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II.1.6

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This problem has Hartshorne Height 2; II.1.2 can be used to solve it.

Sketch

  • a) Show exactness on stalks (this is II.1.2); show F_p/F'_p \cong (F/F')_p. This can be done by a universal property argument since you know what the universal property of $(F/F')_p$ is.
  • b) use a) to show 0\to \mbox{im}\alpha \to F \to F/\mbox{im}\alpha \to 0 is exact and use the 5-lemma with the original exact sequence to get the desired result.

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