I have written almost everything and then I figured out where my (idiotic) problem has been... So I will post this question anyway, just so other people who are looking for this computation online can find an answer. Anything that is written here can be found at Cartan&Eilenberg.
My goal is to compute (as explicitly as possible) the cup product structure in Tate's cohomology of a cyclic group $G$.
I have searched a lot for an answer online and in many books, and the only place I have found what I was looking for is in Cartan&Eilenberg- Homological Algebra. When I tried to verify the equations given I ran across a problem, which was due to some silly mistake by me. The solution there goes something like this:
Let $X=X_\bullet$ be a complete resolution for $G$. By $X\otimes X$ we mean the double complex $X_r\otimes X_s$ with differentials $d'=d\otimes \text{id}$ and $d''=\text{id}\otimes d$. Then a mapping $\phi:X\to X\otimes X$ is defined to be a family of $G$-homomorphisms $\phi_{r,s}:X_{r+s}\to X_r\otimes X_s$ satisfying
- $\phi_{r,s}d=d'\phi_{r+1,s}+(-1)^rd''\phi_{r,s+1}$.
- $\epsilon=(\epsilon\otimes\epsilon)\phi_{0,0}$
where $\epsilon$ is the augmentation map $X_0\to\mathbb{Z}$ given in the complete resolution. (In the book they did not add the sign $(-1)^r$ but they have made some comment regarding the sign which I did not understand. The formula written here is correct).
From such a mapping we get a cup product defined on cochains by $f\smile g:=(f\otimes g)\phi_{r,s}$. My question is about the explicit construction of $\phi$ in the cyclic case.
So assume that $G$ is cyclic of order $n$ with generator $x$. We give a complete resolution by $X_i=\mathbb{Z}[G]$ and differentials $d_{2i}=N\cdot:X_{2i}\to X_{2i-1}$ (multiplication by $N=\sum x^k$), $d_{2i+1}=T\cdot:X_{2i+1}\to X_{2i}$ (multiplication by $T=x-1$), and the augmentation map $\epsilon:x_0\to \mathbb{Z}[G]$ by $\epsilon(1)=1$.
Now the definition of $\phi_{r,s}$ is given in the book by $$ \phi_{r,s}(1) := \begin{cases} 1\otimes 1 & \text{if } r \equiv 0 (\text{mod }2)\\ 1\otimes x & \text{if } r+1\equiv s \equiv 0 (\text{mod }2)\\ \sum_{0\le l< k\le n-1}x^l\otimes x^k & \text{if } r\equiv s \equiv 1 (\text{mod }2) \end{cases}$$
My problem was that It seemd to me that the first property of a mapping does not hold. my mistake was that during my verification of the first property I did not use the correct dimensions for $d',d''$.
I'm currently writing this in more detail in sharelatex.com, I'll send a link here when I'll be done. (This will include the actual equations on the cohomology groups)
