{g} and N = ⨁_{g∈G} N_{g} are G-graded R-modules then the group Hom_{R}(M, N) has a subgroup denoted HOM_{R}(M, N) which can be endowed with a natural G-grading:

HOM_{R}(M, N) = {f ∈ Hom_{R}(M, N) | f(M_{h}) ⊆ N_{hg}∀h ∈ G} = Hom_{R-gr}(M, N(g))

where R-gr denotes the category of G-graded R-modules and grade preserving Rmorphisms and N(g) is the G-graded module, called the g-suspension of N with N(g) = N and N(g)_{h} = N_{hg} for all h ∈ G. In this paper the authors are interested only in modules graded by transitive G-sets, so let H be a subgroup of G and denote by G/H the set {gH | g ∈ G} of left cosets of H in G. An R-module N is G/H-graded if N = ⨁_{x∈G/H} N_{x} (as additive subgroups) and R_{g}N_{x} ⊆ N_{gx} for all g ∈ G and x ∈ G/H. Denote by (G/H, R)-gr the category of G/H-graded R-modules and grade preserving R-morphisms. So, with the above notations the main result of this paper is: Let R be a G-graded ring, M a G-graded R-module, and let K ≤ H be two subgroups of G such that H/K is infinite. Then the following statements are equivalent: (i) M is small in (G/K, R)-gr; (ii) HOM_{G/H,R}(M, N) = HOM_{G/K,R}(M, N) for every G/K-graded R-module N.

Dimitru Busneag (Craiova) Keywords:

groups; graded modules; graded rings; categories; topologies; grade forgetting functors Classification :

∗*16W50 Graded associative rings and modules 16D90 Module categories (assoc. rings and algebras) 18E10 Exact categories, abelian categories 18E15 Grothendieck categories

Zbl. Math. Zbl 1010.20002

Marcus, Andrei Clifford theory of bimodules. (English) Commun. Algebra 28, No.4, 2029-2041 (2000). http://taylorandfrancis.metapress.com/openurl.asp?genre=journalissn=0092-7872

Let G be a finite group and let O be a complete local commutative ring with residue field k = O/J(O) of characteristic p > 0. Let R = ⨁_{g∈G} R_{g} be a strongly G-graded O-algebra and for X ⊆ G, set R_{X} = ⨁_{x∈X} R_{x}. Thus if H ≤ G, then R_{H} is a strongly G-graded O-subalgebra of R. A left R-module M is G/H-graded if M = ⨁_{x∈G/H} M_{x} in O-mod and R_{g}M_{x} ≤ M_{gx} for all g ∈ G and x ∈ G/H (hence R_{g}M_{x} = M_{gx} for g ∈ G and x ∈ G/H). In which case, if X ⊆ G/H set M_{X} = ⨁_{x∈X} M_{x}. Let S also be a strongly G-graded O-algebra and let M be a G-graded (R, S)-bimodule (i.e., R_{g}M_{x}S_{h} = M_{gxh} for all g, x, h ∈ G). For any (R_{1}, S_{1})-submodule N_{1} of M_{1}, set G(N_{1}) = {g ∈ G | R_{g}N_{1}S_{g^{−1}} = N_{1}}. Also the opposite algebra S^{op} will be regarded as a G-graded algebra with (S^{op})_{g} = S_{g^{−1}} for all g ∈ G and set A = R ⊗_{O} S^{op}. Then A is a strongly G × G-graded O-algebra with A(g, h) = R_{g} ⊗ (S^{op})_{h} for all (g, h) ∈ G × G. Set δ(G) = {(g, g) | g ∈ G} ≤ G × G and Δ = A_{δ(G)} = ⨁_{g∈G} R_{g} ⊗ (S^{op})_{g}, so that Δ is a strongly G-graded O-subalgebra of A with Δ_{g} = R_{g} ⊗ (S^{op})_{g^{−1}}. In particular, Δ_{1} = R_{1} ⊗ (S^{op})_{1}. Consequently, the G-graded (R, S)-bimodule M can be viewed as a left (G × G)/δ(G)-graded A-algebra by setting M_{(g,1)δ(G)} = M_{g} for all g ∈ G. If H ≤ K ≤ G × G, U is an A_{H}-module and V is an A_{K}-module, set Ind_{H}^{K}(U) = A_{K} ⊗_{A_{H}} U and Res_{H}^{K}(V) = V regarded as an A_{H}-module. Clearly G acts on the Δ_{1} = R_{1}⊗(S^{op})_{1}-submodules U of M_{1} by (g, U) ↦ ^{g}U = R_{g}US_{g^{−1}} for any g ∈ G. Set G(U) = {g ∈ G | R_{g}US_{g^{−1}} = U}, so that U is a Δ_{G(U)}-submodule of M_{1}. Let N_{1} be an indecomposable (R_{1}, S_{1})-summand of M_{1} and let e ∈ End_{Δ_{1}}(M_{1}) be the primitive idempotent corresponding to N_{1}. Also set N = R_{G(M_{1})}N_{1}S_{G(N_{1})}. By definition, an indecomposable summand Ñ of M lies over N_{1} if N_{1} is a summand of Res_{1×1}^{G×G}(Ñ) and add(M | N_{1}) is the smallest additive subcategory of A-mod that contains all indecomposable summands of M lying over N_{1}. The main result of this paper presented in Section 2 is: Theorem 2.5. Assume that e ∈ Z(End_{Δ_{1}}(M_{1})). Then: (a) add[M | N_{1}] is equivalent to the category of finitely generated projective End_{A}(Ind_{G(N_{1})×G(N_{1})}^{G×G}(N))-modules; and (b) Ind_{G(N_{1})×G(N_{1})}^{G×G}(−): add[M_{G(N_{1})} | N_{1}] → add[M | N_{1}] is an equivalence of categories. Let ′A (A′ resp.) denote the fully G-graded algebra with ′A_{g} = ⨁_{g∈G}(R ⊗_{O} (S^{op})_{g}) (A′_{g} = R_{g} ⊗ S^{op} resp.) for all g ∈ G. Set ′E = End_{′A_{1}}(M)^{op} (E′ = End_{A′_{1}}(M)^{op} resp.) and note that ′A_{1} (A′_{1} resp.) is a G-graded algebra with (′A_{1})_{g} = R_{g} ⊗ (S^{op}) (A′_{1})_{g} = R_{1} ⊗ (S^{op})_{g} for all g ∈ G. Thus ′E and E′ are G-graded algebras such that M becomes a G-graded (′A_{1},′E)-bimodule and a G-graded (A′_{1}, E′)-bimodule. Let J_{gr}(′E) and J_{gr}(E′) denote the graded Jacobson radical of ′E (E′ resp.). Set ′\overline{E} = ′E/J_{gr}(′E) and \overline{E}′ = E′/J_{gr}(E′). The main result of Section 3 is: Theorem 3.6. Assume that M_{1} is an indecomposable (R_{1}, S_{1})-bimodule. Then there are bijections between the (set of isomorphism classes of) indecomposable′A_{1}-summands of M and the projective indecomposable′\overline{E}-modules and between the indecomposable A′_{1}-summands of M and the projective indecomposable \overline{E}′-modules. In the final Section 4, a Green correspondence is investigated and the main result of this section is: Theorem 4.9. Let D be a p-subgroup of G and let H be a subgroup of G such that N_{G}(D) ≤ H ≤ G. Then induction of bimodules gives a bijection between the isomorphism classes of indecomposable A_{H×H}-summands with defect group D of M_{H} and the isomorphism classes of indecomposable A-summands with defect group D of M.

