I would like to know under what condition the morphism $\mathcal{O}\_Y\longrightarrow f\_\ast \mathcal{O}\_X$ induced by a morphism $f:X\longrightarrow Y$ of schemes is injective.


Let me give an example (which I'm not completely sure about though).


I believe, if $X$ and $Y$ are reduced and $f$ is surjective and closed, the morphism $\mathcal{O}\_Y \longrightarrow f\_\ast \mathcal{O}\_X$ is injective.


(Thus, proper flat morphisms of varieties have this property.)


Maybe one could forget about schemes and give a condition for locally ringed spaces? 



Given a morphism from X to Y, when is the morphism from O_Y to the pushforward of O_X injective 
If $f$ is quasi-compact and quasi-separated then the kernel of the map $O\_Y\to f\_\*(O\_X)$ consists of locally nilpotent elements if and only if $f(X)\subset Y$ is a dense set.


