Lemma 29.4.1. Let $i : Z \to X$ be a closed immersion of schemes. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the quasi-coherent sheaf of ideals cutting out $Z$. The functor

$i_* : \mathit{QCoh}(\mathcal{O}_ Z) \longrightarrow \mathit{QCoh}(\mathcal{O}_ X)$

is exact, fully faithful, with essential image those quasi-coherent $\mathcal{O}_ X$-modules $\mathcal{G}$ such that $\mathcal{I}\mathcal{G} = 0$.

Proof. A closed immersion is quasi-compact and separated, see Lemmas 29.2.6 and 29.2.7. Hence Schemes, Lemma 26.24.1 applies and the pushforward of a quasi-coherent sheaf on $Z$ is indeed a quasi-coherent sheaf on $X$.

By Modules, Lemma 17.13.4 the functor $i_*$ is fully faithful.

Now we turn to the description of the essential image of the functor $i_*$. We have $\mathcal{I}(i_*\mathcal{F}) = 0$ for any quasi-coherent $\mathcal{O}_ Z$-module, for example by Modules, Lemma 17.13.4. Next, suppose that $\mathcal{G}$ is any quasi-coherent $\mathcal{O}_ X$-module such that $\mathcal{I}\mathcal{G} = 0$. It suffices to show that the canonical map

$\mathcal{G} \longrightarrow i_* i^*\mathcal{G}$

is an isomorphism1. In the case of schemes and quasi-coherent modules, working affine locally on $X$ and using Lemma 29.2.1 and Schemes, Lemma 26.7.3 it suffices to prove the following algebraic statement: Given a ring $R$, an ideal $I$ and an $R$-module $N$ such that $IN = 0$ the canonical map

$N \longrightarrow N \otimes _ R R/I,\quad n \longmapsto n \otimes 1$

is an isomorphism of $R$-modules. Proof of this easy algebra fact is omitted. $\square$

[1] This was proved in a more general situation in the proof of Modules, Lemma 17.13.4.

Comment #3989 by Jonas Ehrhard on

I'm not sure what is referened by "our local description above" and "exactly the same arguments as above". Though the fact that $\mathcal{G} \rightarrow i_\*i^\*\mathcal{G}$ is an isomorphism is also proved in 17.13.4, maybe just cite this, or extract that fact into another lemma?

Comment #3990 by Jonas Ehrhard on

I meant $\mathcal{G} \rightarrow i_*i^*\mathcal{G}$ is an isomorphism.

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.

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 01QY. Beware of the difference between the letter 'O' and the digit '0'.