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LorenzoMath

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Suppose X and Y are reduced schemes of finite type over a field k and f:X->Y is a morphism. Ox denotes the structural sheaf of X.

Question

What is a stalk of the direct image sheaf f[tex]_{*}[/tex]Ox at y in Y?

Since this question is too general, here is a specific question I encountered.

In addition to the assumptions above, let f be flat and y be a generic point of some irreducible component of Y. In this case, is f[tex]_{*}[/tex]Ox[tex]_{y}[/tex] isomorphic to the direct sum of Ox[tex]_{x}[/tex], where x runs through the preimage of y?

Thanks