The Stacks project

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

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

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