II.1.16

Algebraic_Geometry at Wikia

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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 $p \in U$.