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MR2253589 16W50 Dicu, Camelia ^{(}R-CLUJMI^{)}; Marcus, Andrei ^{(}R-CLUJMI^{)} Group-graded algebras and the relative projectivity of pointed groups. (English. English summary) Q. J. Math. 57 (2006), no. 3, 309–318.

{A review for this item is in process.}

[References]

Note: This list reflects references listed in the original paper as accurately as possible with no attempt to correct errors.

1. J. L. Alperin, M. Linckelmann and R. Rouquier, Source algebras and source modules, J. Algebra 239 (2001), 262–271. MR1827883 (2002b:20011)

2. Y. Fan, L. Puig and Y. Zhou, On a theorem of Fong, J. Algebra 239 (2001), 735–741. MR1832914 (2002b:20013)

3. H. Fottner and B. Külshammer, On indecomposable and imprimitive modules for finite groups—a G-algebra approach, J. London Math. Soc. (2) 59 (1999), 828–844. MR1709083 (2001a:20008)

4. A. Marcus, Projective modules over twisted group algebras of p-solvable groups, Publ. Math. Debrecen 53 (1998), 367–374. MR1657479 (99m:16069)

5. A. Marcus, Representation Theory of Group Graded Algebras, Nova Science Publishers, Commack, New York, 1999. MR1921424 (2003c:16056)

6. L. Puig, Blocks of Finite Groups: The Hyperfocal Subalgebra of a Block, Springer, Berlin, 2002. MR1917233 (2003j:20012)

7. J. Thevenaz, G-Algebras and Modular Representation Theory, Clarendon Press, Oxford, 1995. MR1365077 (96j:20017)

MR2147726 (2006f:16059) 16W50 18E30 20C20

Marcus, Andrei ^{(}R-CLUJMI^{)}

Tilting complexes for group graded algebras. II. (English. English summary) Osaka J. Math. 42 (2005), no. 2, 453–462. In order to verify cases which satisfy Broué’s conjecture [M. Broué, Astérisque No. 181-182 (1990), 61–92; MR1051243 (91i:20006)], for a symmetric algebra Λ over a field k, J. Rickard gave a special set {X_{1}, …, X_{n}} of objects of the bounded derived category D^{b}(mod Λ) inducing a derived equivalence D^{b}(mod Λ) ≅ D^{b}(mod Γ) which sends X_{1}, …, X_{n} to the simple Γ-modules, and he also gave examples of stable equivalences which induce such derived equivalences [J. Rickard, J. Algebra 257 (2002), no. 2, 460–481; MR1947972 (2004a:16023)]. In the present paper, the author gives a similar result for G-graded derived equivalences between symmetric crossed products of finite-dimensional k-algebras and a finite group G, and some stable equivalences which induce G-graded derived equivalences.

{For Part I see [A. Mărcuş, J. Group Theory 6 (2003), no. 2, 175– 193; MR1961566 (2004c:16011)].}

Jun-ichi Miyachi (J-TOKYG)

[References]

Note: This list reflects references listed in the original paper as accurately as possible with no attempt to correct errors.

1. J. Chuang: Derived equivalence in SL_{2}(p^{2}), Trans. Amer. Math. Soc. 353 (2001), 2897–2913. MR1828478 (2002m:20015)

2. M. Holloway: Broué’s conjecture for the Hall-Janko group and its double cover, Proc. London Math. Soc. 86 (2003), 109–130. MR1971465 (2004a:20020)

3. S. König and A. Zimmermann: Derived Equivalences for Group Rings, Lecture Notes in Mathematics, 1685, Springer-Verlag, Berlin, 1998. MR1649837 (2000g:16018)

4. A. Marcus: Equivalences induced by graded bimodules, Comm. Algebra 26 (1998), 713–731. MR1606217 (99a:16039)

5. A. Marcus: Tilting complexes for group graded algebras, J. Group Theory 6 (2003), 175–193. MR1961566 (2004c:16011)

6. J. Rickard: Equivalences of derived categories for symmetric algebras, J. Algebra 257 (2002), 460–481. MR1947972 (2004a:16023)

7. R. Rouquier: Block theory via stable and Rickard equivalences; in Modular Representation Theory of Finite Groups (Charlottesville, VA, 1998) de Gruyter, Berlin (2001) 101–146. MR1889341 (2003g:20018)

8. Y. Usami and N. Yoshida: Preprint (2003).

9. A. Zimmermann: Tilted symmetric orders are symmetric orders, Arch. Math. 73 (1999), 15–17. MR1696576 (2000d:16024)

MR2134255 (2006d:20019) 20C20

Marcus, Andrei ^{(}R-CLUJMI^{)}

Blocks with cyclic defect groups and Clifford extensions. (English. English summary) J. Algebra 287 (2005), no. 1, 1–14. Summary: “We prove that the residual Clifford extension of any nonexceptional character belonging to a p-block with cyclic defect groups of a twisted group algebra is naturally isomorphic to the Clifford extension of any simple module belonging to that block. We also show that Dade’s inductive conjecture holds for blocks with cyclic defect groups.”

Ji Ping Zhang (PRC-BJ-SM)

[References]

Note: This list reflects references listed in the original paper as accurately as possible with no attempt to correct errors.

1. J.L. Alperin, M. Linckelmann, R. Rouquier, Source algebras and source modules, J. Algebra 239 (2001) 262–271. MR1827883 (2002b:20011)

2. M. Broué, Equivalences of blocks of group algebras, in: Finite-Dimensional Algebras and Related Topics, Ottawa, ON, 1992, in: NATO Adv. Sci. Inst. Ser. C, vol. 424, Kluwer Academic, Dordrecht, 1994, pp. 1–26. MR1308978 (97c:20004)

3. E.C. Dade, Counting characters in blocks II.9, in: R. Solomon (Ed.), Representation Theory of Finite Groups, Columbus, OH, 1995, in: Ohio State Univ. Math. Res. Inst. Publ., vol. 6, de Gruyter, Berlin, 1997, pp. 45–59. MR1611009 (99b:20016)

4. E.C. Dade, Counting characters in blocks with cyclic defect groups, I, J. Algebra 186 (1996) 934–969. MR1424599 (98b:20013)

5. T. Ikeda, A remark on socles and normal subgroups, Hokkaido Math. J. 13 (1984) 145–150. MR0746522 (85j:20007)

6. B. Külshammer, Crossed products and blocks with normal defect groups, Comm. Algebra 13 (1985) 147–168. MR0768092 (86c:20015)

7. B. Külshammer, T. Okuyama, A. Watanabe, A lifting theorem with applications to blocks and source algebras, J. Algebra 232 (2000) 299–309. MR1783927 (2001g:20013)

8. M. Linckelmann, Variations sur les blocs à groupes de défaut cycliques, in: Séminaire sur les groupes finis, tome IV, in: Publ. Math. Univ. Paris VII, vol. 34, 1988, pp. 1–83. MR1303138 (96a:20016)

