Lemma 21.19.4. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ be a morphism of ringed topoi. There is a canonical bifunctorial isomorphism

$Lf^*( \mathcal{F}^\bullet \otimes _{\mathcal{O}'}^{\mathbf{L}} \mathcal{G}^\bullet ) = Lf^*\mathcal{F}^\bullet \otimes _{\mathcal{O}}^{\mathbf{L}} Lf^*\mathcal{G}^\bullet$

for $\mathcal{F}^\bullet , \mathcal{G}^\bullet \in \mathop{\mathrm{Ob}}\nolimits (D(\mathcal{O}'))$.

Proof. By Lemma 21.19.1 we may assume that $\mathcal{F}^\bullet$ and $\mathcal{G}^\bullet$ are K-flat complexes of $\mathcal{O}'$-modules such that $f^*\mathcal{F}^\bullet$ and $f^*\mathcal{G}^\bullet$ are K-flat complexes of $\mathcal{O}$-modules. In this case $\mathcal{F}^\bullet \otimes _{\mathcal{O}'}^{\mathbf{L}} \mathcal{G}^\bullet$ is just the total complex associated to the double complex $\mathcal{F}^\bullet \otimes _{\mathcal{O}'} \mathcal{G}^\bullet$. By Lemma 21.18.5 $\text{Tot}(\mathcal{F}^\bullet \otimes _{\mathcal{O}'} \mathcal{G}^\bullet )$ is K-flat also. Hence the isomorphism of the lemma comes from the isomorphism

$\text{Tot}(f^*\mathcal{F}^\bullet \otimes _{\mathcal{O}} f^*\mathcal{G}^\bullet ) \longrightarrow f^*\text{Tot}(\mathcal{F}^\bullet \otimes _{\mathcal{O}'} \mathcal{G}^\bullet )$

whose constituents are the isomorphisms $f^*\mathcal{F}^ p \otimes _{\mathcal{O}} f^*\mathcal{G}^ q \to f^*(\mathcal{F}^ p \otimes _{\mathcal{O}'} \mathcal{G}^ q)$ of Modules on Sites, Lemma 18.26.1. $\square$

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