Tate cohomology of periodic $K$-theory with reality is trivial

Tate Cohomology in Axiomatic Stable Homotopy Theory

Completions of Z=(p)-Tate cohomology of periodic spectra

We construct splittings of some completions of the Z/(p)–Tate cohomology of E(n) and some related spectra. In particular, we split (a completion of) tE(n) as a (completion of) a wedge of E(n−1)’s as a spectrum, where t is shorthand for the fixed points of the Z/(p)–Tate cohomology spectrum (ie the Mahowald inverse limit lim ←− k (P−k ∧ΣE(n))). We also give a multiplicative splitting of tE(n) af...

Tate Cohomology and Periodic Localization of Polynomial Functors

In this paper, we show that Goodwillie calculus, as applied to functors from stable homotopy to itself, interacts in striking ways with chromatic aspects of the stable category. Localized at a fixed prime p, let T (n) be the telescope of a vn self map of a finite S–module of type n. The Periodicity Theorem of Hopkins and Smith implies that the Bousfield localization functor associated to T (n)∗...

When is Galois cohomology free or trivial?

Let p be a prime, F a field containing a primitive pth root of unity, and E/F a cyclic extension of degree p. Using the Bloch–Kato Conjecture we determine precise conditions for the cohomology group Hn(E) := H(GE ,Fp) to be free or trivial as an Fp[Gal(E/F )]-module, and we examine when these properties for Hn(E) are inherited by Hk(E), k > n. By analogy with cohomological dimension, we introdu...

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...