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Let G be a finite group and k a field of characteristic p. It is conjectured in a paper of the first author and John Greenlees that the thick subcategory of the stable module category StMod (k G) consisting of modules whose cohomology is finitely generated over H ⁎ (G , k) is generated by finite dimensional modules and modules with no cohomology. If the centraliser of every element of order p in G is p -nilpotent, this statement follows from previous work. Our purpose here is to prove this conjecture in two cases with non p -nilpotent centralisers. The groups involved are Z / 3 r × Σ 3 (r ⩾ 1) in characteristic three and Z / 2 × A 4 in characteristic two. As a consequence, in these cases the bounded derived category of C ⁎ B G (cochains on BG with coefficients in k) is generated by C ⁎ B S , where S is a Sylow p -subgroup of G. [ABSTRACT FROM AUTHOR]

Let G be a finite p -group, and α an automorphism of the group algebra F p G. Then α fixes the socle of F p G pointwise. More generally, if k is a field of characteristic p , and α is a k -algebra automorphism of kG , then α induces a linear action on the dimension subquotients of the group, and the action on the socle is scalar multiplication by the (p − 1) st power of the product of the determinants of this action. The scalar is thus an element of (k ×) p − 1. [ABSTRACT FROM AUTHOR]

We answer in the negative a question of Hartley about representations of finite groups, by constructing examples of finite simple groups with arbitrarily large representations whose endomorphism ring consists of just the scalars. We show as a consequence that there are finite simple groups of automorphisms of the locally finite simple group S L (∞ , F q) with trivial centralizer. The smallest of our examples is A 6 with q = 9 . [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field of prime characteristic p. Let kge be a block of a group algebra of a finite group G, with normal defect group P and abelian p' inertial quotient L. Then we show that kge is a matrix algebra over a quantised version of the group algebra of a semidirect product of P with a certain subgroup of L. To do this, we first examine the associated graded algebra, using a Jennings-Quillen style theorem. As an example, we calculate the associated graded of the basic algebra of the nonprincipal block in the case of a semidirect product of an extraspecial p-group P of exponent p and order -3 with a quaternion group of order eight with the centre acting trivially. In the case of p = 3, we give explicit generators and relations for the basic algebra as a quantised version of kp. As a second example, we give explicit generators and relations in the case of a group of shape 21+4 : 31+2 in characteristic two. [ABSTRACT FROM AUTHOR]

Let G be a finite group and k a field of characteristic p. We conjecture that if M is a kG -module with H ⁎ (G , M) finitely generated as a module over H ⁎ (G , k) then as an element of the stable module category StMod (k G) , M is contained in the thick subcategory generated by the finitely generated kG -modules and the modules M ′ with H ⁎ (G , M ′) = 0. We show that this is equivalent to a conjecture of the second author about generation of the bounded derived category of cochains C ⁎ (B G ; k) , and we prove the conjecture in the case where the centraliser of every element of G of order p is p -nilpotent. In this case some stronger statements are true, that probably fail for more general finite groups. [ABSTRACT FROM AUTHOR]

Let k be a field of characteristic p > 0. For G an elementary abelian p-group, there exist collections of permutation modules such that if C* is any exact bounded complex whose terms are sums of copies of modules from the collection, then C* is contractible. A consequence is that if G is any finite group whose Sylow p-subgroups are not cyclic or quaternion, and if C* is a bounded exact complex such that each Ci is a direct sum of one dimensional modules and projective modules, then C* is contractible. [ABSTRACT FROM AUTHOR]

We introduce the concept of a nearly null map in the stable module category, and relate it to the notion of virtual projectivity. We show that the trivial module k is virtually M -projective if and only if the map f : N → k in the triangle N → f k → M ⊗ M ⁎ is nearly null. If these conditions hold, then M generates the stable module category in fewer than ℓ steps, where ℓ is the radical length of the group algebra. We give examples to show that if M generates the stable module category then k is not necessarily virtually M -projective. We also give examples to show that for a given group, the degree of virtual projectivity of k with respect to a module M is not bounded. Finally, we develop a theory of virtual M -variety of a module X. This is a subset of the spectrum of the cohomology ring which controls virtual projectivity of X with respect to M. [ABSTRACT FROM AUTHOR]

Let |$k$| be an algebraically closed field of characteristic |$p$|⁠ , and let |${\mathcal{O}}$| be either |$k$| or its ring of Witt vectors |$W(k)$|⁠. Let |$G$| be a finite group and |$B$| a block of |${\mathcal{O}} G$| with normal abelian defect group and abelian |$p^{\prime}$| inertial quotient |$L$|⁠. We show that |$B$| is isomorphic to its second Frobenius twist. This is motivated by the fact that bounding Frobenius numbers is one of the key steps towards Donovan's conjecture. For |${\mathcal{O}}=k$|⁠ , we give an explicit description of the basic algebra of |$B$| as a quiver with relations. It is a quantized version of the group algebra of the semidirect product |$P\rtimes L$|⁠. [ABSTRACT FROM AUTHOR]