9. A. Marcus, Representation Theory of Group Graded Algebras, Nova Science, Commack, NY, 1999. MR1921424 (2003c:16056)

10. A. Marcus, Twisted group algebras, normal subgroups and derived equivalences, Algebr. Represent. Theory 4 (2001) 25–54. MR1825806 (2002b:20016)

11. A. Marcus, Tilting complexes for group graded algebras, J. Group Theory 6 (2003) 175–193. MR1961566 (2004c:16011)

12. U. Reinhardt, P. Schmid, Invariant lattices and modular decomposition of irreducible representations, J. Algebra 87 (1984) 89–104. MR0736770 (85k:20029)

13. R. Rouquier, The derived category of blocks with cyclic defect groups, in: S. König, A. Zimmermann (Eds.), Derived Equivalences for Group Rings, Springer-Verlag, Berlin, 1998, pp. 199– 220. MR1649846

14. J. Thévenaz, G-Algebras and Modular Representation Theory, Clarendon Press, Oxford, 1995. MR1365077 (96j:20017)

MR2021243 (2005a:20018) 20C20 20D06

Marcus, Andrei ^{(}R-CLUJMI^{)}

Broué’s abelian defect group conjecture for alternating groups. (English. English summary) Proc. Amer. Math. Soc. 132 (2004), no. 1, 7–14 (electronic). Broué conjectures that a block of a finite group algebra is derived equivalent to the corresponding block of the normalizer of a defect group, whenever defect groups are abelian. This paper shows that the conjecture holds for alternating groups, under the strongest version of Rickard’s splendid equivalences.

The starting point is the result of J. Chuang and the reviewer [“Derived equivalences for symmetric groups and sl_{2}-categorification”, preprint, arXiv.org/abs/math.RT/0407205] that the conjecture holds for symmetric groups. The author shows that the equivalences for symmetric groups descend to equivalences for alternating groups.

Raphaël Rouquier (4-LEED-PM)

[References]

Note: This list, extracted from the PDF form of the original paper, may contain data conversion errors, almost all limited to the mathematical expressions.

1. J. Chuang, The derived categories of some blocks of symmetric groups and a conjecture of Brou’e, J. Algebra 217 (1999), 114–155. MR1700479 (2001a:20024)

2. J. Chuang and R. Kessar, Symmetric groups, wreath products, Morita equivalences, and Brou’e’s abelian defect group conjecture, Bull. London Math. Soc. 34 (2002), 174–184. MR1874244 (2002m:20016)

3. J. Chuang and R. Rouquier, Talks given at the Durham Symposium on Representations of Finite Groups and Related Algebras (1-11 July 2002) .

4. E. C. Dade, Another way to count characters, J. Reine Angew. Math. 510 (1999), 1–55. MR1696089 (2000e:20017)

5. P. Fong and M. E. Harris, On perfect isometries and isotypies in alternating groups, Trans. Amer. Math. Soc. 349 (1997), 3469– 3516. MR1390981 (97k:20009)

6. A. Marcus, On equivalences between blocks of group algebras: reduction to the simple compo- nents, J. Algebra 184 (1996), 372–396. MR1409219 (97m:20012)

7. A. Marcus, Equivalences induced by graded bimodules, Comm. Algebra 26 (1998), 713-731. MR1606217 (99a:16039)

8. A. Marcus, Tilting complexes for group graded algebras, J. Group Theory 6 (2003), 175–193. MR1961566 (2004c:16011)

9. T. Okuyama, Some examples of derived equivalent blocks of finite groups, preprint 1998.

10. J. Rickard, MSRI talk, Berkeley, November 6, 1990.

11. R. Rouquier, Isom’etries parfaites dans les blocs ‘a d’efaut ab’elien des groupes sym’etriques et sporadiques, J. Algebra 168 (1994), 648–694. MR1292784 (95h:20015)

12. R. Rouquier, The derived category of blocks with cyclic defect groups, in S. K”onig and A. Zimmermann, Derived Equivalences for Group Rings, Springer-Verlag, Berlin, 1998, pp. 199– 220. MR1649846

13. R. Rouquier, Block theory via stable and Rickard equivalences, Modular representation theory of finite groups (Charlottesville, VA, 1998), de Gruyter, Berlin, 2001, pp. 101–146. MR1889341 (2003g:20018)

14. J. Scopes, Cartan matrices and Morita equivalence for blocks of the symmetric groups, J. Algebra 142 (1991), 441–455. MR1127075 (92h:20023)

MR1990270 (2004f:16075) 16W50 16D90 16S50 Marcus, Andrei ^{(}R-CLUJMI^{)}; Modoi, Ciprian ^{(}R-CLUJMI^{)} Graded endomorphism rings and equivalences. (English. English summary) Comm. Algebra 31 (2003), no. 7, 3219–3249. The purpose of this paper is to extend to categories of modules graded by G-sets (where G is a group) certain results on equivalences between module categories. The authors consider, as a starting point, equivalences between subcategories of R-Mod and subcategories of S-Mod (where R is a ring, M a Σ-quasiprojective R-module, and S = End_{R}(M)) that were studied by J. L. García Hernández and the reviewer [Israel J. Math. 58 (1987), no. 3, 324–350; MR0917363 (89b:16041)]. Moreover, they also look at similar graded equivalences that appear in the study of Clifford theory for group-graded rings, where R = ⨁_{g∈G} R_{g} is a ring graded by a group G and M is a simple object of the category of G-graded R-modules and also at equivalences arising in the case when M is taken to be the canonical generator ⨁_{g∈G} R(g) of the category of graded modules, studied by T. Albu and C. Năstăsescu [J. Pure Appl. Algebra 59 (1989), no. 2, 125–150; MR1007917 (90i:16001)].

In order to obtain equivalences that unify these previous results, with graded modules on both sides of the equivalence, the authors consider a G-graded ring R, a G-graded Σ-quasiprojective R-module M, the graded endomorphism ring E = End_{R}(M), and a subgroup H of G. In this setup, they show that certain full subcategories of G/Hgraded R-modules associated with M are equivalent to a quotient category of G/H-graded E-modules determined by the idempotent G-graded ideal of E consisting of the endomorphisms which factor through a finitely generated submodule of M. The authors give some applications of these equivalences and show how many of the previous results fit in this more general setting.

J. L. Gómez Pardo (E-SACOM-AL)

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MR1961566 (2004c:16011) 16E30 16S34 20C20

Marcus, Andrei ^{(}R-CLUJMI^{)}

Tilting complexes for group graded algebras. J. Group Theory 6 (2003), no. 2, 175–193. Let G be a finite group, k a field of characteristic p > 0, and A a block of kG with abelian defect group D. A conjecture due to M. Broué states that if B is the Brauer correspondent block of kN_{G}(D), then the bounded derived categories of A and B are equivalent as triangulated categories.

One of the main tools for studying this conjecture is J. Rickard’s result [J. London Math. Soc. (2) 39 (1989), no. 3, 436–456; MR1002456 (91b:18012)] asserting that such a derived equivalence exists if and only if B is the endomorphism ring of a tilting complex T over A. The present paper gives a graded version of this fundamental theorem. It is obtained by using Keller’s alternative proof of Rickard’s result.