M.E.Harris (Minneapolis) Keywords:

bimodules; finite groups; graded algebras; additive subcategories; indecomposable summands; equivalences of categories; graded Jacobson radical; Green correspondence; defect groups Classification :

∗*20C05 Group rings of finite groups and their modules (group theory)

16W50 Graded associative rings and modules 16D20 Bimodules (assoc. rings and algebras) 16D90 Module categories (assoc. rings and algebras) 20C20 Modular representations and characters of groups

Zbl. Math. Zbl 0965.20001

Marcus, Andrei Blocks of normal subgroups and Morita equivalences. (English) Ital. J. Pure Appl. Math. 7, 77-84 (2000).

Let G be a finite group, let (K, O, F) be a suitable p-modular system, and let R be a Ggraded O-algebra which is a crossed product. Every block b of the identity component R_{1} of R determines a Clifford extension E(b), a twisted group algebra of a subgroup G[b] of G over F. Similarly, every simple K ⊗_{O} R_{1}-module X determines a Clifford extension E(X), a twisted group algebra of a subgroup G_{X} of G over K, and every indecomposable F ⊗_{O} R_{1}-module U determines a Clifford extension E(U), a twisted group algebra of a subgroup G_{U} of G over F. In the first part of the paper, the author shows that these Clifford extensions are invariants of the graded Morita equivalence class of R. In the second part of the paper, the author gives a graded version of a result of M. Linckelmann [Turk. J. Math. 22, No. 1, 93-107 (1998; Zbl 0922.20017)] concerning splendid Morita equivalences between blocks with isomorphic defect groups and equivalent Brauer categories.

Burkhard Külshammer (Jena) Keywords:

crossed products; Clifford extensions; Brauer categories; Morita equivalences; splendid equivalences; finite groups; p-modular systems; graded algebras; twisted group algebras; blocks; simple modules Classification :

∗*20C05 Group rings of finite groups and their modules (group theory)

16D90 Module categories (assoc. rings and algebras) 20C20 Modular representations and characters of groups 16W50 Graded associative rings and modules 16D60 (Semi-)simple modules, etc. (assoc. rings and algebras) 16S35 Twisted and skew group rings, etc.

Zbl. Math. Zbl 1014.16043

Marcus, Andrei Representation theory of group graded algebras. (English) Commack, NY: Nova Science Publishers. xiii, 212 p. $ 89.00 (1999).

