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.

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