As an application, the author considers certain constructions of tilting complexes due to T. Okuyama [Sūrikaisekikenkyūsho Kōkyūroku No. 1149 (2000), 53–59; see MR1798398 (2001f:20025)MR1796364] which were employed to verify Broué’s conjecture for the principal p-block of SL(2, p^{n}) and for principal blocks with elementary abelian defect group of order 9 [see S. Koshitani and N. Kunugi, J. Algebra 248 (2002), no. 2, 575–604; MR1882112 (2002j:20021)]. The author discusses whether the derived equivalences given by these tilting complexes are compatible with p′-extensions of the group G. Moreover, he verifies Broué’s conjecture for the principal p-block of GL(2, p^{n}).

Lidia Angeleri Hügel (Varese)

[References]

Note: This list reflects references listed in the original paper as accurately as possible with no attempt to correct errors.

1. M. Broué. Equivalences of blocks of group algebras. In Finitedimensional algebras and related topics (Ottawa, 1992) (Kluwer Academic Publishers, 1991), pp. 1–26. MR1308978 (97c:20004)

2. E. C. Dade. Extending group-modules in a relatively prime case.

Math. Z. 186 (1984), 81–98. MR0735053 (85k:20019)

3. B. Keller. On the construction of triangle equivalences. In Derived equivalences for group rings (Springer-Verlag, 1998), pp. 155–176. MR1649844

4. B. Keller. Bimodule complexes via strong homotopy actions. Algebr. Represent. Theory 3 (2000), 357–376. MR1808132 (2002b:16018)

5. S. Koshitani and N. Kunugi. Broué’s conjecture holds for principal 3-blocks with elementary abelian defect group of order 9. J. Algebra 248 (2002), 575–604. MR1882112 (2002j:20021)

6. S. König and A. Zimmermann. Derived equivalences for group rings (Springer-Verlag, 1998). MR1649837 (2000g:16018)

7. A. Marcus. On equivalences between blocks of group algebras: reduction to the simple components. J. Algebra 184 (1996), 372– 396. MR1409219 (97m:20012)

8. A. Marcus. Equivalences induced by graded bimodules. Comm. Algebra 26 (1998), 713–731. MR1606217 (99a:16039)

9. A. Marcus. Representation theory of group graded algebras (Nova Science Publishers, 1999). MR1921424 (2003c:16056)

10. C. Năstăsescu. Some constructions over graded rings: applications. J. Algebra 120 (1989), 119–138. MR0977864 (90e:16004)

11. T. Okuyama. Some examples of derived equivalent blocks of finite groups. Preprint (1998).

12. T. Okuyama. Remarks on splendid tilting complexes. In Representation theory of finite groups and related topics (RIMS Kokyuroku, Kyoto University), pp. 53–59. MR1796364

13. T. Okuyama. Derived equivalence in SL(2, q). Preprint (2000).

14. J. Rickard. Morita theory for derived categories. J. London Math. Soc. (2) 39 (1989), 436–456. MR1002456 (91b:18012)

15. J. Rickard. Derived equivalences as derived functors. J. London Math. Soc. (2) 43 (1991), 37–48. MR1099084 (92b:16043)

16. J. Rickard. Bousfield localization for representation theorists. In Infinite length modules (Bielefeld, 1998) (Birkhäuser, 2000), pp. 273–283. MR1789220 (2002c:20016)

17. R. Rouquier. From stable equivalences to Rickard equivalences for blocks with cyclic defect. In Groups ’93 Galway/St. Andrews, vol. 2, London Math. Soc. Lecture Note Ser. 212 (Cambridge University Press, 1995), pp. 512–523. MR1337293 (96h:20021)

18. R. Rouquier. The derived category of blocks with cyclic defect groups. In Derived equivalences for group rings (Springer-Verlag, 1998), pp. 199–220. MR1649846

19. A. Zimmermann. Two-sided tilting complexes for Gorenstein or-

ders. J. Algebra 209 (1998), 585–621. MR1659883 (99m:16024)

MR1874441 (2002k:18004) 18A40 16W30 Madar, Adrian ^{(}R-CLUJMI^{)}; Marcus, Andrei ^{(}R-CLUJMI^{)} Frobenius functors and transfer. (English. English summary) Publ. Math. Debrecen 59 (2001), no. 3-4, 407–427. Two functors F: A → B and G: B → A are called a Frobenius pair if G is both a right and a left adjoint of F. They will define natural transformations Tr_{F} and Tr_{G}. The authors prove certain implications among eleven statements about an object of A, which they call Higman’s criterion. Applications are given in the settings of strongly graded rings, Frobenius extensions of rings, essential submodules, Hopf-Galois extensions, Hopf subalgebras, and modules over Hopf algebras. Finally they characterize Frobenius pairs between module categories.

E. J. Taft (1-RTG)

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MR1825806 (2002b:20016) 20C20 16S35 16W50 20C25

Marcus, Andrei ^{(}R-CLUJMI^{)}

Twisted group algebras, normal subgroups and derived equivalences. (English. English summary) Special issue dedicated to Klaus Roggenkamp on the occasion of his 60th birthday. Algebr. Represent. Theory 4 (2001), no. 1, 25–54. The following notations are used in the paper under review: H is a finite group; K its normal subgroup in H; G = H/K; 𝒪 is a complete discrete valuation ring with algebraically closed field k = 𝒪/J(𝒪) of characteristic p > 0 and other natural additional properties; 𝒪_{α}H is a twisted group algebra of the group H, where α ∈ Z^{2}(H, 𝒪^{×}) is a cocycle, and 𝒪_{α}H is regarded in the paper as a G-graded algebra; G_{b} = N/K, where b is a block idempotent of 𝒪_{α}K with a defect group D, N = {x ∈ H| ^{x}b = b}; D̃ = D ∩ K; Ĥ is the group of homogeneous (graded) units of the algebra 𝒪_{α}H. The author assumes that if D̂ is abelian, then there is a G_{b}-graded Rickard correspondence between b𝒪_{α}Hb and b′𝒪_{α}N_{H}(D̃)b′, where b′ is the Brauer correspondent of b. This hypothesis is called an extended version of Broué’s conjecture. The author proves that this conjecture implies Dade’s inductive conjecture if D is abelian and also that these conjectures hold for p′-extensions of blocks with cyclic defect groups.