The theory of rings graded by groups and semigroups and their representations is a very important theory which has become very useful in many branches of algebra, algebraic geometry and number theory. This book is concentrated on the part of the theory closely connected with some aspects of the finite group representation theory. The book is divided into five chapters as follows: 1. Modules over graded rings; 2. Clifford theory; 3. Clifford extensions; 4. Auslander-Reiten theory; 5. Equivalences induced by graded bimodules. The first chapter is dedicated to the study of the category of modules graded by a G-set presenting the basic properties. In Chapter 2, the author studies the endomorphism ring E of a graded module M and also the functors Hom_{R}(M, −) and M ⊗_{E} −. Several applications to Clifford theory are obtained. An important problem in this theory is to study the structure of a graded simple (graded indecomposable) module when it is considered without gradation. The chapter ends with the Clifford theory of bimodules and blocks. The next chapter investigates the problem of extendability for strongly graded rings over a finite group. Chapter 4 contains the extension of the results of the preceding chapters to functor categories. The last chapter contains the characterization of several types of equivalences (Morita equivalences, Rickard’s derived equivalences and stable Morita equivalences) between two fully G-graded k-algebras projective over k with G finite induced by G-graded bimodules. The book is very well written. It is suitable for a graduate course on graded rings and significant and important for researchers in this area.

Blas Torrecillas (Almeria) Keywords:

strongly graded rings; Clifford theory; functor categories; induction; coinduction; graded modules; representations of finite groups; equivalences of categories; graded bimodules; Morita equivalences; derived equivalences Classification :

∗*16W50 Graded associative rings and modules 16-02 Research monographs (assoc. rings and algebras) 16D90 Module categories (assoc. rings and algebras) 16G70 Auslander-Reiten sequences and quivers 20C05 Group rings of finite groups and their modules (group theory) 20C15 Ordinary representations and characters of groups

Zbl. Math. Zbl 0944.20006

Marcus, Andrei Derived equivalences and Dade’s invariant conjecture. (English) J. Algebra 221, No.2, 513-527 (1999). doi:10.1006/jabr.1999.7982

The paper under review deals with the conjectures of Broué and Dade, for modular representations of finite groups. The approach is via graded rings and equivariant equivalences between their derived module categories. It is first shown that such equivalences induce equivariant maps between Grothendieck groups. Then the author formulates an equivariant version of Broué’s conjecture for blocks with Abelian defect groups and shows that this conjecture implies Dade’s invariant conjecture for such blocks. Finally, it is checked that the equivariant form of Broué’s conjecture holds for blocks with cyclic defect groups, by generalizing an argument of Rouquier.

Burkhard Külshammer (Jena) Keywords:

splendid equivalences; modular representations; finite groups; graded rings; equivariant equivalences; derived module categories; equivariant maps; Grothendieck groups; Broué’s conjecture; blocks; Abelian defect groups; Dade’s invariant conjecture Classification :

∗*20C05 Group rings of finite groups and their modules (group theory)

20C20 Modular representations and characters of groups

18E30 Derived categories, etc.

16D90 Module categories (assoc. rings and algebras)

Zbl. Math. Zbl 1010.20004

Marcus, Andrei Induction of graded interior algebras. (English) Stud. Univ. Babeş-Bolyai, Math. 43, No.1, 45-56 (1998).

Summary: We introduce interior O^{α}H-algebras graded by a finite group Γ and generalized induction for these algebras. This situation occurs in the study of source algebras of blocks of normal subgroups and our construction unifies various constructions introduced by Lluis Puig.

Keywords:

graded interior algebras; blocks; source algebras; induction Classification :

∗*20C20 Modular representations and characters of groups

16W50 Graded associative rings and modules

Zbl. Math. Zbl 0973.20008

Marcus, Andrei Groups, algebras, representations. (English) Linckelmann, Markus (ed.), Proceedings of the summer school on representation theory of algebras, finite and reductive groups, Cluj-Napoca, Romania, September 15-25, 1997. Cluj-Napoca: “Babeş Bolyai” University, Faculty of Mathematics and Computer Science, 1-32 (1998).

This is an introduction to the modular representation theory of finite groups at a level suitable for graduate students. The course starts with the definition of a group ring and ends with a number of conjectures like Dade’s, Alperin-McKay, height zero conjecture and others.

Alexander Zimmermann (Amiens) Keywords:

modular representations; finite groups Classification :

∗*20C20 Modular representations and characters of groups

Zbl. Math. Zbl 0931.16008

Marcus, Andrei Homology of fully graded algebras, Morita and derived equivalences. (English) J. Pure Appl. Algebra 133, No.1-2, 209-218 (1998). http://www.sciencedirect.com/science/journal/00224049

This paper deals with Hochschild and cyclic (co)homology of strongly group graded algebras R = ⨁_{g∈G} R_{g}, R_{g}R_{h} = R_{gh}. As is well-known, these homology theories are Morita invariant for arbitrary algebras. In the graded case, there are spectral sequences converging to the homology of R, with second page the homology of G with coefficients the homology of the identity component R_{1}. It is shown here that, if two strongly Ggraded algebras R and S are graded Morita equivalent (i.e., the equivalence is afforded by a G-graded (R, S)-bimodule), the isomorphism between the homologies of R and S comes from an isomorphism of the second pages of their respective spectral sequences. For Hochschild cohomology, a corresponding result is also proved under the weaker assumption of graded derived equivalence.

