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 Mahowald's inverse limit of P_{-k} smash SE(n)). We also give a multiplicative splitting of tE(n) after a suitable base extension.Comment: 30 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol2/paper8.abs.htm

Higher 𝑝-torsion in the 𝛽-family

We prove the existence of new families of υ 2 {\upsilon _2} -periodic elements of the stable homotopy of the sphere detected in the second filtration of the Adams-Novikov Spectral Sequence for primes greater than 3. Our main corollary is that the p p -component of π ∗ s \pi _ * ^s contains any finite abelian p p -group as a subgroup in some dimension ( for p ≥ 5 ) ({\text {for}}p \geq 5) .</p

Invertible Spectra In The E(n)-Local Stable Homotopy Category

this paper). Here the smash product of two E-local spectra need not be E-local, so one must relocalize the result by applying the Bousfield localization functor LE . The most well-known case is E = K(n), the nth Morava K-theory, considered in [HMS94]. In this paper we study the case E = E(n), where E(n) is the Johnson-Wilson spectrum. In this case the E-localization functor is universally denoted Ln , and we denote the category of E-local spectra by L. Our main theorem is the following result

Cohomology of finite-dimensional connected cocommutative Hopf algebras

AbstractThe May spectral sequence for the cohomology of an algebra is based on filtering the algebra by powers of its augmentation ideal. We discuss conditions for this spectral sequence to stop at a finite stage when applied to a connected cocommutative Hopf algebra. These conditions are satisfied for the subalgebras of the Steenrod algebra A (n) for p = 2 and P(n) for p > 2.The Ivanovskiĭ spectral sequence for the cohomology of a Hopf algebra is conceptually dual to the May spectral sequence. It is obtained by filtering by copowers of the unit co-ideal. If A is a finitely generated connected cocommutative Hopf algebra, then the E1-term is easily computable. Under these circumstances we show this spectral sequence stops at a finite stage

Tate cohomology of theories with one-dimensional coefficient ring

AbstractFor a finite group G we calculate the E-Tate cohomology t(E)G∗ and the E-homology E∗(BG+) as functors of the augmented commutative ring E∗(BG+) when E∗(.) is a complex oriented, vn-periodic cohomology theory with one-dimensional graded coefficient ring E∗