Sergey P. Strunkov (Moscow)

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7. Dade, E. C.: Block extensions, Illinois J. Math. 17 (1973), 198– 273. MR0327884 (48 #6226)

8. Dade, E. C.: Extending group-modules in a relatively prime case, Math. Z. 186 (1984), 81–98. MR0735053 (85k:20019)

9. Dade, E. C.: Counting characters in blocks, II, J. Reine Angew. Math. 448 (1994), 97–190. MR1266748 (95a:20007)

10. Dade, E. C.: Counting characters in blocks, 2.9, In: R. Solomon (ed.), Representation Theory of Finite Groups, Ohio State Univ. Math. Res. Inst. Publ. 6, de Gruyter, Berlin, 1997. MR1611009 (99b:20016)

11. Dade, E. C.: Counting characters in blocks with cyclic defect groups, I, J. Algebra 186 (1996), 934–969. MR1424599 (98b:20013)

12. Fan, Y.: Relative local control and the block source algebras, Science in China (Ser. A) 40 (1997), 785–798. MR1481678 (98i:20004)

13. Harris, M. E.: Splendid derived equivalences for blocks of finite groups, J. London Math. Soc. 60(2) (1999), 71–82. MR1721816 (2000k:20014)

14. Harris, M. E. and Knörr, R.: Brauer correspondences for covering blocks of finite groups, Comm. Algebra 13 (1985), 1213–1218. MR0780645 (86e:20012)

15. Knörr, R. and Robinson, G. R.: Some remarks on a conjecture of Alperin, J. London Math. Soc. 39 (1989), 48–60. MR0989918 (90k:20020)

16. König, S. and Zimmermann, A.: Derived Equivalences for Group Rings, Springer-Verlag, Berlin, 1998. MR1649837 (2000g:16018)

17. Külshammer, B., Okuyama, T. and Watanabe, A.: A lifting theorem with applications to blocks and source algebras, Preprint, 1999. MR1783927 (2001g:20013)

18. Külshammer, B. and Puig, L.: Extensions of nilpotent blocks, Invent. Math. 102 (1990), 17–71. MR1069239 (91i:20009)

19. Linckelmann, M.: On derived equivalences and local structure of blocks of finite groups, Turkish J. Math. 22 (1998), 93–107. MR1631770 (99k:20021)

20. Linckelmann, M.: On splendid derived and stable equivalences between blocks of finite groups, Preprint, 1998. MR1848975 (2002i:20013)

21. Marcus, A.: On equivalences between blocks of group algebras: reduction to the simple components, J. Algebra 184 (1996), 372– 396. MR1409219 (97m:20012)

22. Marcus, A.: Equivalences induced by graded bimodules, Comm. Algebra 26 (1998), 713–731. MR1606217 (99a:16039)

23. Marcus, A.: Derived equivalences and Dade’s invariant conjecture, J. Algebra 221 (1999), 513–527. MR1726712 (2001b:20019)

24. Marcus, A.: Representation Theory of Group Graded Algebras, Nova Science Publishers, New York, 1999. MR1921424 (2003c:16056)

25. Puig, L.: On the Local Structure of Morita and Rickard Equivalences between Brauer Blocks, Birkhäuser Verlag, Basel, 1999. MR1707300 (2001d:20006)

26. Puig, L.: Notes on O^{∗}-groups, Preprint, 1998.

27. Rickard, J.: Derived categories and stable equivalences, J. Pure Appl. Algebra 61 (1989), 303–317. MR1027750 (91a:16004)

28. Rickard, J.: Splendid equivalences: Derived categories and permutation modules, Proc. London Math. Soc. 72 (1996), 331–358. MR1367082 (97b:20011)

29. Rouquier, R.: The derived category of blocks with cyclic defect groups, In: S. König and A. Zimmermann (eds), Derived Equivalences for Group Rings, Springer-Verlag, Berlin, 1998, pp. 199–220. MR1649846

30. Thévenaz, J.: G-Algebras and Modular Representation Theory, Clarendon Press, Oxford, 1995. MR1365077 (96j:20017)

MR1953670 (2003m:18004) 18A40 Madar, Adrian ^{(}R-CLUJMI^{)}; Marcus, Andrei ^{(}R-CLUJMI^{)} Frobenius functors and functor categories. (English. English summary) Studia Univ. Babeş-Bolyai Math. 45 (2000), no. 4, 75–85. 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.”

MR1793749 (2002a:16049) 16W50 Marcus, Andrei ^{(}R-CLUJM^{)}; Modoi, Ciprian ^{(}R-CLUJM^{)} Groups of homomorphism graded by G-sets. Ital. J. Pure Appl. Math. No. 8 (2000), 115–126. Let G be a group, R a G-graded ring, and M and N G-graded Rmodules. In general, Hom_{R}(M, N) is not a G-graded module, but we can construct a G-graded submodule, namely

HOM_{R}(M, N) = ⨁_{g∈G} Hom_{R-gr}(M, N(g)).

The subscript R-gr refers to grade preserving homomorphisms, and N(g) is the g-suspension of N. J. L. Gómez Pardo, G. Militaru and C. Năstăsescu [Comm. Algebra 22 (1994), no. 8, 3171–3181; MR1272380 (95c:16054)] proved that Hom_{R}(M, N) is the completion of HOM_{R}(M, N) in the finite topology, and their results lead to necessary and sufficient conditions for the two Hom’s to be equal.

Now let X be a left G-set, let M be G-graded as before, and N be graded by the G-set X, i.e.

N = ⨁_{x∈X} N_{x} with R_{g}M_{x} ⊂ M_{gx}.

A particular case is X = G/K, where K is a (not necessarily normal) subgroup of G. In this case, we can define an R-submodule HOM_{X,R}(M, N) of Hom_{R}(M, N) that is graded by X, and the main result states that Hom_{R}(M, N) is the completion of HOM_{X,R}(M, N) with respect to the finite topology.

For an intermediate subgroup K ⊂ H ⊂ G, the authors also study the relation between HOM_{G/K,R}(M, N) and HOM_{G/H,R}(M, N). They happen to be equal if and only if M is small as an R-module.

Stefaan Caenepeel (Brussels)

MR1784549 (2001j:20014) 20C20 16D90

Marcus, Andrei ^{(}R-CLUJMI^{)}

Blocks of normal subgroups and Morita equivalences. International Algebra Conference (Iaşi, 1998). Ital. J. Pure Appl. Math. No. 7 (2000), 77–84. Let G be a finite group and N a normal subgroup of G. It is proved in the paper that the Clifford extensions resulting from the inductive conjecture by E. C. Dade [in Representation theory of finite groups (Columbus, OH, 1995), 45–59, de Gruyter, Berlin, 1997; MR1611009 (99b:20016)] associated to blocks of the twisted group algebra of N over a completed discrete valuation ring F are invariant under Morita equivalences induced by graded bimodules, where F is big enough for G. Also investigated is the local structure of the graded Morita equivalences between indecomposable graded subalgebras of the group algebra F G.

Ji Ping Zhang (PRC-BJ-SM)

MR1747369 (2001c:16077) 16W50 16D20

Marcus, Andrei ^{(}R-CLUJMI^{)}

Clifford theory of bimodules. Comm. Algebra 28 (2000), no. 4, 2029–2041. The work relates to the structural theory of bimodules over algebras strongly graded by a finite group. Initially a Clifford correspondence for indecomposable bimodules and indecomposable projective modules over some endomorphism algebra is obtained. Then these results and the Green correspondence are applied to situations over a complete local commutative Noetherian ring with residue field of positive characteristic. For a p-subgroup of G, Brauer’s theorem is proved.

Andrew Borisovich Verëvkin (Simbirsk)

[References]

Note: This list reflects references listed in the original paper as accurately as possible with no attempt to correct errors.

