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II.1.16

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This problem has Hartshorne Height 1

HAPPY

  • a) In the discrete topology all points are open and closed, use this with the fact that any  U \subset X is irreducible so show all maps  U \to A must be constant.
  • b) The hard part is to show the last surjection  0 \to F'(U) \to F(U) \to F''(U) \to 0
    • Since the map is locally surjective, the plan is to locally find a pre image that glues together.
    • For a fixed  c \in F''(U) , take two local preimages on V\subset U and W\subset U say b, b' respectively. Then  b - b' is in the kernel of F(W \cap V) so comes from some section of F'(W\cap V). (This is all a bit of a diagram chase)
    • Use flasqueness to find a correction element for b,b'. Then use Zorn's lemma to show that all the local preimages can be corrected to be glued together.
  • c) Surround the map you want to be surjective by a bunch of other surjective maps.
  • d) Follows from definitions.
  • e) Straightforward, the map F \to G on an open set U sends a section s to the product of all its restrictions s_p \in F_p for  p \in U.

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