II.1.16

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).  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 (This is all a bit of a diagram chase).
 * 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 $$ p \in U$$.