M.Lorenz (Philadelphia) Keywords:

Hochschild cohomologies; group graded rings; derived equivalences; cyclic cohomologies; Morita invariants; Morita equivalences; strongly group graded algebras; spectral sequences; Hochschild homologies; cyclic homologies Classification :

∗*16E40 Homology and cohomology theories for assoc. rings 16W50 Graded associative rings and modules 18E30 Derived categories, etc. 16D90 Module categories (assoc. rings and algebras) 18G40 Spectral sequences (homological algebra)

Zbl. Math. Zbl 0930.20009

Marcus, Andrei Projective modules over twisted group algebras of p-solvable groups. (English) Publ. Math. 53, No.3-4, 367-374 (1998). http://www.math.klte.hu/publi/contents.php?szam=40

The result in this paper may be viewed as an application of the author’s earlier papers [Publ. Math. 42, No. 3-4, 303-314 (1993; Zbl 0801.16042) and Commun. Algebra 23, No. 12, 4393-4404 (1995; Zbl 0847.16024)] on Clifford theory for strongly graded algebras. Using this theory the author is able to generalize some results of P. Fong [Trans Am. Math. Soc. 103, 484-494 (1962; Zbl 0105.25603)] and I. M. Isaacs [J. Algebra 99, 89-107 (1986; Zbl 0587.20005)] on modular representation theory for p-solvable groups.

Henning Haahr Andersen (Aarhus) Keywords:

p-solvable groups; twisted group algebras; strongly graded algebras; Clifford theory; modular representations Classification :

∗*20C20 Modular representations and characters of groups 16W50 Graded associative rings and modules 20C05 Group rings of finite groups and their modules (group theory) 20D10 Solvable finite groups 16S35 Twisted and skew group rings, etc. 16D40 Free, etc. modules and ideals (assoc. rings and algebras)

Zbl. Math. Zbl 0911.16023

Marcus, Andrei On Picard groups and graded rings. (English) Commun. Algebra 26, No.7, 2211-2219 (1998). http://taylorandfrancis.metapress.com/openurl.asp?genre=journalissn=0092-7872

Let k be a commutative ring, G a group (possibly infinite), and R a strongly G-graded kalgebra with identity graded component R_{1}. The group G acts on Pic_{k}(R_{1}) by mapping the class of a module M to the class of R_{g} ⊗_{R_{1}} M ⊗_{R_{1}} R_{g^{−1}}, and Pic_{k}(R_{1})^{G} is the subgroup of Pic_{k}(R_{1}) of invariants under this action. There is also a “Picard group” of graded R-bimodules, i.e. R-bimodules M such that M is G-graded and R_{g}M_{h}R_{t} = M_{ght}; the group of isomorphism classes of invertible G-graded R-bimodules is denoted Pic_{k}^{gr}(R). In this paper the author gives a direct proof of the exact sequence

1 → H^{1}(G, U(Z(R_{1}))) ⟶^{Φ} Pic_{k}^{gr}(R) ⟶^{Ψ} Pic_{k}(R_{1})^{G} ⟶^{Θ} H^{2}(G, U(Z(R_{1}))),

derived by M. Beattie and A. del Río [Graded equivalences and Picard groups, J. Pure Appl. Algebra (to appear)]. In this sequence, Φ and Ψ are group homomorphisms but Θ is not. The author’s techniques avoid the use of rings with local units and the smash product R#(kG)^{∗rat}, so that the argument is more straightforward.

Margaret Beattie (Sackville) Keywords:

Picard groups; strongly graded algebras; graded bimodules; exact sequences Classification :

∗*16W50 Graded associative rings and modules

14C22 Picard groups

16W20 Morphisms of associative rings

16D20 Bimodules (assoc. rings and algebras)

Zbl. Math. Zbl 0904.16005

Marcus, Andrei Equivalences induced by graded bimodules. (English) Commun. Algebra 26, No.3, 713-731 (1998). http://taylorandfrancis.metapress.com/openurl.asp?genre=journalissn=0092-7872

Let G be a finite group, k a commutative ring, and let R and S be two fully G-graded k-algebras. The algebras R and S are called graded Morita equivalent if there are Ggraded bimodules _{R}M_{S} and _{S}N_{R} inducing a Morita equivalence R ∼ S such that the bimodule isomorphisms M ⊗_{S} N → R and N ⊗_{R} M → S are grade preserving. Let M_{1} be an R_{1}-S_{1}-bimodule, N_{1} an S_{1}-R_{1}-bimodule, and denote M = R ⊗_{R_{1}} M_{1} and N = N_{1} ⊗_{S_{1}} S. The following statements are equivalent: (i) There is a structure of a G-graded R-S-bimodule on M and a structure of a G-graded S-R-bimodule on N (extending the given structures), such that M and N induce a graded Morita equivalence between R and S. (ii) M_{1} and N_{1} induce a Morita equivalence between R_{1} and S_{1}, and M_{1} extends to a Δ-module, where Δ = ⨁_{g∈G}(R_{g} ⊗_{k} S_{g}^{op}). – Similar results are obtained for derived equivalences induced by complexes of G-graded bimodules and for stable equivalences of Morita type, which are induced by graded bimodules.

A.I.Kashu (Kishinev) Keywords:

graded Morita equivalences; derived equivalences; stable equivalences; graded bimodules Classification :

∗*16D90 Module categories (assoc. rings and algebras)

16W50 Graded associative rings and modules 16D20 Bimodules (assoc. rings and algebras)

Zbl. Math. Zbl 0910.16025

Marcus, Andrei Green functors associated to a strongly graded algebra. (English) Mathematica 39, No.2, 257-263 (1997).

