Lemma 36.3.9 (08DX)—The Stacks project

Table of contents
Part 2: Schemes
Chapter 36: Derived Categories of Schemes
Section 36.3: Derived category of quasi-coherent modules
Lemma 36.3.9 (cite)

previous lemma

Lemma 36.3.9. Let $X$ be a scheme.

For objects $K, L$ of $D_\mathit{QCoh}(\mathcal{O}_ X)$ the derived tensor product $K \otimes ^\mathbf {L}_{\mathcal{O}_ X} L$ is in $D_\mathit{QCoh}(\mathcal{O}_ X)$.
If $X = \mathop{\mathrm{Spec}}(A)$ is affine then

\[ \widetilde{M^\bullet } \otimes _{\mathcal{O}_ X}^\mathbf {L} \widetilde{K^\bullet } = \widetilde{M^\bullet \otimes _ A^\mathbf {L} K^\bullet } \]

for any pair of complexes of $A$-modules $K^\bullet $, $M^\bullet $.

Proof. The equality of (2) follows immediately from Lemma 36.3.6 and the construction of the derived tensor product. To see (1) let $K, L$ be objects of $D_\mathit{QCoh}(\mathcal{O}_ X)$. To check that $K \otimes ^\mathbf {L} L$ is in $D_\mathit{QCoh}(\mathcal{O}_ X)$ we may work locally on $X$, hence we may assume $X = \mathop{\mathrm{Spec}}(A)$ is affine. By Lemma 36.3.5 we may represent $K$ and $L$ by complexes of $A$-modules. Then part (2) implies the result. $\square$

The Stacks project

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