TITLE: Kunneth formula for sheaf cohomology of varieties
QUESTION [46 upvotes]: What is a good reference for the following fact (the hypotheses may not be quite right):

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
  $$H^m(X \times Y, \mathcal{F} \boxtimes \mathcal{G}) \cong \bigoplus_{p+q=m} H^p(X,\mathcal{F}) \otimes_k H^q(Y, \mathcal{G}).$$

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.

REPLY [18 votes]: One can find this in section 9.2 of Kempf's book "Algebraic Varieties". 
The slightly more general case where $X, Y$ are over an affine scheme $\operatorname{Spec} R$ and $\mathcal{F}, \mathcal{G}$ are quasi-coherent sheaves flat over $\operatorname{Spec} R$ can be found in 
Kempf: "Some elementary proofs of basic theorems in the cohomology of quasi-coherent sheaves"
Rocky Mountain J. Math. Volume 10, Number 3 (1980), 637-646. link