The tensor product of two modules over a group ring serves as a model for a similar construction (using bimodules) in the case of modules over a strongly G-graded algebra over a complete discrete valuation ring with residue field of positive characteristic. This is done in an attempt to find “invariants” of graded derived equivalences.

S.Raianu (Bucureşti) Keywords:

strongly graded algebras; tensor products; bimodules; graded derived equivalences Classification :

∗*16W50 Graded associative rings and modules

16D90 Module categories (assoc. rings and algebras)

Zbl. Math. Zbl 1010.20003

Marcus, Andrei Group-graded algebras, adjoint functors and the Green correspondence. (English) Stud. Univ. Babeş-Bolyai, Math. 41, No.2, 87-94 (1996).

Summary: Let G be a finite group and k a commutative ring. In the context of fully G-graded k-algebras, we give versions of the Green correspondence and of the Burry-Carlson-Puig theorem, generalizing recent results of M. Auslander and M. Kleiner [Adv. Math. 104, No. 2, 297-314 (1994; Zbl 0807.20009)].

Keywords:

group-graded algebras; Green correspondence; Burry-Carlson-Puig theorem Classification :

∗*20C20 Modular representations and characters of groups 16W50 Graded associative rings and modules 18A40 Adjoint functors

Zbl. Math. Zbl 0880.16025

Marcus, Andrei Graded equivalences and Broué’s conjecture. (English) An. Ştiinţ. Univ. “Ovidius” Constanţa, Ser. Mat. 4, No.2, 107-126 (1996).

This is an expository paper, presenting results obtained in two papers of the author on the reduction of Broué’s conjecture to the case of simple groups. The methods and tools used are: Morita equivalences, derived equivalences and stable equivalences of Morita type between two group-graded algebras, induced by group-graded bimodules or complexes of group-graded bimodules.

S.Raianu (Bucureşti) Keywords:

Broué’s conjecture; Morita equivalences; derived equivalences; stable equivalences; group-graded algebras; group-graded bimodules Classification :

∗*16W50 Graded associative rings and modules

20C20 Modular representations and characters of groups

20C05 Group rings of finite groups and their modules (group theory)

18E30 Derived categories, etc. 16S34 Group rings (assoc. rings) 16D90 Module categories (assoc. rings and algebras)

Zbl. Math. Zbl 0873.16027

Marcus, Andrei Tensor products of fully graded algebras and applications. (English) Mathematica 38, No.1-2, 133-139 (1996).

The note uses some constructions over fully (or strongly) graded algebras in order to generalize Mackey’s tensor product formula. A graded version of Nagao’s theorem is also obtained.

S.Raianu (Bucureşti) Keywords:

strongly graded algebras; Mackey’s tensor product formula; Nagao’s theorem Classification :

∗*16W50 Graded associative rings and modules

Zbl. Math. Zbl 0866.20007

Marcus, Andrei On equivalences between blocks of group algebras: Reduction to the simple components. (English) J. Algebra 184, No.2, 372-396 (1996). http://www.sciencedirect.com/science/journal/00218693

Let p be a prime and let O be a complete discrete valuation ring with quotient field K of characteristic zero and with residue field k = O/J(O) of characteristic p. All Oalgebras and O-algebra modules will be assumed to be unitary, finitely generated and O-free. Let G be a finite group and let R = ⨁_{g∈G} R_{g} and S = ⨁_{g∈G} S_{g} be G-crossed product O-algebras. The opposite ring S^{op} is a G^{op}-crossed product O-algebra and A = R ⊗_{O} S^{op} is a G × G^{op}-crossed product O-algebra. Let δ : G → G × G^{op} be defined by: g ↦ (g, g^{−1}) for all g ∈ G. Then Δ ≡ A_{δ(G)} = ⨁_{g∈G}(R_{g} ⊗_{O} S_{g^{−1}}^{op}) is a δ(G)-crossed product O-subalgebra of A. One of the main points of this paper is that there are three naturally isomorphic Olinear functors A ⊗_{Δ} −, R ⊗_{R_{1}} −, − ⊗_{S_{1}} S : Δ-mod → A-mod (Section 3). Let M be an (R_{1}, S_{1})-bimodule. Then its O-dual M^{∗} = Hom_{O}(M, O) is an (S_{1}, R_{1})-bimodule. One of the main results of this paper is: Theorem 3.4. (a) Assume that (M, M^{∗}) defines a Morita equivalence between R_{1} and S_{1} and that M is δ(G)-invariant and extends to a Δmodule. Then R⊗_{R_{1}} M (resp. S⊗_{S_{1}} M^{∗}) is a naturally G-graded (R, S)-bimodule (resp. (S, R)-bimodule) and (R ⊗_{R_{1}} M, S ⊗_{S_{1}} M^{∗}) defines a Morita equivalence between R and S. (b) Assume that G is a p′-group and that (C, C^{∗}) defines a derived equivalence between R_{1} and S_{1}. If (C, d) extends to a complex of Δ-modules, then (R ⊗_{R_{1}} C, S ⊗_{S_{1}} C^{∗}) defines a derived equivalence between R and S. These ideas are applied, in this paper, to modular group representation theory as follows. Assume also that K and k are “big enough” for all of the finite groups under consideration. Let X̃, Ỹ be finite groups with normal subgroups X ⊴ Ỹ, Y ⊴ Ỹ and with an isomorphism α : X̃/X → Ỹ/Y. Let e ∈ Z(OX), f ∈ Z(OY) be X̃, Ỹinvariant block idempotents, resp. and set R = eOX̃ and S = fOỸ. Set G = X̃/X and identify G with Ỹ/Y via α. Then R and S are G-crossed product O-algebras with R_{1} = eOX and S_{1} = fOY. In this context, results on Puig equivalences and Rickard tilting complexes are obtained (Corollary 3.9). Section 4 presents applications to wreath products and Section 5 contains applications to isomorphic blocks, groups with cyclic Sylow p-subgroups, the principal 3-block of A_{5} and symmetric groups, etc. The basic idea of examining the consequences of the existence of an extension to the “diagonal” is related to earlier work of P. Fong and the reviewer on the character level. (Also submitted to MR).

