Let k be a field and let G be a finite group. By a theorem of D. Benson, H. Krause and S. Schwede, there is a canonical element in the Hochschild cohomology of the Tate cohomology γG ∈ HH 3,−1Ĥ∗(G) with the following property: Given any graded Ĥ∗(G)-module X, the image of γG in Ext 3,−1 Ĥ∗(G) (X,X) is zero if and only if X is isomorphic to a direct summand of Ĥ(G,M) for some kG-module M . Suppo...

Introduction 2 1. Preliminaries on conformal algebras and modules 4 2. Basic definitions 9 3. Extensions and deformations 13 4. Homology 17 5. Exterior multiplication, contraction, and module structure 18 6. Cohomology of conformal algebras and their annihilation Lie algebras 19 6.1. Cohomology of the basic complex 19 6.2. Cohomology of the reduced complex 21 6.3. Cohomology of conformal algebr...

Consider a field k of characteristic p > 0, G(r) the r-th Frobenius kernel of a smooth algebraic group G, DG(r) the Drinfeld double of G(r), and M a finite dimensional DG(r)-module. We prove that the cohomology algebra H(DG(r), k) is finitely generated and that H(DG(r),M) is a finitely generated module over this cohomology algebra. We exhibit a finite map of algebras θr : H(G(r), k) ⊗ S(g) → H(...

In this note we describe aspects of the cohomology of coherent sheaves on a complete toric variety X over a field k and, more generally, the local cohomology, with supports in a monomial ideal, of a finitely generated module over a polynomial ring S. This leads to an efficient way of computing such cohomology, for which we give explicit algorithms. The problem is finiteness. The i local cohomol...

Let p ≥ 3 be a prime. We consider the cyclotomic extension Z(p)[ζp2] of Z(p), with Galois group G := (Z/p2)∗. Since this extension is wildly ramified, Z(p)[ζp2] is not projective as a module over the group ring Z(p)G (Speiser). Extending this module structure, we can regard Z(p)[ζp2 ] as a module over the twisted group ring Z(p)[ζp2] ≀G; as such, it remains faithful and non projective. We calcu...

We study H∗(P ), the mod p cohomology of a finite p–group P , viewed as an Fp[Out(P )]–module. In particular, we study the conjecture, first considered by Martino and Priddy, that, if eS ∈ Fp[Out(P )] is a primitive idempotent associated to an irreducible Fp[Out(P )]–module S, then the Krull dimension of eSH ∗(P ) equals the rank of P . The rank is an upper bound by Quillen’s work, and the conj...

Following our approach to metric Lie algebras developed in a previous paper we propose a way of understanding pseudo-Riemannian symmetric spaces which are not semi-simple. We introduce cohomology sets (called quadratic cohomology) associated with orthogonal modules of Lie algebras with involution. Then we construct a functorial assignment which sends a pseudo-Riemannian symmetric space M to a t...

BRST-resolution is studied for the principally graded Wakimoto module of ŝl2 recently found in [9]. The submodule structure is completely determined and irreducible representations can be obtained as the zero-th cohomology group. e-mail:[email protected]

Let k be a field and let G be a finite group. There is a canonical element in the Hochschild cohomology of the Tate cohomology γG ∈ HH3,−1Ĥ∗(G, k) with the following property. Given a graded Ĥ∗(G, k)-module X, the image of γG in Ext 3,−1 Ĥ∗(G,k) (X,X) vanishes if and only if X is isomorphic to a direct summand of Ĥ∗(G,M) for some kG-module M . The description of the realizability obstruction wo...