This problem has Harthshorne Height 1. This problem shows why its useful to talk about stalks.
- a) : in words, the things that map to zero near p are the same as the things near p that map to zero. It can be proved by showing an element of one group defines and element of the other group and vice versa. Similarly for im. But in this case it will be better to work on the level of presheaves.
- injectivity: when the map is injective, use a) to show . When you have injectivity on stalks, use the sheaf axioms to show the map is injective on every open set.
- surjectivity: when the map is surjective, use a) to show im. For the other direction, argue and use the proposition II.1.1.
- c) another application of part a for one direction and proposition II.1.1 for the other direction.