Question

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.

Answer 1

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 and $X$ and $Y$ by affine open subsets ${U_i}$, and ${V_j}$, and write down the Cech complex for $\mathcal{F}$ and $\mathcal{G}$ with respect to these two coverings, and the Cech 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-Zilberg (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).

Answer 2

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