M.E.Harris (Minneapolis) Keywords:

complete discrete valuation rings; finite groups; crossed product algebras; Morita equivalences; derived equivalences; block idempotents; Puig equivalences; Rickard tilting complexes; groups with cyclic Sylow p-subgroups Classification :

∗*20C20 Modular representations and characters of groups 16D90 Module categories (assoc. rings and algebras) 16S34 Group rings (assoc. rings) 20C05 Group rings of finite groups and their modules (group theory) 16S35 Twisted and skew group rings, etc.

Zbl. Math. Zbl 0886.16029

Marcus, Andrei Modules graded by G-sets and Clifford theory. (English) Mathematica 37, No.1-2, 131-154 (1995).

Modules graded by G-sets occur in group-representation theory when dealing with induced modules. If K is a field, H is a subgroup of the group G, G/H = {gH | g ∈ G}, and M is a left KH-module, then the KG-module KG ⊗_{KH} M induced from M has a decomposition

KG ⊗_{KH} M = ⨁_{x∈G/H}(Kx ⊗_{KH} M)

into additive subgroups, where Kx = ⨁_{g∈x} Kg, such that for g ∈ G, Kg(Kx⊗_{KH} M) = Kgx ⊗_{KH} M. Gradings by transitive G-sets were introduced by E. C. Dade. In order to set our context, in the first section of this paper we give more general definitions. We fix a homomorphism θ: G → H of groups and a homomorphism φ: A → B of G-sets, where B is an H-set, hence a G-set by restriction of scalars. We define first the “ungrading” functor for graded rings U(θ): G-grring → H-grring and its right adjoint, the “truncation” functor T (θ): H-grring → G-grring. Then we construct the ungrading functor for graded modules U(θ, φ): (G, A, R)-gr → (H, B, U(θ)(R))-gr and its right adjoint F(θ, φ), and the truncation functor for graded modules T (θ, φ): (H, B, S)-gr → (G, A, T (θ)(S))-gr and its left adjoint, the induction functor I(θ, φ). These functors behave well with respect to compositions of homomorphisms of groups and G-sets. In the second section we restrict to the case of modules graded by transitive G-sets. If H is a subgroup of G, then (G/H, R)-gr is the category (G, G/H, R)-gr. We establish some properties of the functors defined above and we construct the coinduction functor Coind: (H/K, R_{H})-gr → (G/K, R)-gr (where K ≤ H ≤ G), which is a right adjoint of the truncation functor. The main result of this section is Theorem 2.16, from which we obtain correspondences between the simple objects of the categories (G/K, R)-gr, (H/K, R_{H})-gr and (G/H, R)-gr. A well-known theorem of E. C. Dade states that the categories R_{1}-mod and R-gr are equivalent via the induction and the truncation functors if and only if 1 ∈ R_{g}R_{g^{−1}} for all g ∈ G. We generalize this in Section 3 by showing that the categories (H/K, R_{H})-gr and (G/K, R)-gr are equivalent via the induction and the truncation functors if and only if 1 ∈ R_{gH}R_{Hg^{−1}} for all g ∈ G. The main ingredient in the construction of the coinduction functor mentioned above is the following observation: if M is a finitely generated G-graded R-module and N is a G/H-graded R-module, then Hom_{R}(M, N) is in a natural way a G/H-graded left E-module, where E = End_{R}(M)^{op}. This observation is also used in Section 3, where we show that the equivalences (which we shall call Clifford correspondences) CD[M] ⇄ E-mod and (R|M)-mod ⇄ (E|E)-mod preserve the G/H-graded modules (and also the relative H-projective modules in the strongly graded case). Here M is again a finitely generated G-graded R-module, CD[M] is the full subcategory of R-mod consisting of R-modules N for which there is an exact sequence M^{(J)} → M^{(I)} → N → 0 in R-mod, (R|M)-mod consists of R-modules which divide a direct sum of copies of M, and (E|E)-mod is the category of projective E-modules. As consequences of our result, we obtain a graded version of Morita’s theorem, and in particular, some equivalences involving the smash product. We also deduce Clifford correspondences for arbitrary subgroups for both simple and indecomposable modules.

