The Stacks project

Lemma 26.24.2. Let $f : X \to Y$ be a morphism of schemes. Suppose that

  1. $f$ induces a homeomorphism of $X$ with a closed subset of $Y$, and

  2. $f^\sharp : \mathcal{O}_ Y \to f_*\mathcal{O}_ X$ is surjective.

Then $f$ is a closed immersion of schemes.

Proof. Assume (1) and (2). By (1) the morphism $f$ is quasi-compact (see Topology, Lemma 5.12.3). Conditions (1) and (2) imply conditions (1) and (2) of Lemma 26.23.7. Hence $f : X \to Y$ is a monomorphism. In particular, $f$ is separated, see Lemma 26.23.3. Hence Lemma 26.24.1 above applies and we conclude that $f_*\mathcal{O}_ X$ is a quasi-coherent $\mathcal{O}_ Y$-module. Therefore the kernel of $\mathcal{O}_ Y \to f_*\mathcal{O}_ X$ is quasi-coherent by Lemma 26.7.8. Since a quasi-coherent sheaf is locally generated by sections (see Modules, Definition 17.10.1) this implies that $f$ is a closed immersion, see Definition 26.4.1. $\square$

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