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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.

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