The Stacks project

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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0AZS. Beware of the difference between the letter 'O' and the digit '0'.