Examples of Tate cohomology rings

Question

If $G$ is a finite group with periodic cohomology then the Tate cohomology ring can be easily computed to be the localization $\hat{H}^\ast(G,\mathbb{Z}) = H^\ast(G,\mathbb{Z})_{(z)}$ where $z$ is a unit of minimal positive degree. Examples are

• Cyclic group:

  $\hat{H}^\ast(C_n,\mathbb{Z}) = \mathbb{Z}[z,z^{-1}]/(nz), |z|=2$

• Quaternion group:

  $\hat{H}^\ast(Q_{2^n},\mathbb{Z}) = \mathbb{Z}[z,z^{-1},a,b]/(2^nz, 2a,2b), |z| =4, |a| = |b| =2$

• Binary icosahedral group:

  $\hat{H}^\ast(I,\mathbb{Z}) = \mathbb{Z}[z,z^{-1}]/(120z), |z|=4$

The same formular holds in mod-$p$ cohomology, if the cohomology of $G$ is $p$-periodic.

Question: Are there computations of integral or mod-$p$ Tate cohomology rings of finite groups with non-periodic cohomology in the literature ?

Answer 1

Antonio Bellezza's PhD thesis (Pisa, 2002) computes the ring structure of the Tate cohomology for $\mathbb{Z}/p^a\times\mathbb{Z}/p^b$, and also the mod-p Tate cohomology of $\mathbb{Z}_p^2$. The title is Integral Duality and the Structure of Tate Cohomology Rings.

Also, there is an unpublished/unfinished paper of Weiss, found here: http://www.math.uwo.ca/~schebolu/research/Jan/tateprop.pdf , which computes $\hat{H}^*(\mathbb{Z}_2^r,\mathbb{Z}_2)$.

And just to add to your current list (periodic groups): $\hat{H}^*(S_3,\mathbb{Z}_3)=\Lambda[x]\otimes\mathbb{Z}_3[z,z^{-1}]$ with $|x|=3$ and $|z|=4$.

Answer 2

The ring structure on mod $p$ Tate cohomology can be split up into 4 parts:

1. Products $\hat{H}^n\otimes\hat{H}^m\to\hat{H}^{n+m}$ for $n,m\geq0$. These just come from products on ordinary cohomology.
2. Products $\hat{H}^n\otimes\hat{H}^m\to\hat{H}^{n+m}$ for $n\geq0$, $m<0$, and $n+m<0$. These are given by cap products when we identify $\hat{H}^m$ with the dual of $H^{-m-1}$.
3. Products $\hat{H}^n\otimes\hat{H}^m\to\hat{H}^{n+m}$ for $n\geq0$, $m<0$, and $n+m\geq 0$.
4. Products $\hat{H}^n\otimes\hat{H}^m\to\hat{H}^{n+m}$ for $n,m<0$.

In fact, for most groups, all products of type 3 and 4 vanish. This is true, for instance, if the center of a $p$-Sylow subgroup of $G$ has rank greater than 1: see this paper of Benson and Carlson (they also show that all products of type 3 vanish iff all products of type 4 vanish). By Chouinard's theorem, this implies that for arbitrary non-periodic groups, all elements in negative mod $p$ Tate cohomology are nilpotent.

Answer 3

Bellezza's thesis is here

http://www.abdn.ac.uk/~mth192/html/archive/bellezza.html

Answer 4

Artin and Tate showed in their class field theory book that a finite group has periodic mod $p$ cohomology if and only if its Sylow $p$–group is either a cyclic group or a generalised quaternion group. Swan proved later that a group has periodic cohomology for all primes if and only if it acts freely on a finite CW-complex which has the homotopy type of a sphere.