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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