1. J.L. Alperin, Local representation theory, Cambridge Univ. Press, 1986. MR0860771 (87i:20002)

2. P. Boisen, The representation theory of fully group-graded algebras, J. Algebra 151 (1992), 160–179. MR1182020 (93i:20005)

3. E.C. Dade, Block extensions, Illinois J. Math. 17 (1973), 198–272. MR0327884 (48 #6226)

4. E.C. Dade, The equivalence of various generalizations of grouprings and modules, Math. Z. 181 (1982), 335–344. MR0678889 (84a:16018)

5. E.C. Dade, Block of fully graded rings, Pacific J. Math., Olga Taussky-Todd memorial issue (1998), 85–122. MR1610839 (99b:16066)

6. J. Haefner, Graded equivalence theory with applications, J. Algebra 172 (1995), 385–424. MR1322411 (96f:16052)

7. A. Marcus, Static modules and Clifford theory for strongly graded rings, Publ. Math. Debrecen 43 (1993), 303–314. MR1229677 (94h:16079)

8. A. Marcus, Thesis, Bucharest University (in Romanian), 1994.

9. A. Marcus, Equivalences induced by graded bimodules, Comm. Algebra 26 (1998), 713–731. MR1606217 (99a:16039)

10. C. Năstăsescu, Ş. Raianu, F. van Oystaeyen, Graded modules over G-sets, Math. Z. 203 (1990), 605–627. MR1044067 (91h:16071)

11. C. Năstăsescu and F. van Oystaeyen, Graded Ring Theory, North-Holland, Amsterdam, 1982. MR0676974 (84i:16002)

12. S.J. Witherspoon, A module-theoretic approach to Clifford theory

for blocks, preprint (1997). MR1646212 (2000e:20022)

MR1921424 (2003c:16056) 16W50 16G70 20C15 Marcus, Andrei

⋆Representation theory of group graded algebras. Nova Science Publishers, Inc., Commack, NY, 1999. xiv+212 pp. ISBN 1-56072-750-0 From the preface: “The use of strongly graded algebras in the representation theory of finite groups was originated by E. Dade in the late 1960s in his important work on Clifford theory.

“Let G = H/N and choose a system {h_{x}| x ∈ G} of representatives for the left cosets of N in H. Denoting kN = R_{1}, kNh_{x} = R_{x}, then R = ⨁_{x∈G} R_{x} and R_{x}R_{y} = R_{xy}. Dade called a ring with these properties strongly G-graded, and established many of their basic properties. Another important example of a G-graded algebra (not necessarily strongly graded) is the endomorphism algebra E = End_{kH} (kH ⊗_{kN} V) of the kH-module induced from V.

“Rings graded by groups or semigroups have become interesting objects of study in their own right, and the following may be regarded as the principal guiding problem.

“If (P) is a property concerning modules, what is the relationship between the properties (P) in R-mod, (P) in R_{1}-Mod and (gr-P) in the category R-Gr of G-graded modules?

“Moreover, many results have been extended to the more general context of Hopf algebra actions on rings. We concentrate, however, on presenting results which are closely connected to the representation theory of finite groups. In this respect, the book is fairly self-contained, but we rely on various sources for basic facts on related topics. References are given every time when this problem occurs.”

Contents: Chapter I. Modules over graded rings; Chapter II. Clifford theory; Chapter III. Clifford extensions; Chapter IV. Auslander-Reiten theory; Chapter V. Equivalences induced by graded bimodules.

MR1726712 (2001b:20019) 20C20 20C05

Marcus, Andrei ^{(}R-CLUJ-MI^{)}

Derived equivalences and Dade’s invariant conjecture. J. Algebra 221 (1999), no. 2, 513–527. The author defines in the paper under review a special kind of equivalence between derived categories of group rings, so-called Gequivariant derived equivalence. For G-equivariant splendid equivalences of derived categories of blocks of group rings with abelian defect group he proves that Dade’s invariant conjecture holds.

The setting is the following: Let H be a finite group with normal subgroup K and quotient G = H/K. Let R be a complete discrete valuation domain of characteristic 0 with residue field of characteristic p and suppose that the residue field is algebraically closed and that the field of fractions is big enough. Let b be a block of RK with defect group P and let c be its Brauer correspondent in RN_{K}(P). If b is G-invariant, then RHb and RN_{G}(P)c are G-graded with identity component the above-defined blocks of K and N_{K}(P). A tilting complex C in D^{b}(RKb ⊗_{R} RN_{K}(P)c^{op}) is G-equivariant if “conjugation” by the homogeneous components of RHb and RN_{H}(P)c fixes C.

Finally the author shows that the complexes constructed by R. Rouquier [in Derived equivalences for group rings, 199–220, Lecture Notes in Math., 1685, Springer, Berlin, 1998; see MR1649837 (2000g:16018)MR1649846] for blocks of cyclic defect groups are G-equivariant.

Alexander Zimmermann (F-PCRD-LFD)

[References]

Note: This list reflects references listed in the original paper as accurately as possible with no attempt to correct errors.

1. M. Broué, Equivalences of blocks of group algebras, in ”Proceedings of the NATO Advanced Research Workshop on Representation of Algebras and Related Topics,” Kluwer Academic, Dordrecht, 1991. MR1308978 (97c:20004)

2. E. C. Dade, Block extensions, Illinois, J. Math. 17 (1973), 198– 273. MR0327884 (48 #6226)

3. E. C. Dade, Extending group-modules in a relatively prime case, Math. Z. 186 (1984), 81–98. MR0735053 (85k:20019)

4. E. C. Dade, Counting characters in blocks, 2.9, in ”Proceedings of the 1995 Conference on Representation Theory at Ohio State University.” MR1266748 (95a:20007)

5. E. C. Dade, Counting characters in blocks with cyclic defect groups, J. Algebra 186 (1996), 934–969. MR1424599 (98b:20013)

6. D. Happel, ”Triangulated Categories in the Representation Theory of Finite Dimensional Algebras,” Cambridge Univ. Press, Cambridge, UK, 1986. MR0935124 (89e:16035)

7. M. Harris, Splendid derived equivalences for blocks of finite groups, J. London Math. Soc., to appear. MR1721816 (2000k:20014)

8. B. Külshammer and L. Puig, Extensions of nilpotent blocks, Invent. Math. 102 (1990), 17–71. MR1069239 (91i:20009)

9. B. Knörr and G. R. Robinson, Some remarks on a conjecture of Alperin, J. London Math. Soc. 39 (1989), 48–60. MR0989918 (90k:20020)

10. M. Linckelmann, On derived equivalences and local structure of blocks of finite groups, preprint, 1996. MR1631770 (99k:20021)

11. M. Linckelmann, On splendid derived and stable equivalences between blocks of finite groups, preprint, 1997. MR1848975 (2002i:20013)

12. A. Marcus, On equivalences between blocks of group algebras: Reduction to the simple components, J. Algebra 184 (1996), 372– 396. MR1409219 (97m:20012)

13. A. Marcus, Equivalences induced by graded bimodules, Comm. Algebra 26 (1998), 713–731. MR1606217 (99a:16039)

14. J. Rickard, Derived categories and stable equivalences, J. Pure Appl. Algebra 61 (1989), 303–317. MR1027750 (91a:16004)

15. J. Rickard, Splendid equivalences: derived categories and permutation modules, Proc. London Math. Soc. 72 (1994), 331–358. MR1367082 (97b:20011)

16. R. Rouquier, The derived category of blocks with cyclic defect groups (S. König and A. Zimmermann, Eds.), in ”Derived Equivalences for Group Rings,” Springer-Verlag, Berlin, 1998. MR1649846

17. J. Thévenaz, ”G-Algebras and Modular Representation Theory,” Clarendon Press, Oxford, 1995. MR1365077 (96j:20017)

MR1855326 (2002g:16070) 16W50 20C20

Marcus, Andrei ^{(}R-CLUJMI^{)}

Induction of graded interior algebras. (English. English summary) Studia Univ. Babeş-Bolyai Math. 43 (1998), no. 1, 45–56. 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 by Lluis Puig.”

