# II.1.2

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

### HAPPY

• 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.