The Stacks project

Lemma 33.29.2. Let $k$ be a field. Let $X$ and $Y$ be schemes over $k$ and let $\mathcal{F}$, resp. $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_ X$-module, resp. $\mathcal{O}_ Y$-module. Then we have a canonical isomorphism

\[ H^ n(X \times _{\mathop{\mathrm{Spec}}(k)} Y, \text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _{\mathop{\mathrm{Spec}}(k)} Y}} \text{pr}_2^*\mathcal{G}) = \bigoplus \nolimits _{p + q = n} H^ p(X, \mathcal{F}) \otimes _ k H^ q(Y, \mathcal{G}) \]

provided $X$ and $Y$ are quasi-compact and quasi-separated.

Proof. If $X$ and $Y$ are separated or more generally have affine diagonal, then please see Lemma 33.29.1 for “better” proof (the feature it has over this proof is that it identifies the maps as pullbacks followed by cup products). Let $X'$, resp. $Y'$ be the infinitesimal thickening of $X$, resp. $Y$ whose structure sheaf is $\mathcal{O}_{X'} = \mathcal{O}_ X \oplus \mathcal{F}$, resp. $\mathcal{O}_{Y'} = \mathcal{O}_ Y \oplus \mathcal{G}$ where $\mathcal{F}$, resp. $\mathcal{G}$ is an ideal of square zero. Then

\[ \mathcal{O}_{X' \times Y'} = \mathcal{O}_{X \times Y} \oplus \text{pr}_1^*\mathcal{F} \oplus \text{pr}_2^*\mathcal{G} \oplus \text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} \text{pr}_2^*\mathcal{G} \]

as sheaves on $X \times Y$. In this way we see that it suffices to prove that

\[ H^ n(X \times Y, \mathcal{O}_{X \times Y}) = \bigoplus \nolimits _{p + q = n} H^ p(X, \mathcal{O}_ X) \otimes _ k H^ q(Y, \mathcal{O}_ Y) \]

for any pair of quasi-compact and quasi-separated schemes over $k$. Some details omitted.

To prove this statement we use cohomology and base change in the form of Cohomology of Schemes, Lemma 30.7.3. This lemma tells us there exists a bounded below complex of $k$-vector spaces, i.e., a complex $\mathcal{K}^\bullet $ of quasi-coherent modules on $\mathop{\mathrm{Spec}}(k)$, which universally computes the cohomology of $Y$ over $\mathop{\mathrm{Spec}}(k)$. In particular, we see that

\[ R\text{pr}_{1, *}(\mathcal{O}_{X \times Y}) \cong (X \to \mathop{\mathrm{Spec}}(k))^*\mathcal{K}^\bullet \]

in $D(\mathcal{O}_ X)$. Up to homotopy the complex $\mathcal{K}^\bullet $ is isomorphic to $\bigoplus _{q \geq 0} H^ q(Y, \mathcal{O}_ Y)[-q]$ because this is true for every complex of vector spaces over a field. We conclude that

\[ R\text{pr}_{1, *}(\mathcal{O}_{X \times Y}) \cong \bigoplus \nolimits _{q \geq 0} H^ q(Y, \mathcal{O}_ Y)[-q] \otimes _ k \mathcal{O}_ X \]

in $D(\mathcal{O}_ X)$. Then we have

\begin{align*} R\Gamma (X \times Y, \mathcal{O}_{X \times Y}) & = R\Gamma (X, R\text{pr}_{1, *}(\mathcal{O}_{X \times Y})) \\ & = R\Gamma (X, \bigoplus \nolimits _{q \geq 0} H^ q(Y, \mathcal{O}_ Y)[-q] \otimes _ k \mathcal{O}_ X) \\ & = \bigoplus \nolimits _{q \geq 0} R\Gamma (X, H^ q(Y, \mathcal{O}_ Y) \otimes \mathcal{O}_ X)[-q] \\ & = \bigoplus \nolimits _{q \geq 0} R\Gamma (X, \mathcal{O}_ X) \otimes _ k H^ q(Y, \mathcal{O}_ Y)[-q] \\ & = \bigoplus \nolimits _{p, q \geq 0} H^ p(X, \mathcal{O}_ X)[-p] \otimes _ k H^ q(Y, \mathcal{O}_ Y)[-q] \end{align*}

as desired. The first equality by Leray for $\text{pr}_1$ (Cohomology, Lemma 20.13.1). The second by our decomposition of the total direct image given above. The third because cohomology always commutes with finite direct sums (and cohomology of $Y$ vanishes in sufficiently large degree by Cohomology of Schemes, Lemma 30.4.4). The fourth because cohomology on $X$ commutes with infinite direct sums by Cohomology, Lemma 20.19.1. The final equality by our remark on the derived category of a field above. $\square$

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