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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.

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A couple of pages further on in EGA III.2 is Theorem 6.7.8 which is exactly the kind of thing you are looking for, I imagine. –  Tony Scholl Aug 5 '10 at 19:30
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Use Cech covering of $X \times Y$ built from finite open affine covers of $X$ and $Y$, together with elementary stuff on homology of tensor product of bnded complexes over a field & that can compute sheaf cohom. cup products using pairings of resolutions (explained in Godemenet's book; EGA says where Godement relates cup product to Cech theory); surely you want isom. to be def'd by cup product! Using a touch of homological alg. with Tor, get same over any ring if assume qcoh sheaves & cohomologies of factors are flat. Better to work it out for yourself than to waste time with that part of EGA. –  BCnrd Aug 5 '10 at 20:35
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Thanks! The more I looked at that part of EGA, the worse it looked. –  Charles Staats Aug 5 '10 at 21:24
    
Charles, I know this is settled now, but I think Thm. 6.7.8 in EGA III$_2$ would be a better and slightly less intimidating reference for this. –  Sándor Kovács Apr 11 '13 at 1:44

2 Answers 2

up vote 23 down vote accepted

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-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).

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"It is not hard to see that the last is the tensor product of the first two". Do you mean homotopy-equivalent? –  Martin Brandenburg Aug 27 '13 at 16:44

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

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