The Stacks project

First proof. This follows from the following sequence of isomorphisms

The first equality holds because $f = a \circ p$. The second equality by Lemma 36.22.1. The third equality by Lemma 36.22.5. The fourth equality by Lemma 36.22.1. We omit the verification that the composition of these isomorphisms is the same as the map (36.23.0.1). If $S$ is affine, then the source and target of the arrow (36.23.0.2) are the result of applying $R\Gamma (S, -)$ to the source and target of (36.23.0.1) and we obtain the final statement; details omitted. $\square$

Second proof. The construction of the arrow (36.23.0.1) is compatible with restricting to open subschemes of $S$ as is immediate from the construction of the relative cup product. Thus it suffices to prove that (36.23.0.1) is an isomorphism when $S$ is affine.

Assume $S = \mathop{\mathrm{Spec}}(A)$ is affine. By Leray we have $R\Gamma (S, Rf_*K) = R\Gamma (X, K)$ and similarly for the other cases. By Cohomology, Lemma 20.31.7 the map (36.23.0.1) induces the map (36.23.0.2) on taking $R\Gamma (S, -)$. On the other hand, by Lemmas 36.4.1 and 36.3.9 the source and target of the map (36.23.0.1) are in $D_\mathit{QCoh}(\mathcal{O}_ S)$. Thus, by Lemma 36.3.5, it suffices to prove that (36.23.0.2) is an isomorphism.

Assume $S = \mathop{\mathrm{Spec}}(A)$ and $X = \mathop{\mathrm{Spec}}(B)$ and $Y = \mathop{\mathrm{Spec}}(C)$ are all affine. We will use Lemma 36.3.5 without further mention. In this case we can choose a K-flat complex $K^\bullet $ of $B$-modules whose terms are flat such that $K$ is represented by $\widetilde{K}^\bullet $. Similarly, we can choose a K-flat complex $M^\bullet $ of $C$-modules whose terms are flat such that $M$ is represented by $\widetilde{M}^\bullet $. See More on Algebra, Lemma 15.59.10. Then $\widetilde{K}^\bullet $ is a K-flat complex of $\mathcal{O}_ X$-modules and similarly for $\widetilde{M}^\bullet $, see Lemma 36.3.6. Thus $La^*K$ is represented by

and similarly for $Lb^*M$. This in turn is a K-flat complex of $\mathcal{O}_{X \times _ S Y}$-modules by the lemma cited above and More on Algebra, Lemma 15.59.3. Thus we finally see that the complex of $\mathcal{O}_{X \times _ S Y}$-modules associated to

represents $La^*K \otimes _{\mathcal{O}_{X \times _ S Y}}^\mathbf {L} Lb^*M$ in the derived category of $X \times _ S Y$. Taking global sections we obtain $\text{Tot}(K^\bullet \otimes _ A M^\bullet )$ which of course is also the complex representing $R\Gamma (X, K) \otimes _ A^\mathbf {L} R\Gamma (Y, M)$. The fact that the isomorphism is given by cup product follows from the relationship between the genuine cup product and the naive one in Cohomology, Section 20.31 (and in particular Cohomology, Lemma 20.31.3 and the discussion following it).

Assume $S = \mathop{\mathrm{Spec}}(A)$ and $Y$ are affine. We will use the induction principle of Cohomology of Schemes, Lemma 30.4.1 to prove the statement. To do this we only have to show: if $X = U \cup V$ is an open covering with $U$ and $V$ quasi-compact and if the map (36.23.0.2)

for $U$ and $Y$ over $S$, the map (36.23.0.2)

for $V$ and $Y$ over $S$, and the map (36.23.0.2)

for $U \cap V$ and $Y$ over $S$ are isomorphisms, then so is the map (36.23.0.2) for $X$ and $Y$ over $S$. However, by Cohomology, Lemma 20.33.7 these maps fit into a map of distinguished triangles with (36.23.0.2) the final leg and hence we conclude by Derived Categories, Lemma 13.4.3.

Assume $S = \mathop{\mathrm{Spec}}(A)$ is affine. To finish the proof we can use the induction principle of Cohomology of Schemes, Lemma 30.4.1 on $Y$. Namely, by the above we already know that our map is an isomorphism when $Y$ is affine. The rest of the argument is exactly the same as in the previous paragraph but with the roles of $X$ and $Y$ switched. $\square$

Proof. Parts (1), (2), (3) are local on $S$ hence we may and do assume $S$ is affine. Since $a$ is quasi-compact we conclude that $X$ is quasi-compact. Since $\mathcal{F}^\bullet $ is locally bounded, we conclude that $\mathcal{F}^\bullet $ is bounded.

For (1) and (2) we can use the first spectral sequence $R^ pa_*\mathcal{F}^ q \Rightarrow R^{p + q}a_*\mathcal{F}^\bullet $ of Derived Categories, Lemma 13.21.3. Combining Cohomology of Schemes, Lemma 30.4.5 and Homology, Lemma 12.24.11 we conclude.

