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I'm not sure that this is generally true, but Harthorne p73 seems to suggest it. If it is true could someone give me a hint for the proof?

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    @EdwardHughes:I am sorry but I don't quite see what you mean. See if you agree with this: $f_{*,x}:\mathcal O_{Y,f(P)} \to f_*(\mathcal O_X)_{f(P)}$ is an isomorphism.2012-10-16

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Yes. See Stacks project, (21.2.2).