Lemma 21.18.5 (08I6)—The Stacks project

Lemma 21.18.5. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ be a morphism of ringed topoi. There is a canonical bifunctorial isomorphism

$\mathcal{F}^\bullet \otimes _\mathcal {O}^{\mathbf{L}} Lf^*\mathcal{G}^\bullet = \mathcal{F}^\bullet \otimes _{f^{-1}\mathcal{O}_ Y}^{\mathbf{L}} f^{-1}\mathcal{G}^\bullet$

for $\mathcal{F}^\bullet$ in $D(\mathcal{O})$ and $\mathcal{G}^\bullet$ in $D(\mathcal{O}')$ .

Proof. Let $\mathcal{F}$ be an $\mathcal{O}$ -module and let $\mathcal{G}$ be an $\mathcal{O}'$ -module. Then $\mathcal{F} \otimes _{\mathcal{O}} f^*\mathcal{G} = \mathcal{F} \otimes _{f^{-1}\mathcal{O}'} f^{-1}\mathcal{G}$ because $f^*\mathcal{G} = \mathcal{O} \otimes _{f^{-1}\mathcal{O}'} f^{-1}\mathcal{G}$ . The lemma follows from this and the definitions. $\square$

Comments (1)

Comment #9814 by Amos Elsworthy on October 21, 2024 at 13:20

Small typo in the statement of the lemma: the derived tensor product on the right side should be over $f^{-1}\mathcal{O}'$ , not $f^{-1}\mathcal{O}_Y$ .

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