The Stacks project

Situation 42.50.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. Let $i : Z \to X$ be a closed immersion. Let $E \in D(\mathcal{O}_ X)$ be an object. Let $p \geq 0$. Assume

  1. $E$ is a perfect object of $D(\mathcal{O}_ X)$,

  2. the restriction $E|_{X \setminus Z}$ is zero, resp. isomorphic to a finite locally free $\mathcal{O}_{X \setminus Z}$-module of rank $< p$ sitting in cohomological degree $0$, and

  3. at least one1 of the following is true: (a) $X$ is quasi-compact, (b) $X$ has quasi-compact irreducible components, (c) there exists a locally bounded complex of finite locally free $\mathcal{O}_ X$-modules representing $E$, or (c) there exists a morphism $X \to X'$ of schemes locally of finite type over $S$ such that $E$ is the pullback of a perfect object on $X'$ and the irreducible components of $X'$ are quasi-compact.

[1] Please ignore this technical condition on a first reading; see discussion in Remark 42.50.5.

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Your email address will not be published.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