LorenzoMath
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I do not know if this counts as number theory, but I came to this question while studying number theory, so I post this here.
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_{*}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_{*}Ox_{y} isomorphic to the direct sum of Ox_{x}, where x runs through the preimage of y?
Thanks
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_{*}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_{*}Ox_{y} isomorphic to the direct sum of Ox_{x}, where x runs through the preimage of y?
Thanks