II.1.2

This problem has Harthshorne Height 1. This problem shows why its useful to talk about stalks.

Getting Started/Hints

 * a) $$(\ker \phi)_p = \ker (\phi_p)$$: 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$$(\phi_p) = (\mbox{im} \phi)_p$$.  But in this case it will be better to work on the level of presheaves.


 * b)
 * injectivity: when the map is injective, use a) to show $$\ker(\phi_p) = 0$$. 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$$(\phi_p) = G_p$$. For the other direction, argue $$(\mbox{im}\phi)_p = G_p$$ 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.