Tate (co)homology via pinched complexes

Tate Cohomology for Arbitrary Groups via Satellites

We define cohomology groups Ĥn(G;M), n ∈ Z, for an arbitrary group G and G-module M , using the concept of satellites. These cohomology groups generalize the Farrell-Tate groups for groups of finite virtual cohomological dimension and form a connected sequence of functors, characterized by a natural universal property. The classical Tate cohomology groups of finite groups have been generalized ...

Tate Cohomology in Commutative Algebra.

Finite Generation of Tate Cohomology

Let G be a finite group and let k be a field of characteristic p. Given a finitely generated indecomposable non-projective kG-module M , we conjecture that if the Tate cohomology Ĥ ∗ (G, M) of G with coefficients in M is finitely generated over the Tate cohomology ring Ĥ ∗ (G, k), then the support variety VG(M) of M is equal to the entire maximal ideal spectrum VG(k). We prove various results w...

Tate Cohomology over Fairly General Rings

Tate cohomology was originally defined over finite groups. More recently, Avramov and Martsinkovsky showed how to extend the definition so that it now works well over Gorenstein rings. This paper improves the theory further by giving a new definition that works over more general rings, specifically, those with a dualizing complex. The new definition of Tate cohomology retains the desirable prop...

Tate Cohomology Lowers Chromatic Bousfield Classes

Let G be a finite group. We use the results of [5] to show that the Tate homology of E(n) local spectra with respect to G produces E(n− 1) local spectra. We also show that the Bousfield class of the Tate homology of LnX (for X finite) is the same as that of Ln−1X. To be precise, recall that Tate homology is a functor from G-spectra to G-spectra. To produce a functor PG from spectra to spectra, ...