The Stacks project

Lemma 37.59.12. Let $i : Z \to X$ be a perfect closed immersion of schemes. Then $i_*\mathcal{O}_ Z$ is a perfect $\mathcal{O}_ X$-module, i.e., it is a perfect object of $D(\mathcal{O}_ X)$.

Proof. This is more or less immediate from the definition. Namely, let $U = \mathop{\mathrm{Spec}}(A)$ be an affine open of $X$. Then $i^{-1}(U) = \mathop{\mathrm{Spec}}(A/I)$ for some ideal $I \subset A$ and $A/I$ has a finite resolution by finite projective $A$-modules by More on Algebra, Lemma 15.82.2. Hence $i_*\mathcal{O}_ Z|_ U$ can be represented by a finite length complex of finite locally free $\mathcal{O}_ U$-modules. This is what we had to show, see Cohomology, Section 20.47. $\square$

Comments (2)

Comment #6652 by Brian Shin on

Very small typo: should be

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Your email address will not be published.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 0G2E. Beware of the difference between the letter 'O' and the digit '0'.