Vladimir V. Bavula (4-SHEF-PM)

MR1811346 (2001k:20017) 20C20

Marcus, Andrei ^{(}R-CLUJMI^{)}

Groups, algebras, representations. (English. English summary) Proceedings of the Summer School on Representation Theory of Algebras, Finite and Reductive Groups (Cluj-Napoca, 1997), 1–32, Babeş-Bolyai Univ. Fac. Math. Comput. Sci. Res. Semin., ‘‘Babeş-Bolyai’’ Univ., Cluj-Napoca, 1998. This paper is a summary of an introductory talk given in 1997 at the Summer School on Representation Theory of Algebras, Finite and Reductive Groups in Cluj, Romania. It is concerned with representation theory of finite groups, starts at the very beginning and ends with some of the main conjectures which are at the center of research. To be more precise the author deals with p-modular systems and modules, lifting idempotents, self-injective and symmetric algebras, induction and restriction, relative projectivity, vertices and sources, defect groups, Green and Brauer correspondences, Clifford theory, characters and numerical invariants of blocks.

Dealing with so many different topics means that the paper cannot contain many proofs, so only short but instructive proofs are given. To understand the concepts and methods the reader has to have a good background in algebra. This is particularly necessary since much of the material is presented in its most general form. However, in reading the paper the non-experts may pick up some flavour of the theory.

Wolfgang Willems (D-MAGM-AG)

MR1746526 (2001c:16014) 16D90 20C20

Marcus, Andrei ^{(}R-CLUJ-MI^{)}

Equivariant Morita equivalences between blocks of group algebras. Mathematica 40(63) (1998), no. 2, 227–234. As this paper continues the author’s earlier work [Comm. Algebra 26 (1998), no. 3, 713–731; MR1606217 (99a:16039)], the contents of the earlier review will not be duplicated here.

So, with 𝒪 denoting a complete discrete valuation ring with algebraically closed residue field k of characteristic p > 0 and field of quotients 𝒦 of characteristic 0, and with G denoting a finite group, the concept of a strongly G-graded algebra R is available. Such an object has a subalgebra R_{1} corresponding to the identity element of G. The concept of an equivariant Morita equivalence between R_{1} and S_{1} is developed for R and S strongly G-graded algebras.

The first result is that such a Morita equivalence induces a Gisomorphism between the Cartan-decomposition triangles, also known as the cde triangles. The applications of this to modular representation theory are discussed. The relationship of this material and the socalled splendid Morita equivalences is developed.

P. Donovan (5-NSW)

MR1701257 (2000f:16011) 16D90

Marcus, Andrei ^{(}R-CLUJ-MI^{)}

Stable equivalences, crossed products and symmetric algebras. Mathematica 40(63) (1998), no. 1, 123–129. Let 𝒪 be a complete discrete valuation ring with residue field k of positive characteristic p. All 𝒪-algebras are assumed to be finitely generated 𝒪-free modules. Suppose that G is a finite group and R, S are strongly graded 𝒪-algebras. Suppose that M is an (R_{1}, S_{1})- bimodule which is projective as R_{1}- and as S_{1}-module. Also, let N be an (S_{1}, R_{1})-bimodule with similar properties. The aim of the paper is to present two cases in which the pair (M, N) induces an 𝒪-stable equivalence between R_{1}, S_{1} if the pair

(R ⊗_{𝒪} S^{op}) ⊗_{Δ(R,S)} M, (S ⊗_{𝒪} R^{op}) ⊗_{Δ(S,R)} N

induces a graded Morita equivalence between R, S. Here Δ(R, S) is a G-graded algebra such that Δ(R, S)_{g} = R_{g} ⊗ S_{g^{−1}}, g ∈ G.

V. A. Artamonov (RS-MOSCM-AL)

MR1657479 (99m:16069) 16W50 16D40 20C05

Marcus, Andrei ^{(}R-CLUJMI^{)}

Projective modules over twisted group algebras of p-solvable groups. (English. English summary) Publ. Math. Debrecen 53 (1998), no. 3-4, 367–374. Let 𝒪 be a complete discrete valuation ring with algebraically closed residue field of characteristic p > 0, let G be a finite psolvable group with Hall p′-subgroup H, and let 1 = N_{0} < N_{1} < ⋯ < N_{r} = G be a chain of normal subgroups of G such that N_{i}/N_{i−1} is either a p-group or a p′-group, for i = 1, ⋯, r. Moreover, let R = ⨁_{g∈G} R_{g} be a strongly G-graded 𝒪-algebra, let R_{H} = ⨁_{h∈H} R_{h}, and let M be an indecomposable R_{1}-module. Finally, let (R|R ⊗_{R_{1}} M)-mod denote the full subcategory of R-mod consisting of direct summands of finite direct sums of copies of R ⊗_{R_{1}}M. Marcus shows the following: (i) Let V ∈ (R|R ⊗_{R_{1}}M)-mod be an indecomposable R-module. Then there exists an indecomposable R_{H}-module W ∈ (R_{H}|R_{H} ⊗_{R_{1}} M)-mod such that V ≅ R ⊗_{R_{H}} W. (ii) Let W, W′ ∈ (R_{H}|R_{H} ⊗_{R_{1}} M)-mod be indecomposable R_{H}-modules such that the restrictions of W and W′ to R_{H∩N_{i}} are isotypic for i = 0, ⋯, r, and R ⊗_{R_{H}} W ≅ R ⊗_{R_{H}} W′. Then R ⊗_{R_{H}} W is an indecomposable R-module, and there is g ∈ N_{G}(H) such that W′ ≅ R_{gH} ⊗_{R_{H}} W. These results are graded versions of theorems proved by H. Fottner and the reviewer [“Remarks on G-algebras”, J. London Math. Soc. (2), to appear] which in turn generalize results by P. Fong and I. M. Isaacs.

