This problem has Hartshorne Height 1
- a) In the discrete topology all points are open and closed, use this with the fact that any is irreducible so show all maps must be constant.
- b) The hard part is to show the last surjection
- Since the map is locally surjective, the plan is to locally find a pre image that glues together.
- For a fixed , take two local preimages on and say b, b' respectively. Then is in the kernel of so comes from some section of . (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 on an open set sends a section s to the product of all its restrictions for .