Let $\phi : \mathcal F \to \mathcal G$ be a homomorphism of quasicoherent sheaves on a scheme $X$. What are the necessary properties of the scheme $X$, so that in order to show the surjectivity of $\phi$, would it be enough to show the surjectivity of $\phi$ on stalks only for closed points of $X$ ?
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4$\begingroup$ Why "necessary", not "sufficient"? Passing to coker, for sufficiency it's same to ask for sufficient conditions so q-coh sheaf with vanishing stalks at all closed points is 0. Every q-coh sheaf on a q-compact q-separated scheme is the direct limit of q-coh subsheaves of finite type, so enough to treat the latter case. Then the sheaf vanishes on open around any pt where stalk vanishes, so suffices that open containing all closed pts is whole space; i.e., a non-empty closed set has a closed pt. True in any q-c scheme (EGA 0$_{\rm{I}}$, 2.1.3; I, 2.1.4). So "q-compact, q-sep'td" suffices. $\endgroup$– BCnrdCommented Jul 11, 2010 at 7:42
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4$\begingroup$ A separate comment: I don't know the motivation for the question, but unless it is the role of closed pts in proof of Serre's cohomological criterion for affineness, I recommend letting go from attachment to closed pts when away from classical-type situations (such as loc. finite type over a field) because the property of a pt being closed is not local (e.g., consider generic pt in spec of dvr). I was told that there are schemes without closed pts, but it doesn't really matter: in general non-local schemes, "closed pt" is not such an important concept (unlike classical cases). $\endgroup$– BCnrdCommented Jul 11, 2010 at 8:30
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