Cohomology of rings with algebraic group action

Algebraic Elements in Group Rings

In this brief note, we study algebraic elements in the complex group algebra C[G]. Specifically, suppose £ e C[G] satisfies /(£) = 0 for some nonzero polynomial f(x) e C[x]. Then we show that a certain fairly natural function of the coefficients of Z is bounded in terms of the complex roots of f(x). For G finite, this is a recent observation of [HLP], Thus the main thrust here concerns infinite...

Generalized Burnside rings and group cohomology

We define the cohomological Burnside ring B(G,M) of a finite group G with coefficients in a ZG-module M as the Grothendieck ring of the isomorphism classes of pairs [X, u] where X is a G-set and u is a cohomology class in a cohomology group H X(G,M). The cohomology groups H ∗ X(G,M) are defined in such a way that H∗ X(G, M) ∼= ⊕iH∗(Hi,M) when X is the disjoint union of transitive G-sets G/Hi. I...

Cohomology of Λ-rings

A cohomology theory for λ-rings is developed. This is then applied to study deformations of λ-rings.

Cohomology rings of arrangements

The Coherent Cohomology Ring of an Algebraic Group

Let G be a group scheme of finite type over a field, and consider the cohomology ring H∗(G) with coefficients in the structure sheaf. We show that H∗(G) is a free module of finite rank over its component of degree 0, and is the exterior algebra of its component of degree 1. When G is connected, we determine the Hopf algebra structure of H∗(G).