# Cup product in Tate Cohomology Ring

Let $$G$$ be a finite cyclic group of order $$p$$. The tate cohomology groups $$\hat{H}^*(G, \mathbb{F}_p)$$ are defined using a complete resolution of $$\mathbb{F}$$ as $$\mathbb{F}_pG$$-module.

There is a cup product on $$\hat{H}^*(G,\mathbb{F_p})$$, so I want to find the structure of this ring.

In this MO question

Examples of Tate cohomology rings

it is stated the integer case; however, I have not been able to find a reference to that statement.

For the integer case, there is an isomorphism $$\hat{H^n}(G,\mathbb{Z}) \cong \hat{H^{n+d}}(G,\mathbb{Z})$$ given by the cup product with some element $$u$$ of degree $$d$$ (the period). Does this also holds for the case $$\mathbb{F}_p$$?

Also, what techinque is used to compute the period of the tate cohomology and how to describe the ring?. For example, if $$G$$ is a cyclic group of order $$2$$, I think that $$\hat{H}^*(G, \mathbb{F}_2) \cong \mathbb{F}_2[t,t^{-1}]$$ because degreewise, $$\hat{H^n}(G,\mathbb{F}_2) = \mathbb{F}_2$$.

• The periodicity is proved in extreme generality in section 16 of Greenlees-May's book on Tate cohomology, see in particular Corollary 16.7: for any commutative ring $A$, if $G$ acts freely and orientation preservingly on a sphere of dimension $n$, then the cup product with a certain class $[\chi_V] \in H^{n+1}(G;A)$ induces an isomorphism $\hat H^*(G;A) \cong \hat H^{*+n+1}(G;A)$. You can drop the oriented assumption if $A$ is an $\Bbb F_2$-algebra. I don't remember if they talk about the ring structure though. – Mike Miller Sep 11 at 22:39

(starting on page 250) they calculate the cup products $$\hat{H}^*(G;A)\otimes \hat{H}^*(G;A')\to \hat{H}^*(G;A\otimes A')$$ explicitly, where $$A,A'$$ are $$\mathbb{Z}G$$-modules, using a map of resolutions. This gives a reference for the integer case, and probably it's not too hard to adapt thier argument to get the calculation for $$\mathbb{F}_p$$.