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?


2 Answers 2


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.


This is the condition of $f$ being scheme-theoretically dominant. (If $f$ satisfies the closely related condition considered in LRG's answer, of having dense image, one says that $f$ is dominant.) These conditions are discussed very carefully in the stacks project. (Google "stacks project" if you don't know the link already.) (Also, the precise definitions may involve finiteness conditions that I am omitting here; the stacks project write-up will have complete details.)

  • 1
    $\begingroup$ I looked at the chapters "Morphisms" and "More on morphisms", but I couldn't find it. Which chapter should I look at? $\endgroup$ Apr 11, 2010 at 14:03
  • $\begingroup$ The best reference for this question I have found is Grothendieck, EGA, I, 9.5. $\endgroup$ Jun 17, 2010 at 14:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.