I would like to write down explicitly the generating cocycles of this second cohomology group, $H^2(Z\_n \times Z\_n, k^\*)$. Here $k$ is an algebraically closed field of characteristic zero and $Z\_n$ is the cyclic group with $n$ elements.


I need to know what resolution to use and how to get the formulas!



Describe the second cohomology group $H^2(Z_n \times Z_n. k^*)$.
I would take the standard cyclic resolution of $G = Z/nZ$:
$$
\dots \stackrel{1-t}\to Z[G] \stackrel{\sum t^i}\to Z[G] \stackrel{1-t}\to Z[G] \to Z \to 0,
$$
where $t$ is the generator of $G$, and then take the tensor square of two such --- this would give a resolution
$$
\dots \to Z[G\_1\times G\_2]^3 \stackrel{d\_2}\to Z[G\_1\times G\_2]^2 \stackrel{d\_1}\to Z[G\_1\times G\_2] \to Z \to 0,
$$
where $G\_1 = G\_2 = Z/nZ$ and the maps are given by
$$
d\_1 = (1-t\_1,1-t\_2),
\qquad
d\_2 = \left(\begin{array}{ccc}
\sum t\_1^i & 1-t\_2 & 0 \cr
0 & 1-t\_1 & \sum t\_2^i 
\end{array}\right) 
$$
($t\_1$ and $t\_2$ are the generators of $G\_1$ and $G\_2$ respectively). 
I think you can use this for the calculations.


