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If $f \colon X \to Y$ is a proper monomorphism of schemes [monomorphism in the sense of category theory], then the underlying map of topological spaces is continuous, closed, and injective, hence a homeomorphism onto its image. Properness also implies finite type, which eliminates some of the more egregious monomorphisms of schemes. But I don't see any obvious argument (or counterexample) for why the corresponding morphism of sheaves $\mathcal{O}_Y \to f_* \mathcal{O}_X$ should be surjective.

Motivation: $X \to Y$ is a monomorphism iff the corresponding morphism of functors $\mathrm{Hom}(-,X) \to \mathrm{Hom}(-,Y)$ is injective. Thus, a positive answer to my question here would, in nice situations, allow for a valuative criterion as to whether a subfunctor is closed.

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If $f$ is of finite presentation then the answer is yes; cf. EGA IV${}_3$ 8.11.5. – Sam Lichtenstein Nov 10 '10 at 18:22
Without assuming finite presentation, it is EGA IV$_4$, 18.12.6 (and 18.12 has plenty of other nifty results removing finite presentation restrictions). – BCnrd Nov 10 '10 at 18:59

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