Lemma 21.20.9 (0GL1)—The Stacks project

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Part 1: Preliminaries
Chapter 21: Cohomology on Sites
Section 21.20: Some properties of K-injective complexes
Lemma 21.20.9 (cite)

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Lemma 21.20.9. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ . For $L$ in $D(\mathcal{O}_ U)$ and $K$ in $D(\mathcal{O})$ we have $j_!L \otimes _\mathcal {O}^\mathbf {L} K = j_!(L \otimes _{\mathcal{O}_ U}^\mathbf {L} K|_ U)$ .

Proof. Represent $L$ by a complex of $\mathcal{O}_ U$ -modules and $K$ by a K-flat complexe of $\mathcal{O}$ -modules and apply Modules on Sites, Lemma 18.27.9. Details omitted. $\square$

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