Burkhard Külshammer (D-FSUMI-IM)

MR1653709 (99m:16011) 16E40 16W50

Marcus, Andrei ^{(}R-CLUJM^{)}

Homology of fully graded algebras, Morita and derived equivalences. (English. English summary) Ring theory (Miskolc, 1996). J. Pure Appl. Algebra 133 (1998), no. 1-2, 209–218. Let K be a commutative ring and let G be a group. A G-graded K-algebra R = ⨁_{g∈G} R_{g} is called a strongly G-graded K-algebra if R_{g}R_{h} = R_{gh} for all g, h ∈ G.

The main result of the paper is to prove that if strongly graded K-algebras R and S are graded Morita equivalent, then the isomorphisms between their Hochschild and cyclic homologies are induced by isomorphisms between the second terms of the corresponding spectral sequences.

Vladimir V. Bavula (4-SHEF-PM)

MR1626590 (99c:16043) 16W50 16D20 16S50 16W20

Marcus, Andrei ^{(}R-CLUJMI^{)}

On Picard groups and graded rings. Comm. Algebra 26 (1998), no. 7, 2211–2219. This paper gives an alternate and possibly more direct proof of a previous result of Beattie and del Río concerning Picard groups and strongly graded rings. Specifically, let k be a commutative ring, G a group, R a strongly G-graded k-algebra so that R_{1} denotes the coefficient ring of R. In general, the Picard group of a ring consists of those isomorphism classes of invertible bimodules. When the ring is graded, such as R in this case, there are additional Picard type groups to be considered. Beattie and del Río used Pic_{k}^{gr}(R) to denote the group of isomorphism classes of G-graded invertible bimodules. In addition, since G acts on Pic_{k}(R_{1}) by ^{g}[P] = [R_{g} ⊗ P ⊗ R_{g^{−1}}], the notation Pic_{k}(R_{1})^{G} refers to the group of G-invariant elements of Pic_{k}(R_{1}). Using U(Z(R_{1})) to denote the units of the center of R_{1} and H^{1}, H^{2} to denote the first and second cohomology groups, the aforementioned result of Beattie and del Río is as follows: Theorem. There is an 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}))).

Beattie and Río proved this result by investigating the various actions of the group G and by using the smash product. They also proved this result for rings with local units. In the paper under review, the author restricts his attention to unital rings and re-proves the above result using graded bimodules (over G-graded rings R and S), graded Morita equivalence, extendibility of modules and the construction of a strongly G-graded ring Δ(R, S). These tools are developed in an earlier paper by Marcus and a paper by Dade.

Jeremy Haefner (1-CO2)

MR1606217 (99a:16039) 16W50 16D20

Marcus, Andrei ^{(}R-CLUJMI^{)}

Equivalences induced by graded bimodules. (English. English summary) Comm. Algebra 26 (1998), no. 3, 713–731. Let k be a commutative ring, G a finite group, and R and S strongly (or fully) G-graded rings. The author investigates graded equivalences, graded derived equivalences and graded stable equivalences between R and S. These equivalences are induced by G-graded R-S bimodules, and the author gives new characterizations of these equivalences by viewing these bimodules as left modules over the strongly G-graded k-algebra

Δ = ⨁_{g∈G}(R_{g} ⊗_{k} S_{g^{−1}}).

Such equivalences also produce a Clifford theory; i.e, the induction, restriction and truncations functors commute with these equivalences.

Jeremy Haefner (1-CO2)

MR1621824 (99e:16058) 16W50

Marcus, Andrei ^{(}R-CLUJ-MI^{)}

Green functors associated to a strongly graded algebra. Mathematica 39(62) (1997), no. 2, 257–263. Let G be a finite group, C a complete discrete valuation ring with residue field k of characteristic p > 0 and R a strongly G-graded C-algebra. Denote by Δ = Δ(R, R) = ⨂_{x∈G}(R_{g} ⊗_{C} R_{g}^{op}). If H is a subgroup of G, denote by A_{Z}(R, H) the Green ring of Δ_{H} and by R_{Z}(R, H) the Grothendieck ring of Δ_{H}. The main results are: (1) R_{Z}(R, –) and A_{Z}(R, –) are Green functors over Z; (2) the Green functors R_{Z}(R, –) and A_{Z}(R, –) are invariant under derived equivalences induced by certain two-sided tilting complexes.

P. Wauters (Diepenbeek)

MR1623156 (99b:16067) 16W50 20C20

Marcus, Andrei ^{(}R-CLUJ-MI^{)}

Group-graded algebras, adjoint functors and the Green correspondence. (English. English summary) Studia Univ. Babeş-Bolyai Math. 41 (1996), no. 2, 87–94. Summary: “Let G be a finite group and k a commutative ring. In the context of fully G-graded k-algebras, versions of the Green correspondence and of the Burry-Carlson-Puig theorem are presented, generalizing recent results of M. Auslander and M. Kleiner.”

MR1606844 16W50

Marcus, Andrei ^{(}R-CLUJ-MI^{)}

Tensor products of fully graded algebras and applications. Mathematica 38(61) (1996), no. 1-2, 133–139.

{There will be no review of this item.}

MR1428460 (97i:20012) 20C20 16W50

Marcus, Andrei ^{(}R-CLUJ-C^{)}

Graded equivalences and Broué’s conjecture. (English. English summary) Representation theory of groups, algebras, and orders (Constanţa, 1995). An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat. 4 (1996), no. 2, 107–126. Summary: “This paper is an extended version of a talk given at the Constan’ta conference. Our aim is to present some results obtained in two papers of ours [J. Algebra 184 (1996), no. 2, 372–396MR 97m:20012; “Blocks of normal subgroups and equivalences induced by graded bimodules”, to appear] concerning the reduction of Broué’s conjecture to the case of simple groups. This reduction involves an investigation of Morita equivalences, derived equivalences and stable equivalences of Morita type between two G-graded algebras R and S, which are induced by G-graded (R, S)-bimodules or complexes of G-graded bimodules.”

MR1409219 (97m:20012) 20C20

Marcus, Andrei ^{(}R-CLUJ-C^{)}

On equivalences between blocks of group algebras: reduction to the simple components. (English. English summary) J. Algebra 184 (1996), no. 2, 372–396. 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 𝒪-algebras 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 𝒪-linear 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 Δ-module. 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, Ỹ -invariant block idempotents, respectively, and set R = e𝒪X̃ and S = f𝒪Ỹ. Set G = X̃/X and identify G with Ỹ /Y via α. Then R and S are Gcrossed 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.

M. E. Harris (1-ILCC)

MR1607848 (98k:16059) 16W50 16D90 16S90

Marcus, Andrei ^{(}R-CLUJ-MI^{)}

Modules graded by G-sets and Clifford theory. Mathematica 37(60) (1995), no. 1-2, 131–154. The author studies modules graded by G-sets, where G is a group, and their applications to induced representations. The main tools used in this research are certain functors between the categories of graded rings and graded modules, defined by homomorphisms of groups and G-sets, and special attention is devoted to these functors in the case of modules graded by transitive G-sets. In the last part of the paper the author investigates Clifford theory, extending results of C. Năstăsescu, S. X. Liu and F. M. J. Van Oystaeyen [Math. Z. 207 (1991), no. 3, 341–358; MR1115168 (93b:16078)] and B. R. Zhou [J. Algebra 139 (1991), no. 1, 172–189; MR1106346 (92f:16051)], and also obtaining a graded version of Morita’s theorem on equivalences.