Nilpotence has been studied in stable homotopy theory and algebraic geometry. We study the corresponding notion in modular representation theory of finite groups, and apply the discussion to the study of ghosts, and generation of the stable module category. In particular, we show that for a finitely generated kG -module M , the tensor M -generation number and the tensor M -ghost number are both equal to the degree of tensor nilpotence of a certain map associated with M . [ABSTRACT FROM AUTHOR]

Let k be a field of odd prime characteristic p . We calculate the Lie algebra structure of the first Hochschild cohomology of a class of quantum complete intersections over k . As a consequence, we prove that if B is a defect 2-block of a finite group algebra kG whose Brauer correspondent C has a unique isomorphism class of simple modules, then a basic algebra of B is a local algebra which can be generated by at most 2 I elements, where I is the inertial index of B , and where we assume that k is a splitting field for B and C . [ABSTRACT FROM AUTHOR]

Abstract: We develop a duality theory for localizations in the context of ring spectra in algebraic topology. We apply this to prove a theorem in the modular representation theory of finite groups. Let be a finite group and be an algebraically closed field of characteristic . If is a homogeneous nonmaximal prime ideal in , then there is an idempotent module which picks out the layer of the stable module category corresponding to , and which was used by Benson, Carlson and Rickard [D.J. Benson, J.F. Carlson, J. Rickard, Thick subcategories of the stable module category, Fund. Math. 153 (1997) 59–80] in their development of varieties for infinitely generated -modules. Our main theorem states that the Tate cohomology is a shift of the injective hull of as a graded -module. Since can be constructed using a version of the stable Koszul complex, this can be viewed as a statement of localized Gorenstein duality in modular representation theory. Various consequences of this theorem are given, including the statement that the stable endomorphism ring of the module is the -completion of cohomology , and the statement that is a pure injective -module. In the course of proving the theorem, we further develop the framework introduced by Dwyer, Greenlees and Iyengar [W.G. Dwyer, J.P.C. Greenlees, S. Iyengar, Duality in algebra and topology, Adv. Math. 200 (2006) 357–402] for translating between the unbounded derived categories and . We also construct a functor to the full stable module category, which extends the usual functor and which preserves Tate cohomology. The main theorem is formulated and proved in , and then translated to and finally to . The main theorem in can be viewed as stating that a version of Gorenstein duality holds after localizing at a prime ideal in . This version of the theorem holds more generally for a compact Lie group satisfying a mild orientation condition. This duality lies behind the local cohomology spectral sequence of Greenlees and Lyubeznik for localizations of . In a companion paper [D.J. Benson, Idempotent -modules with injective cohomology, J. Pure Appl. Algebra 212 (7) (2008) 1744–1746], a more recent and shorter proof of the main theorem is given. The more recent proof seems less natural, and does not say anything about localization of the Gorenstein condition for compact Lie groups. [Copyright &y& Elsevier]

Abstract: Let Λ be a finite dimensional indecomposable weakly symmetric algebra over an algebraically closed field k, satisfying . Let be representatives of the isomorphism classes of simple Λ-modules, and let E be the matrix whose entry is . If there exists an eigenvalue λ of E satisfying then the minimal resolution of each non-projective finitely generated Λ-module has exponential growth, with radius of convergence . On the other hand, if all eigenvalues λ of E satisfy then the dimensions of the modules in the minimal projective resolution of each finitely generated Λ-module are either bounded or grow linearly. In this case, we classify the possibilities for the matrix E. The proof is an application of the Perron–Frobenius theorem. [Copyright &y& Elsevier]

Abstract: Let G be a finite group, k a perfect field, and V a finite-dimensional kG-module. We let G act on the power series by linear substitutions and address the question of when the invariant power series form a unique factorization domain. We prove that for a permutation module for a p-group in characteristic p, the answer is always positive. On the other hand, if G is a cyclic group of order p, k has characteristic p, and V is an indecomposable kG-module of dimension r with , we show that the invariant power series form a unique factorization domain if and only if r is equal to 1, 2, or p. This contradicts a conjecture of Peskin. [Copyright &y& Elsevier]

Abstract: The block variety of a finitely generated indecomposable module over the block algebra of a -block of a finite group , introduced in (J. Algebra 215 (1999) 460), can be computed in terms of a vertex and a source of . We use this to show that is connected, and that every closed homogeneous subvariety of the affine variety defined by block cohomology (cf. Algebras Rep. Theory 2 (1999) 107) is the variety of a module over the block algebra. This is analogous to the corresponding statements on Carlson''s cohomology varieties in (Invent. Math. 77 (1984) 291). [Copyright &y& Elsevier]

The author comments on Richard P. Stanley's article "Invariants of finite groups and their applications to combinatorics." The author discusses the proximity and power of commutative algebra techniques in examining objects associated with finite group actions. He cites Stanley's emphasis on Cohen-Macaulay, Daniel Gorenstein, and complete intersection conditions in commutative algebra, and his impact on other mathematicians in such subjects as algebraic topology and commutative algebra.