The Stacks project

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.

Comments (3)

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 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 is an isomorphism.

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