Lemma 43.14.1. Let $X$ be a locally Noetherian scheme.

1. If $\mathcal{F}$ and $\mathcal{G}$ are coherent $\mathcal{O}_ X$-modules, then $\text{Tor}_ p^{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$ is too.

2. If $L$ and $K$ are in $D^-_{\textit{Coh}}(\mathcal{O}_ X)$, then so is $L \otimes _{\mathcal{O}_ X}^\mathbf {L} K$.

Proof. Let us explain how to prove (1) in a more elementary way and part (2) using previously developed general theory.

Proof of (1). Since formation of $\text{Tor}$ commutes with localization we may assume $X$ is affine. Hence $X = \mathop{\mathrm{Spec}}(A)$ for some Noetherian ring $A$ and $\mathcal{F}$, $\mathcal{G}$ correspond to finite $A$-modules $M$ and $N$ (Cohomology of Schemes, Lemma 30.9.1). By Derived Categories of Schemes, Lemma 36.3.9 we may compute the $\text{Tor}$'s by first computing the $\text{Tor}$'s of $M$ and $N$ over $A$, and then taking the associated $\mathcal{O}_ X$-module. Since the modules $\text{Tor}_ p^ A(M, N)$ are finite by Algebra, Lemma 10.75.7 we conclude.

By Derived Categories of Schemes, Lemma 36.10.3 the assumption is equivalent to asking $L$ and $K$ to be (locally) pseudo-coherent. Then $L \otimes _{\mathcal{O}_ X}^\mathbf {L} K$ is pseudo-coherent by Cohomology, Lemma 20.47.5. $\square$

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