Keywords:

modules graded by G-sets; projective modules; induced modules; transitive Gsets; graded rings; ungrading functors for graded modules; truncation functors for graded modules; induction functors; coinduction functors; simple objects; Clifford correspondences; Morita’s theorem; equivalences; smash products; indecomposable modules Classification :

∗*16W50 Graded associative rings and modules 16D90 Module categories (assoc. rings and algebras) 20C05 Group rings of finite groups and their modules (group theory) 16S34 Group rings (assoc. rings)

Zbl. Math. Zbl 0847.16024

Marcus, Andrei Clifford theory for projective modules over strongly graded rings. (English) Commun. Algebra 23, No.12, 4393-4404 (1995). http://taylorandfrancis.metapress.com/openurl.asp?genre=journalissn=0092-7872 Let G be a finite group and let R = ⨁_{x∈G} R_{x} be a strongly G-graded ring. All modules in this paper are finitely generated left modules. Let R-gr denote the abelian category of (finitely generated) G-graded R-modules and grade-preserving R-morphisms. The intersection of the maximal graded left ideals of R is the graded Jacobson radical, J_{gr}(R), of R and J_{gr}(R) = J(R_{1})R = RJ(R_{1}) ⊆ J(R). If W is an R-module, let H(w) = W/(J(R)W) and H_{gr}(W) = W/(J_{gr}(R)W) denote the head and gr-head of W, resp. It is well-known that R ⊗_{R_{1}} − : R_{1}-mod → R-gr and (−)_{1} : R-gr → R_{1}-mod form an equivalence between the abelian categories R_{1}-mod and R-gr. Let M = ⨁_{x∈G} M_{x} be an indecomposable G-graded R-module and let E = End_{R}(M)^{op}. As is well-known, E has a natural G-grading and set G(M) = {x ∈ G | R_{x} ⊗_{R_{1}} M_{1}, M_{1} in R_{1}-mod} and let (R|M)-mod denote the full subcategory of R-modules that “divide” a finite direct sum of copies of M and let D = E/J_{gr}(E). The main results of this paper present extensions and generalizations of the general theory to obtain: functors that induce isomorphisms between Grothendieck groups associated to the categories (E|E)-mod and (D|D)-mod (Theorem 2.3) and results on heads and projective covers (Corollary 2.4, Propositions 2.6 and 2.7 and Corollary 2.8). Section 3 contains extensions and generalizations of results on crossed products.

M.E.Harris (Minneapolis) Keywords:

equivalences between Abelian categories; finite groups; strongly G-graded rings; finitely generated left modules; maximal graded left ideals; graded Jacobson radical; Grothendieck groups; heads; projective covers; crossed products Classification :

∗*16W50 Graded associative rings and modules 16D40 Free, etc. modules and ideals (assoc. rings and algebras) 16D90 Module categories (assoc. rings and algebras) 20C05 Group rings of finite groups and their modules (group theory) 16E20 K-theory of noncommutative rings 16S35 Twisted and skew group rings, etc.

Zbl. Math. Zbl 0832.16027

Marcus, Andrei Indecomposable modules and Clifford extensions. (English) Mathematica 35, No.2, 169-173 (1993).

See the preceding review Zbl 0832.16026.

Classification :

∗*16S35 Twisted and skew group rings, etc. 16W50 Graded associative rings and modules 20C05 Group rings of finite groups and their modules (group theory) 20C20 Modular representations and characters of groups Zbl. Math. Zbl 0832.16026

Marcus, Andrei Indecomposable modules and Clifford extensions. (English) Proceedings of the international conference on group theory, Timişoara, Romania, 17-20 September, 1992. Timişoara: Univ. Timişoara, Analele Universităţii din Timişoara. 125-131 (1993).

Summary: Let N ↣ H ↠ G be an extension of finite groups, k a field, V a Ginvariant indecomposable kN-module, E = End_{kH}(V^{H})^{op}, E_{1} = End_{kN}(V)^{op} and K = Z(E_{1}/J(E_{1})). If (Ind(E_{1}/J(E_{1})), |G/C_{G}(K)|) = 1 there is a class ω_{H}(V) ∈ H^{2}(G, K^{∗}) associated to E/J(E_{1})E. Under suitable assumptions, if G is a p-group, we reduce the computation of this class to the case of a subextension Y ↣ X ↠ G of p_{F} -elementary subgroups and we conclude that the order of ω_{H}(V) divides the number of exp(N)th roots of unity contained in K.

Keywords:

extensions of finite groups; indecomposable modules; p-groups Classification :

∗*16S35 Twisted and skew group rings, etc.

20C05 Group rings of finite groups and their modules (group theory)

16W50 Graded associative rings and modules

20C20 Modular representations and characters of groups

Zbl. Math. Zbl 0801.16042

Marcus, Andrei Static modules and Clifford theory for strongly graded rings. (English) Publ. Math. 42, No.3-4, 303-314 (1993). http://www.math.klte.hu/publi/contents.php?szam=40

Let A be a ring, M a left A-module and let D = End_{A}(M)^{o}. An A-module V is called M-static if the natural homomorphism M ⊗_{D} Hom_{A}(M, V) → V, m ⊗ f ↦ mf = f(m) is an isomorphism. There exists a number of useful characterizations of M-static modules. The author uses these modules for an investigation of Clifford theory for strongly graded rings. In particular he obtains the direct (non-stable) Clifford correspondence for strongly graded rings in the case of simple and of indecomposable modules. It turns out that these correspondences are compatible with induction, which allows the author to obtain some results of Clifford theory in a transparent fashion. The paper is very well written and contains a number of interesting results.

