Sunday, 7 June 2009

ag.algebraic geometry - Given a morphism from X to Y, when is the morphism from O_Y to the pushforward of O_X injective

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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