The Stacks project

Lemma 36.28.1. Assumptions and notation as in Lemma 36.27.2. Then there are functorial isomorphisms

\[ H^ i(S, K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}) \longrightarrow H^ i(X, E \otimes _{\mathcal{O}_ X}^\mathbf {L} (\mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F})) \]

for $\mathcal{F}$ quasi-coherent on $S$ compatible with boundary maps (see proof).

Proof. We have

\[ \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*\mathcal{F} = \mathcal{G}^\bullet \otimes _{f^{-1}\mathcal{O}_ S}^\mathbf {L} f^{-1}\mathcal{F} = \mathcal{G}^\bullet \otimes _{f^{-1}\mathcal{O}_ S} f^{-1}\mathcal{F} = \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F} \]

the first equality by Cohomology, Lemma 20.27.4, the second as $\mathcal{G}^ n$ is a flat $f^{-1}\mathcal{O}_ S$-module, and the third by definition of pullbacks. Hence we obtain

\begin{align*} H^ i(X, E \otimes ^\mathbf {L}_{\mathcal{O}_ X} (\mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F})) & = H^ i(X, E \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*\mathcal{F}) \\ & = H^ i(S, Rf_*(E \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{G}^\bullet \otimes ^\mathbf {L}_{\mathcal{O}_ X} Lf^*\mathcal{F})) \\ & = H^ i(S, Rf_*(E \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{G}^\bullet ) \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}) \\ & = H^ i(S, K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}) \end{align*}

The first equality by the above, the second by Leray (Cohomology, Lemma 20.13.1), and the third equality by Lemma 36.22.1. The statement on boundary maps means the following: Given a short exact sequence $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ of quasi-coherent $\mathcal{O}_ S$-modules, the isomorphisms fit into commutative diagrams

\[ \xymatrix{ H^ i(S, K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}_3) \ar[r] \ar[d]_\delta & H^ i(X, E \otimes ^\mathbf {L}_{\mathcal{O}_ X} (\mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}_3)) \ar[d]^\delta \\ H^{i + 1}(S, K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}_1) \ar[r] & H^{i + 1}(X, E \otimes ^\mathbf {L}_{\mathcal{O}_ X} (\mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}_1)) } \]

where the boundary maps come from the distinguished triangle

\[ K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}_1 \to K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}_2 \to K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}_3 \to K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \mathcal{F}_1[1] \]

and the distinguished triangle in $D(\mathcal{O}_ X)$ associated to the short exact sequence

\[ 0 \to \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}_1 \to \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}_2 \to \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}_3 \to 0 \]

of complexes of $\mathcal{O}_ X$-modules. This sequence is exact because $\mathcal{G}^ n$ is flat over $S$. We omit the verification of the commutativity of the displayed diagram. $\square$

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