The Stacks project

Proof. In this proof unadorned products and tensor products are over $k$. As maps

we use functoriality of cohomology to get maps $H^ p(X, \mathcal{F}) \to H^ p(X \times Y, \text{pr}_1^*\mathcal{F})$ and $H^ p(Y, \mathcal{G}) \to H^ p(X \times Y, \text{pr}_2^*\mathcal{G})$ and then we use the cup product

The result is true when $X$ and $Y$ are affine by the vanishing of higher cohomology groups on affines (Cohomology of Schemes, Lemma 30.2.2) and the definitions (of pullbacks of quasi-coherent modules and tensor products of quasi-coherent modules).

Choose finite affine open coverings $\mathcal{U} : X = \bigcup _{i \in I} U_ i$ and $\mathcal{V} : Y = \bigcup _{j \in J} V_ j$. This determines an affine open covering $\mathcal{W} : X \times Y = \bigcup _{(i, j) \in I \times J} U_ i \times V_ j$. Note that $\mathcal{W}$ is a refinement of $\text{pr}_1^{-1}\mathcal{U}$ and of $\text{pr}_2^{-1}\mathcal{V}$. Thus by Cohomology, Lemma 20.15.1 we obtain maps

compatible with pullback maps on cohomology. In Cohomology, Equation (20.25.3.2) we have constructed a map of complexes

defining the cup product on cohomology. Combining the above we obtain a map of complexes

We warn the reader that this map is not an isomorphism of complexes. Recall that we may compute the cohomologies of our quasi-coherent sheaves using our coverings (Cohomology of Schemes, Lemmas 30.2.5 and 30.2.6). Thus on cohomology (33.29.1.1) reproduces the map of the lemma.

Consider a short exact sequence $0 \to \mathcal{F} \to \mathcal{F}' \to \mathcal{F}'' \to 0$ of quasi-coherent modules. Since the construction of (33.29.1.1) is functorial in $\mathcal{F}$ and since the formation of the relevant Čech complexes is exact in the variable $\mathcal{F}$ (because we are taking sections over affine opens) we find a map between short exact sequence of complexes

(we have dropped the outer zeros). Looking at long exact cohomology sequences we find that if the result of the lemma holds for $2$-out-of-$3$ of $\mathcal{F}, \mathcal{F}', \mathcal{F}''$, then it holds for the third.

Observe that $X$ has finite cohomological dimension for quasi-coherent modules, see Cohomology of Schemes, Lemma 30.4.2. Using induction on $d(\mathcal{F}) = \max \{ d \mid H^ d(X, \mathcal{F}) \not= 0\}$ we will reduce to the case $d(\mathcal{F}) = 0$. Assume $d(\mathcal{F}) > 0$. By Cohomology of Schemes, Lemma 30.4.3 we have seen that there exists an embedding $\mathcal{F} \to \mathcal{F}'$ such that $H^ p(X, \mathcal{F}') = 0$ for all $p \geq 1$. Setting $\mathcal{F}'' = \mathop{\mathrm{Coker}}(\mathcal{F} \to \mathcal{F}')$ we see that $d(\mathcal{F}'') < d(\mathcal{F})$. Then we can apply the result from the previous paragraph to see that it suffices to prove the lemma for $\mathcal{F}'$ and $\mathcal{F}''$ thereby proving the induction step.

Arguing in the same fashion for $\mathcal{G}$ we find that we may assume that both $\mathcal{F}$ and $\mathcal{G}$ have nonzero cohomology only in degree $0$. Let $V \subset Y$ be an affine open. Consider the affine open covering $\mathcal{U}_ V : X \times V = \bigcup _{i \in I} U_ i \times V$. It is immediate that

(equality of complexes). We conclude that

on $Y$. Here $A$ is a basis for the $k$-vector space $\Gamma (X, \mathcal{F})$. Cohomology on $Y$ commutes with direct sums (Cohomology, Lemma 20.19.1). Using the Leray spectral sequence for $\text{pr}_2$ (via Cohomology, Lemma 20.13.6) we conclude that $H^ n(X \times Y, \text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} \text{pr}_2^*\mathcal{G})$ is zero for $n > 0$ and isomorphic to $H^0(X, \mathcal{F}) \otimes H^0(Y, \mathcal{G})$ for $n = 0$. This finishes the proof (except that we should check that the isomorphism is indeed given by cup product in degree $0$; we omit the verification). $\square$