Let us prove (3) by the induction principle of Cohomology of Schemes, Lemma 30.4.1. Namely, for a quasi-compact open of $U$ of $X$ consider the condition that $R(a|_ U)_*(\mathcal{F}^\bullet |_ U)$ has finite tor dimension. If $U, V$ are quasi-compact open in $X$, then we have a relative Mayer-Vietoris distinguished triangle

by Cohomology, Lemma 20.33.5. By the behaviour of tor amplitude in distinguished triangles (see Cohomology, Lemma 20.48.6) we see that if we know the result for $U$, $V$, $U \cap V$, then the result holds for $U \cup V$. This reduces us to the case where $X$ is affine. In this case we have

by Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7) and the vanishing of higher direct images of quasi-coherent modules under an affine morphism (Cohomology of Schemes, Lemma 30.2.3). Since $\mathcal{F}^ n$ is $S$-flat by assumption and $X$ affine, the modules $a_*\mathcal{F}^ n$ are flat for all $n$. Hence $a_*\mathcal{F}^\bullet $ is a bounded complex of flat $\mathcal{O}_ S$-modules and hence has finite tor dimension.

Proof of part (5). Denote $a' : (X, a^{-1}\mathcal{O}_ S) \to (S, \mathcal{O}_ S)$ the obvious flat morphism of ringed spaces. Part (5) says that

Thus Cohomology, Equation (20.54.2.1) gives a functorial map from the left to the right and we want to show this map is an isomorphism. This question is local on $S$ hence we may and do assume $S$ is affine. The rest of the proof is exactly the same as the proof of Lemma 36.22.1 except that we have to show that the functor $K \mapsto Ra'_*(L(a')^*K \otimes _{a^{-1}\mathcal{O}_ S}^\mathbf {L} \mathcal{F}^\bullet )$ commutes with direct sums. This is where we will use $\mathcal{F}^ n$ has the structure of a quasi-coherent $\mathcal{O}_ X$-module. Namely, observe that $K \mapsto L(a')^*K \otimes _{a^{-1}\mathcal{O}_ S}^\mathbf {L} \mathcal{F}^\bullet $ commutes with arbitrary direct sums. Next, if $\mathcal{F}^\bullet $ consists of a single quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}^\bullet = \mathcal{F}^ n[-n]$ then we have $L(a')^*G \otimes _{a^{-1}\mathcal{O}_ S}^\mathbf {L} \mathcal{F}^\bullet = La^*K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{F}^ n[-n]$, see Cohomology, Lemma 20.27.4. Hence in this case the commutation with direct sums follows from Lemma 36.4.5. Now, in general, since $S$ is affine (hence $X$ quasi-compact) and $\mathcal{F}^\bullet $ is locally bounded, we see that

is bounded. Arguing by induction on $b - a$ and considering the distinguished triangle

the proof of this part is finished. Some details omitted.

Proof of part (4). Let $a' : (X, a^{-1}\mathcal{O}_ S) \to (S, \mathcal{O}_ S)$ be as above. Since $\mathcal{F}^\bullet $ is a locally bounded complex of flat $a^{-1}\mathcal{O}_ S$-modules we see the complex $a^{-1}\mathcal{G} \otimes _{a^{-1}\mathcal{O}_ S} \mathcal{F}^\bullet $ represents $L(a')^*\mathcal{G} \otimes _{a^{-1}\mathcal{O}_ S}^\mathbf {L} \mathcal{F}^\bullet $ in $D(a^{-1}\mathcal{O}_ S)$. Hence (4) follows from (5). $\square$

Proof. We may write

Arguing by induction on $b - a$ and considering the distinguished triangle

and using More on Algebra, Lemma 15.59.5 we reduce to the case where $\mathcal{F}^\bullet $ consists of a single quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ placed in degree $0$. In this case the Čech complex for $\mathcal{F}$ and $\mathcal{U}$ is homotopy equivalent to the alternating Čech complex, see Cohomology, Lemma 20.23.6. Since $U_{i_0 \ldots i_ p}$ is always affine, we see that $\mathcal{F}(U_{i_0 \ldots i_ p})$ is $A$-flat. Hence $\check{\mathcal{C}}_{alt}^\bullet (\mathcal{U}, \mathcal{F})$ is a bounded complex of flat $A$-modules and hence K-flat by More on Algebra, Lemma 15.59.7. $\square$

Proof. We strongly urge the reader to read the proof of Varieties, Lemma 33.29.1 first. 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

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

which is compatible with the cup product on cohomology by Cohomology, Lemma 20.31.4. Combining the above we obtain a map of complexes

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 (36.23.3.1). Namely, by Cohomology of Schemes, Lemma 30.2.2 and Cohomology, Lemma 20.25.2 the canonical maps

and

are isomorphisms. On the other hand, the complex $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$ is K-flat by Lemma 36.23.3 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.

We still have to show that (36.23.4.1) 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. The short exact sequence $0 \to \mathcal{F} \to \mathcal{F}' \to \mathcal{F}'' \to 0$ of flat quasi-coherent $\mathcal{O}_ X$-modules produces a short exact sequence of complexes

and a short exact sequence of complexes

Moreover, the maps (36.23.4.1) between these are compatible with these short exact sequences. Hence it suffices to prove (36.23.4.1) is an isomorphism for $\mathcal{F}'$ and $\mathcal{F}''$. Finally, we have $d(\mathcal{F}'') < d(\mathcal{F})$. 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 $K$-flat complex $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$ of $A$-modules sitting in degrees $\geq 0$. It follows that $\Gamma (X, \mathcal{F})$ is a flat $A$-module (because we can compute higher Tor's against this module by tensoring with the Cech complex). 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

(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

on $Y$ where the notation on the right hand side indicates the module

Using the Leray spectral sequence for $q$ we find

Using Lemma 36.22.1 for the morphism $b : Y \to S = \mathop{\mathrm{Spec}}(A)$ and using that $\Gamma (X, \mathcal{F})$ 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 $n = 0$. Of course, here we also use that $\mathcal{G}$ only has cohomology in degree $0$. 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). $\square$