Lemma 36.32.3 (0B9S)—The Stacks project

Processing math: 100%

Table of contents
Part 2: Schemes
Chapter 36: Derived Categories of Schemes
Section 36.32: Applications
Lemma 36.32.3 (cite)

numberstags

Lemma 36.32.3. Let $f : X \to S$ be a proper morphism of finite presentation. Let $\mathcal{F}$ be an $\mathcal{O}_ X$ -module of finite presentation, flat over $S$ . Fix $i, r \in \mathbf{Z}$ . Then there exists an open subscheme $U \subset S$ with the following property: A morphism $T \to S$ factors through $U$ if and only if $Rf_{T, *}\mathcal{F}_ T$ is isomorphic to a finite locally free module of rank $r$ placed in degree $i$ .

Proof. By cohomology and base change (more precisely by Lemma 36.30.4) the object $K = Rf_*\mathcal{F}$ is a perfect object of the derived category of $S$ whose formation commutes with arbitrary base change. Thus this lemma follows immediately from Lemma 36.31.3. $\square$

Comments (2)

Comment #7876 by qyk on November 17, 2022 at 09:30

Is the condition that $f$ is flat redundant? We can use (2) in \href{https://stacks.math.columbia.edu/tag/0B91}{Lemma 0B91}, which does not need flatness of $f$ .

Comment #8147 by Aise Johan de Jong on February 27, 2023 at 13:51

THanks and fixed here.

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).

Comments (2)

Post a comment