On the Relative Group Cohomology Ring

Relative Group Cohomology and the Orbit Category

Let G be a finite group and F be a family of subgroups of G closed under conjugation and taking subgroups. We consider the question whether there exists a periodic relative F-projective resolution for Z when F is the family of all subgroups H ≤ G with rkH ≤ rkG − 1. We answer this question negatively by calculating the relative group cohomology FH(G,F2) where G = Z/2 × Z/2 and F is the family o...

The Cohomology of a Coxeter Group with Group Ring Coefficients

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).

On the Cohomology of Relative Hopf Modules

Let H be a Hopf algebra over a field k, and A an Hcomodule algebra. The categories of comodules and relative Hopf modules are then Grothendieck categories with enough injectives. We study the derived functors of the associated Hom functors, and of the coinvariants functor, and discuss spectral sequences that connect them. We also discuss when the coinvariants functor preserves injectives.

Cohomology of Coxeter groupswith group ring coefficients: II

The cohomology of a group G with coefficients in a left G–module M is denoted H .GIM /. We are primarily interested in the case where M is the group ring, ZG . Since ZG is a G–bimodule, H .GIZG/ inherits the structure of a right G– module. When G is discrete and acts properly and cocompactly on a contractible CW complex , there is a natural topological interpretation for this cohomology group:...