What is a good reference for the following fact (the hypotheses may not be quite right):



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> Let $X$ and $Y$ be projective varieties over a field $k$. Let $\mathcal{F}$ and $\mathcal{G}$ be coherent sheaves on $X$ and $Y$, respectively. Let $\mathcal{F} \boxtimes \mathcal{G}$ denote $p\_1^\*(\mathcal{F}) \otimes\_{\mathcal{O}\_{X \times Y}} p\_2^\* \mathcal{G}$. Then
>  $$H^m(X \times Y, \mathcal{F} \boxtimes \mathcal{G}) \cong \bigoplus\_{p+q=m} H^p(X,\mathcal{F}) \otimes\_k H^q(Y, \mathcal{G}).$$
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Note: Wikipedia leads me to believe that this may be related to Theorem 6.7.3 in EGA III2, but I find this theorem quite intimidating. Although I would be willing to study this if there is no more basic reference, I would at least like some confirmation that I am studying the right thing.



Kunneth formula for sheaf cohomology of varieties
The treatment in EGA is indeed intimidating, but in fact over a field the formula is not hard to prove. You only need $X$ and $Y$ to be separated schemes over $k$, and $\mathcal{F}$ and $\mathcal{G}$ to be quasi-coherent. Then cover $X$ and $Y$ by affine open subsets $\{U\_i\}$, and $\{V\_j\}$, and write down the Čech complex for $\mathcal{F}$ and $\mathcal{G}$ with respect to these two coverings, and the Čech complex of $\mathcal{F} \boxtimes \mathcal{G}$ with respect to the covering $U\_i \times V\_j$. It is not hard to see that the last is the tensor product of the first two; then the thesis follows from Eilenberg-Zilber (or however you want to call the fact that the cohomology of the tensor product of two complexes over a field is the tensor product of the cohomlogies).


