Equivariant homology and cohomology of groups

We provide and study an equivariant theory of group (co)homology of a group G with coefficients in a Γ-equivariant G-module A, when a separate group Γ acts on G and A, generalizing the classical Eilenberg-MacLane (co)homology theory of groups. Relationship with equivariant cohomology of topological spaces is established and application to algebraic K-theory is given.     

Localisation and colocalisation of KK-theory

The localisation of an R-linear triangulated category T\mathcal{T} at S ⁻¹ R for a multiplicatively closed subset S is again triangulated, and related to the original category by a long exact sequence involving a version of T\mathcal{T} with coefficients in S ⁻¹ R/R. We examine these theories and, under some assumptions, write the latter as an inductive limit of T\mathcal{T} with torsion coefficients. Our main application is the case where T\mathcal{T} is equivariant bivariant K-theory and R the ring of integers.