Lemma 75.23.1. Assumptions and notation as in Lemma 75.22.2. Then there are functorial isomorphisms
\[ H^ i(B, K \otimes ^\mathbf {L}_{\mathcal{O}_ B} \mathcal{F}) \longrightarrow H^ i(X, E \otimes ^\mathbf {L}_{\mathcal{O}_ X} (\mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F})) \]
for $\mathcal{F}$ quasi-coherent on $B$ compatible with boundary maps (see proof).
Proof.
We have
\[ \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*\mathcal{F} = \mathcal{G}^\bullet \otimes _{f^{-1}\mathcal{O}_ B}^\mathbf {L} f^{-1}\mathcal{F} = \mathcal{G}^\bullet \otimes _{f^{-1}\mathcal{O}_ B} f^{-1}\mathcal{F} = \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F} \]
the first equality by Cohomology on Sites, Lemma 21.18.5, the second as $\mathcal{G}^ n$ is a flat $f^{-1}\mathcal{O}_ B$-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(B, Rf_*(E \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{G}^\bullet \otimes ^\mathbf {L}_{\mathcal{O}_ X} Lf^*\mathcal{F})) \\ & = H^ i(B, Rf_*(E \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{G}^\bullet ) \otimes ^\mathbf {L}_{\mathcal{O}_ B} \mathcal{F}) \\ & = H^ i(B, K \otimes ^\mathbf {L}_{\mathcal{O}_ B} \mathcal{F}) \end{align*}
The first equality by the above, the second by Leray (Cohomology on Sites, Remark 21.14.4), and the third equality by Lemma 75.20.1. The statement on boundary maps means the following: Given a short exact sequence $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ then the isomorphisms fit into commutative diagrams
\[ \xymatrix{ H^ i(B, K \otimes ^\mathbf {L}_{\mathcal{O}_ B} \mathcal{F}_3) \ar[r] \ar[d]_\delta & H^ i(X, E \otimes ^\mathbf {L}_{\mathcal{O}_ X} (\mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}_3)) \ar[d]^\delta \\ H^{i + 1}(B, K \otimes ^\mathbf {L}_{\mathcal{O}_ B} \mathcal{F}_1) \ar[r] & H^{i + 1}(X, E \otimes ^\mathbf {L}_{\mathcal{O}_ X} (\mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}_1)) } \]
where the boundary maps come from the distinguished triangle
\[ K \otimes ^\mathbf {L}_{\mathcal{O}_ B} \mathcal{F}_1 \to K \otimes ^\mathbf {L}_{\mathcal{O}_ B} \mathcal{F}_2 \to K \otimes ^\mathbf {L}_{\mathcal{O}_ B} \mathcal{F}_3 \to K \otimes ^\mathbf {L}_{\mathcal{O}_ B} \mathcal{F}_1[1] \]
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. This sequence is exact because $\mathcal{G}^ n$ is flat over $B$. We omit the verification of the commutativity of the displayed diagram.
$\square$
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