The Stacks project

Lemma 73.5.6. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For objects $K, L$ of $D_\mathit{QCoh}(\mathcal{O}_ X)$ the derived tensor product $K \otimes ^\mathbf {L} L$ is in $D_\mathit{QCoh}(\mathcal{O}_ X)$.

Proof. Let $\varphi : U \to X$ be a surjective ├ętale morphism from a scheme $U$. Since $\varphi ^*(K \otimes _{\mathcal{O}_ X}^\mathbf {L} L) = \varphi ^*K \otimes _{\mathcal{O}_ U}^\mathbf {L} \varphi ^*L$ we see from Lemma 73.5.2 that this follows from the case of schemes which is Derived Categories of Schemes, Lemma 36.3.9. $\square$


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