Narrow Results

The problem of determining a complete set of invariants for characterizing the Clifford class of a central simple G-algebra over a field K is studied under the assumption that the algebra is simple ring. This is successful in the case of interior central simple G-algebras over K, and also when the inertia subgroup of the G-algebra is a direct factor of G. As an application of the latter, it is shown that if G is a finite abelian group, L is a Galois extension of K with Galois group G, and E is a G-algebra isomorphic to End_KV for some KG-module V, then the skew group ring of G over L ⊗_K E is a split matrix algebra over K.

We apply the G-algebra theory to the tensor product of algebras. These considerations are applied to extend the results of Alghamdi and Khammash [1], Khammash [4] and Külshammer [5, Proposition 1.2] on the tensor product of group algebras and modules over an algebraically closed field to lattices over a complete discrete valuation ring. This places these results in the standard integral finite group modular representation theory of G-algebras as pioneered by Puig (cf. [8]). We also study some aspects of covering homomorphisms and the Green correspondence in this context (cf. [8, Sections 20 and 25]).

Let K be a normal subgroup of the finite group H. To a point of a K-interior H-algebra we associate a group extension, and we prove that this extension is isomorphic to an extension associated to the point given by the Brauer homomorphism. This may be regarded as a generalization and an alternative treatment of Dade's results [2, Section 12].