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Let $X$ and $Y$ be finitely presented schemes over $\mathbb{C}$. Let $f\colon X\to Y$ be a proper morphism. Let us assume that for any finitely presented scheme $S$ the induced map $$Mor_{Sch}(S,X)\to Mor_{Sch}(S,Y)$$ is injective.

Question. Is it true that $f$ is a closed imbedding?

The simplest case which I do not understand is the case of $X$ and $Y$ being spectrums of local Artinian rings.

I am also interested in the analogous situation when finitely presented schemes over $\mathbb{C}$ are replaced by complex analytic spaces.

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    $\begingroup$ There is a notion of a closed subfunctor (introduced by Grothendieck), see e.g. [FGA]. Of course $X \to Y$ is a closed embedding if and only if $Mor(-,X)$ is a closed subfunctor in $Mor(-,Y)$. $\endgroup$
    – Sasha
    Commented Nov 24, 2014 at 6:42

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Yes, it is true : this is EGA IV, Cor. 18.12.6. (your condition means by definition that $f$ is a monomorphism).

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  • $\begingroup$ See also the MO question mathoverflow.net/questions/45578/… where where the same question has been solved, and an easier version (with finite presentation hypothesis) of IV.18.12.6 is referenced. $\endgroup$
    – pbelmans
    Commented Nov 24, 2014 at 10:00

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