The Stacks project

Lemma 73.22.2. Let $S$ be a scheme. Let $B$ be a Noetherian algebraic space over $S$. Let $f : X \to B$ be a morphism of algebraic spaces which is locally of finite type and quasi-separated. Let $E \in D(\mathcal{O}_ X)$ be perfect. Let $\mathcal{G}^\bullet $ be a bounded complex of coherent $\mathcal{O}_ X$-modules flat over $B$ with support proper over $B$. Then $K = Rf_*(E \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{G}^\bullet )$ is a perfect object of $D(\mathcal{O}_ B)$.

Proof. The object $K$ is perfect by Lemma 73.22.1. We check the lemma applies: Locally $E$ is isomorphic to a finite complex of finite free $\mathcal{O}_ X$-modules. Hence locally $E \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{G}^\bullet $ is isomorphic to a finite complex whose terms are of the form

\[ \bigoplus \nolimits _{i = a, \ldots , b} (\mathcal{G}^ i)^{\oplus r_ i} \]

for some integers $a, b, r_ a, \ldots , r_ b$. This immediately implies the cohomology sheaves $H^ i(E \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{G})$ are coherent. The hypothesis on the tor dimension also follows as $\mathcal{G}^ i$ is flat over $f^{-1}\mathcal{O}_ Y$. $\square$


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Your email address will not be published.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 0DKJ. Beware of the difference between the letter 'O' and the digit '0'.