In a recent paper [1], Adem, Maginnis and Milgram calculated the mod 2 cohomology of the Mathieu simple group M12. An interesting feature of the answer they obtain is that it is Cohen–Macaulay. There is a subalgebra isomorphic to the rank three Dickson invariants, namely a polynomial ring in generators of degrees 4, 6 and 7, and over this subalgebra the whole cohomology ring is a finitely gener...

The result of this paper is the determination of the cohomology of Artin groups of type An, Bn and Ãn with non-trivial local coefficients. The main result is an explicit computation of the cohomology of the Artin group of type Bn with coefficients over the module Q[q±1, t±1]. Here the first n− 1 standard generators of the group act by (−q)-multiplication, while the last one acts by (−t)-multipl...

Freyd’s generating hypothesis, interpreted in the stable module category of a finite p-group G, is the statement that a map between finite-dimensional kG-modules factors through a projective if the induced map on Tate cohomology is trivial. We show that Freyd’s generating hypothesis holds for a non-trivial finite p-group G if and only if G is either C2 or C3. We also give various conditions whi...

Let G be a finite p-group which does not contain a rank two elementary abelian p-group as a direct factor. Then the ideal of essential classes in the mod-p cohomology ring of G is a Cohen–Macaulay module whose Krull dimension is the p-rank of the centre of G. This basically answers in the affirmative a question posed by J. F. Carlson (Question 5.4 in [7]).

Each Gr-category (or categorical group) is Gr-equivalent to a Grcategory of type (Π, A, ξ), where Π is a group, A is a Π-module and ξ is a 3-cocycle of Π, with coefficients in A. In this paper, first we show that each Gr-functor induces one between Gr-categories of type (Π, C). Then we give results on the existence of Gr-functor and the classification of Gr-functors by cohomology groups H(Π, C)...

We prove that the Drinfeld double of an arbitrary finite group scheme has finitely generated cohomology. That is to say, for G any scheme, and D(G) ring kG, we show self-extension algebra trivial representation a algebra, each D(G)-representation V extensions from form module over aforementioned algebra. As corollary, find all categories \({{\,\mathrm{rep}\,}}(G)^*_\mathscr {M}\) dual \({{\,\ma...

In this note we show that if k is an algebraic number field with algebraic closure I and M is a finitely generated, free Zrmodule with continuous Ga\(k/k)action, then the continuous Galois cohomology group Hl(k, M) is a finitely generated Z,-module under certain conditions on M (see Theorem 1 below). Also, we present a simpler construction of a mapping due to S. Bloch which relates torsion alge...

In this paper, we prove the conjectures made in a joint paper of the author with Carlson and Robinson, on the vanishing of cohomology of a nite group G. In particular, we prove that if k is a eld of characteristic p, then every non-projective kG-module M in the principal block has nontrivial cohomology in the sense that H (G; M) 6 = 0, if and only if the centralizer in G of every element of ord...

0.1. In this paper we construct a number of geometrically meaningful 1– and 2– cocycles of infinite dimensional Lie groups and Lie algebras. The groups are groups of diffeomorphisms of smooth manifolds and their subgroups, such as the group of symplectomorphisms of a symplectic manifold; the Lie algebras are algebras of vector fields. Coefficient spaces of the cocycles are spaces of various ten...

Let H be a bialgebra. An H-module-algebra is an associative algebra A that is also an H-module such that the multiplication map on A becomes an H-module morphism. This algebraic structure arises often in algebraic topology, quantum groups [11, Chapter V.6], Lie and Hopf algebras theory [3, 16, 19], and group representations [1, Chapter 3]. For example, in algebraic topology, the complex cobordi...