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.



When a proper morphism of schemes is a closed imbedding?
Yes, it is true : this is EGA IV, Cor. 18.12.6. (your condition means by definition that $f$ is a monomorphism).