J. L. Gómez Pardo (E-SACOM-AL)

MR1352540 (96h:16030) 16S35 16D90 16W50

Mărcuş, Andrei ^{(}R-CLUJM^{)}

Clifford theory for projective modules over strongly graded rings. Comm. Algebra 23 (1995), no. 12, 4393–4404. This article develops the theory of projective covers and radicals of modules over a ring which is strongly graded by a finite group. The invariance of the Grothendieck group of indecomposable projective modules under reduction by the radical (under some conditions) is proved. The author also studies the behavior of Clifford inertial indexes of such modules. In the last section connections between the semisimplicity of a graded ring and the extendibility of certain modules are established.

Andrew Borisovich Verëvkin (Simbirsk)

MR1324931 (96e:16059) 16W50

Mărcuş, Andrei ^{(}R-CLUJM^{)}

Indecomposable modules and Clifford extensions. Mathematica (Cluj ) 35(58) (1993), no. 2, 169–173. Let G be a finite group, and R a strongly G-graded ring. Consider a left R_{1}-module V satisfying the condition that R_{x} ⊗_{R_{1}} V ≅ V for all x ∈ G. Then E = End_{R}(R ⊗_{R_{1}} V)^{op} is a crossed product of E_{1} ≅ End_{R_{1}}(V)^{op} and G. E. C. Dade [J. Algebra 72 (1981), no. 2, 374–403; MR0641332 (84g:20009); Math. Z. 186 (1984), no. 1, 81–98; MR0735053 (85k:20019)] has studied properties of E. The author puts Dade’s theory into a more general framework. The starting point is that R ⊗ E is a G-graded (R, E)-bimodule.

Let A and B be crossed products; the author constructs another crossed product C such that the category of G-graded (A, B)- bimodules is equivalent to the category of C-modules. The results of Dade, as well as some results of P. Schmid [J. Algebra 119 (1988), no. 1, 185–212; MR0971353 (89k:20014)] appear as special cases. Some other applications are given.

It is a pity that the paper is written in such a compact way. Much of the terminology and notation remains unexplained, and this makes it hard for the reader to read the paper.

Stefaan Caenepeel (Brussels)

MR1274276 (95a:20010) 20C20

Mărcuş, Andrei ^{(}R-CLUJM^{)}

Indecomposable modules and Clifford extensions. (English. English summary) Proceedings of the International Conference on Group Theory (Timişoara, 1992). An. Univ. Timişoara Ser. Ştiinţ. Mat. 1993, Special Issue, 125–131. Let N be a normal subgroup of a finite group H, let G = H/N, let k be a field, and let V be a G-invariant indecomposable kN-module. By standard constructions and under suitable assumptions, there exists a cohomology class ω ∈ H^{2}(G, K^{∗}), where K is some field extension of k, such that V extends to H if ω vanishes. The present paper gives some additional information about ω when G is a p-group.

Jacques Thévenaz (CH-LAUS)

MR1229677 (94h:16079) 16W50 20C15

Mărcuş, Andrei ^{(}R-CLUJ^{)}

Static modules and Clifford theory for strongly graded rings. (English. English summary) Publ. Math. Debrecen 42 (1993), no. 3-4, 303–314. Let R = ⨁_{x∈G} R_{x} be a strongly G-graded ring and let M = ⨁_{x∈G} M_{x} be a G-graded left R-module which is gr-indecomposable of finite length as a left graded module, where G is a finite group; further let E = End_{R}(M)^{op}. Then M is a G-graded (R, E)-bimodule. It is well known that if M is gr-simple, then the category (RgM)-mod of Rmodules generated by M is equivalent with E-mod, via the functors Hom_{R}(M, −) and M ⊗_{E} −. The aim of this paper is to obtain a direct Clifford correspondence in the case M is gr-indecomposable: More precisely, let J_{gr}(E) be the graded Jacobson radical of E and let D = E/J_{gr}(E). By using the notion of static modules and a graded version of Fitting’s lemma, the composition functor D ⊗_{E} Hom_{R}(M, −) induces an isomorphism between the Grothendieck groups associated to the category (R/M)-mod of R-modules which divide a finite direct sum of copies of M and to the category (D/D)-mod of finitely generated projective D-modules. This isomorphism is compatible with induction from subgroups of G.

Hidetoshi Marubayashi (J-NARU)

MR1343293 16W50

Mărcuş, Andrei ^{(}R-CLUJ^{)}

Compounded Clifford correspondences for modules over graded rings. (English. English summary) Proceedings of the Conference on Algebra (Cluj-Napoca, 1991), 73–92, Preprint, 92-1, ‘‘Babeş-Bolyai’’ Univ., Cluj-Napoca, 1992.

{This item will not be reviewed individually. For details of the collection in which this item appears see MR1343278 (96c:16002) .}

MR1343292 16W50

Mărcuş, Andrei ^{(}R-CLUJ^{)}

Extensions of indecomposable modules. (English. English summary) Proceedings of the Conference on Algebra (Cluj-Napoca, 1991), 61–72, Preprint, 92-1, ‘‘Babeş-Bolyai’’ Univ., Cluj-Napoca, 1992.

{This item will not be reviewed individually. For details of the collection in which this item appears see MR1343278 (96c:16002) .} MR1343291 16W50

Mărcuş, Andrei ^{(}R-CLUJ^{)}

Clifford theory and the Loewy length of induced modules. (English. English summary) Proceedings of the Conference on Algebra (Cluj-Napoca, 1991), 55–60, Preprint, 92-1, ‘‘Babeş-Bolyai’’ Univ., Cluj-Napoca, 1992.

{This item will not be reviewed individually. For details of the collection in which this item appears see MR1343278 (96c:16002) .}

MR1202617 (95c:20091) 20M10

Mărcuş, Andrei ^{(}R-CLUJ^{)}

Retract extensions of completely simple semigroups by nil semigroups. Mathematica (Cluj ) 34(57) (1992), no. 1, 37–41. The main theorem reads as follows: S is an Archimedean semigroup with an idempotent such that, for all x, y ∈ S, y ∈ xS^{1} implies y^{m} ∈ x^{2}S^{1} and y ∈ S^{1}x implies y^{n} ∈ S^{1}x^{2} for some m, n if and only if S is a retract extension of a completely simple semigroup by a nil semigroup. This extends earlier results on various kinds of (weakly) exponential semigroups.

L. Márki (Budapest)

MR1138697 (93g:16006) 16D80 Mărcuş, Andrei Invariant extensions of simple modules. Stud. Cerc. Mat. 43 (1991), no. 1-2, 37–42. Summary: “Let R = ⨁_{g∈G} R_{g} be a strongly G-graded ring and M be a G-invariant simple R_{1}-module which extends to an R_{H}-module, where H is a normal subgroup of G. We search for conditions which guarantee the existence of a G-invariant extension of M to R_{H}.”