I would like to know under what condition the morphism mathcalOYlongrightarrowfastmathcalOX induced by a morphism f:XlongrightarrowY 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 mathcalOYlongrightarrowfastmathcalOX 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?
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