G.Karpilovsky (Chico) Keywords:

simple modules; static modules; strongly graded rings; Clifford correspondence; indecomposable modules; induction Classification :

∗*16W50 Graded associative rings and modules 20C20 Modular representations and characters of groups 16D60 (Semi-)simple modules, etc. (assoc. rings and algebras) 16S34 Group rings (assoc. rings) 16D90 Module categories (assoc. rings and algebras)

Zbl. Math. Zbl 0837.16038

Marcus, Andrei Extensions of indecomposable modules. (English) Proceedings of the 9th national conference on algebra held at the University of Cluj-Napoca, Romania, September 18-20, 1991. Cluj-Napoca: Univ. Cluj-Napoca, Fac. of Math. and Computer Science, “Babeş- Bolyai” Univ., Fac. Math. Comput. Sci. Res. Semin., Prepr. 1992(1), 61-72 (1992).

Summary: Let G be a finite group and R = ⨁_{x∈G} R_{x} be a finite dimensional strongly graded F -algebra (F a field). Assume that V is a G-invariant indecomposable R_{1}- module, finite dimensional over F. We find sufficient conditions for the existence of an extension of V to an R-module and for the existence of a G/H-invariant extension of V to R_{H}, where H is a normal subgroup of G.

Keywords:

finite groups; finite dimensional strongly graded algebras; indecomposable modules; existence; extensions; normal subgroups Classification :

∗*16W50 Graded associative rings and modules 16P10 Finite associative rings and finite-dimensional algebras

Zbl. Math. Zbl 0836.16028

Marcus, Andrei Compounded Clifford correspondences for modules over graded rings. (English) Proceedings of the 9th national conference on algebra held at the University of Cluj-Napoca, Romania, September 18-20, 1991. Cluj-Napoca: Univ. Cluj-Napoca, Fac. of Math. and Computer Science, “Babeş- Bolyai” Univ., Fac. Math. Comput. Sci. Res. Semin., Prepr. 1992(1), 73-92 (1992).

Summary: We study the behaviour of Dade’s shifting isomorphism under the following operations: compounding, restriction, induction and conjugation.

Keywords:

Dade’s shifting isomorphism; restriction; induction; conjugation Classification :

∗*16W50 Graded associative rings and modules Zbl. Math. Zbl 0835.16039

Marcus, Andrei Clifford theory and the Loewy length of induced modules. (English) Proceedings of the 9th national conference on algebra held at the University of Cluj-Napoca, Romania, September 18-20, 1991. Cluj-Napoca: Univ. Cluj-Napoca, Fac. of Math. and Computer Science, “Babeş- Bolyai” Univ., Fac. Math. Comput. Sci. Res. Semin., Prepr. 1992(1), 55-60 (1992).

Summary: Let G be a finite group, R = ⨁_{g∈G} R_{g} a strongly G- graded ring such that R_{1} is an artinian ring and M an R_{1}- module. Using Clifford theory we show that the Loewy length of the induced module M^{G} = R ⊗_{R_{1}} M is equal to the nilpotency index of J(End_{R_{H}} (R_{H} ⊗_{R_{1}} M)).

Keywords:

strongly G-graded rings; Artinian rings; Clifford theory; Loewy length; induced modules; nilpotency index Classification :

∗*16W50 Graded associative rings and modules 16D60 (Semi-)simple modules, etc. (assoc. rings and algebras) 16D90 Module categories (assoc. rings and algebras)

Zbl. Math. Zbl 0794.20064

Mărcuş, Andrei Retract extensions of completely simple semigroups by nil semigroups. (English) Mathematica 34, No.1, 37-41 (1992).

A semigroup S is called a P -semigroup iff for every x, y ∈ S, y ∈ S^{1}xS^{1} implies y^{m} ∈ S^{1}x^{2}S^{1} for some positive integer m. S is called a left [right] P -semigroup iff for every x, y ∈ S, y ∈ xS^{1} implies y^{m} ∈ x^{2}S^{1} [y ∈ S^{1}x implies y^{m} ∈ S^{1}x^{2}] for some positive integer m. The following is the main result of this paper. A semigroup S is an archimedean left and right P -semigroup with idempotent if and only if S is a retract extension of a completely simple semigroup by a nil semigroup. With interest the reviewer reads this result since it is a generalization of the reviewer’s result of 1954 for the commutative case [A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, Vol. 1 (Am. Math. Soc., 1961; Zbl 0111.03403)].

T.Tamura (Davis) Keywords:

archimedean left and right P -semigroup; idempotent; retract extension; completely simple semigroup; nil semigroup Classification :

∗*20M10 General structure theory of semigroups

20M12 Ideal theory of semigroups

Zbl. Math. Zbl 0747.16019

Mărcuş, Andrei Invariant extensions of simple modules. (English) Stud. Cercet. Mat. 43, No.1/2, 37-42 (1991).

If R is a strongly G-graded ring and M is a G-invariant simple R_{1}-module (1 is the unit element of the group G) which extends to an R_{H}-module (H is a normal subgroup of G), then necessary and sufficient conditions are given for M to have a G/H-invariant extension to R.

S.Raianu (Bucureşti) Keywords:

strongly G-graded ring; G-invariant simple R_{1}-module; invariant extension Classification :

∗*16W50 Graded associative rings and modules 16D60 (Semi-)simple modules, etc. (assoc. rings and algebras)