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II.1.20

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This problem has Hartshorne Height 2. II.1.2 can be used to solve this.

HAPPY

  • Part a) just requires the zero and gluing axiom; and important point is  0 \in \Gamma_{V\cap Z}(V,F|_V) is equivalent to a section whose support is empty. The gluing axiom from the fact that F is already a sheaf, the only thing to be careful is to make sure the elements you get by gluing have support contained in Z.
  • Part b) can be done by using II.1.2 and showing exactness on stalks.
    • Finally for the last statement, note that the stronger condition that for every U, a sheaf map F(U) \to G(U) is surjective, implies that the map F \to G is surjective.

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