Lemma 36.22.5. The cup product (36.22.4.1) is an isomorphism if $\mathcal{F}^\bullet$, resp. $\mathcal{G}^\bullet$ consist of a single $S$-flat quasi-coherent $\mathcal{O}_ X$, resp. $\mathcal{O}_ Y$-module sitting in degree zero and $X$ and $Y$ are quasi-compact with affine diagonal.

Proof. Assume $\mathcal{F}^\bullet = \mathcal{F}$ and $\mathcal{G}^\bullet = \mathcal{G}$ are both reduced to a single quasi-coherent module placed in degree $0$ and both $\mathcal{F}$ and $\mathcal{G}$ are flat over $S$. 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 $\mathcal{W} : X \times _ S Y = \bigcup _{(i, j) \in I \times J} U_ i \times _ S V_ j$. Note that $\mathcal{W}$ is a refinement of $\text{pr}_1^{-1}\mathcal{U}$ and of $\text{pr}_2^{-1}\mathcal{V}$. Thus by the discussion in Cohomology, Section 20.25 we obtain maps

$\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \to \check{\mathcal{C}}^\bullet (\mathcal{W}, p^*\mathcal{F}) \quad \text{and}\quad \check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{G}) \to \check{\mathcal{C}}^\bullet (\mathcal{W}, q^*\mathcal{G})$

well defined up to homotopy and compatible with pullback maps on cohomology. In Cohomology, Equation (20.25.3.2) we have constructed a map of complexes

$\text{Tot}( \check{\mathcal{C}}^\bullet (\mathcal{W}, p^*\mathcal{F}) \otimes _ A \check{\mathcal{C}}^\bullet (\mathcal{W}, q^*\mathcal{G})) \longrightarrow \check{\mathcal{C}}^\bullet (\mathcal{W}, p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ S Y}} q^*\mathcal{G})$

which is compatible with the cup product on cohomology by Cohomology, Lemma 20.31.3. Combining the above we obtain a map of complexes

36.22.5.1
\begin{equation} \label{perfect-equation-kunneth-on-cech} \text{Tot}( \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \otimes _ A \check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{G})) \to \text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{W}, p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ S Y}} q^*\mathcal{G})) \end{equation}

We claim this is the map in the statement of the lemma, i.e., the source and target of this arrow are the same as the source and target of (36.22.4.1). Namely, by Cohomology of Schemes, Lemma 30.2.2 and Cohomology, Lemma 20.25.2 the canonical maps

$\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \to R\Gamma (X, \mathcal{F}), \quad \check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{G}) \to R\Gamma (Y, \mathcal{G})$

and

$\check{\mathcal{C}}^\bullet (\mathcal{W}, p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ S Y}} q^*\mathcal{G}) \to R\Gamma (X \times _ S Y, p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ S Y}} q^*\mathcal{G})$

are isomorphisms. On the other hand, the complex $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$ is homotopy equivalent to the alternating Čech complex $\check{\mathcal{C}}_{alt}^\bullet (\mathcal{U}, \mathcal{F})$ by Cohomology, Lemma 20.23.6. As the modules $\mathcal{F}^ n(U_{i_0\ldots i_ s})$ are $A$-flat, we see that the complex $\check{\mathcal{C}}_{alt}^\bullet (\mathcal{U}, \mathcal{F})$ is K-flat (More on Algebra, Lemma 15.57.9). Hence $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$ is a K-flat complex of $A$-modules too and we conclude that $\text{Tot}( \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \otimes _ A \check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{G}))$ represents the derived tensor product $R\Gamma (X, \mathcal{F}) \otimes _ A^\mathbf {L} R\Gamma (Y, \mathcal{G})$ as claimed. Finally, we have $p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ S Y}} q^*\mathcal{G} = p^{-1}\mathcal{F} \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}\mathcal{G}$ by Remark 36.22.2.

We still have to show that (36.22.5.1) is a quasi-isomorphism. We will do this using dimension shifting. Set $d(\mathcal{F}) = \max \{ d \mid H^ d(X, \mathcal{F}) \not= 0\}$. Assume $d(\mathcal{F}) > 0$. Set $U = \coprod \nolimits _{i \in I} U_ i$. This is an affine scheme as $I$ is finite. Denote $j : U \to X$ the morphism which is the inclusion $U_ i \to X$ on each $U_ i$. Since the diagonal of $X$ is affine, the morphism $j$ is affine, see Morphisms, Lemma 29.11.11. It follows that $\mathcal{F}' = j_*j^*\mathcal{F}$ is $S$-flat, see Morphisms, Lemma 29.25.4. It also follows that $d(\mathcal{F}') = 0$ by combining Cohomology of Schemes, Lemmas 30.2.4 and 30.2.2. For all $x \in X$ we have $\mathcal{F}_ x \to \mathcal{F}'_ x$ is the inclusion of a direct summand: if $x \in U_ i$, then $\mathcal{F}' \to (U_ i \to X)_*\mathcal{F}|_{U_ i}$ gives a splitting. We conclude that $\mathcal{F} \to \mathcal{F}'$ is injective and $\mathcal{F}'' = \mathcal{F}'/\mathcal{F}$ is $S$-flat as well. Also $d(\mathcal{F}'') < d(\mathcal{F})$. In this way we reduce to the case $d(\mathcal{F}) = 0$.

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 $0$. Observe that this means that $\Gamma (X, \mathcal{F})$ is quasi-isomorphic to the finite complex $\check{\mathcal{C}}_{alt}^\bullet (\mathcal{U}, \mathcal{F})$ of flat $A$-modules sitting in degrees $0, \ldots , |I|$. It follows that $\Gamma (X, \mathcal{F})$ is a flat $A$-module. Let $V \subset Y$ be an affine open. Consider the affine open covering $\mathcal{U}_ V : X \times _ S V = \bigcup _{i \in I} U_ i \times _ S V$. It is immediate that

$\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \otimes _ A \mathcal{G}(V) = \check{\mathcal{C}}^\bullet (\mathcal{U}_ V, p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} q^*\mathcal{G})$

(equality of complexes). By the flatness of $\mathcal{G}(V)$ over $A$ we see that $\Gamma (X, \mathcal{F}) \otimes _ A \mathcal{G}(V) \to \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \otimes _ A \mathcal{G}(V)$ is a quasi-isomorphism. Since the sheafification of $V \mapsto \check{\mathcal{C}}^\bullet (\mathcal{U}_ V, p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} q^*\mathcal{G})$ represents $Rq_*(p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} q^*\mathcal{G})$ by Cohomology of Schemes, Lemma 30.7.1 we conclude that

$Rq_*(p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} q^*\mathcal{G}) \cong \Gamma (X, \mathcal{F}) \otimes _ A \mathcal{G}$

on $Y$ with obvious notation. Using this and Lemma 36.21.1 (which applies as $\Gamma (X, \mathcal{F})$ is $A$-flat) we conclude that $H^ n(X \times _ S Y, p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} q^*\mathcal{G})$ is zero for $n > 0$ and isomorphic to $H^0(X, \mathcal{F}) \otimes _ A H^0(Y, \mathcal{G})$ 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). $\square$

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