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This problem has Hartshorne Height 2; II.1.2 can be used to solve it.
- a) Show exactness on stalks (this is II.1.2); show . 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 is exact and use the 5-lemma with the original exact sequence to get the desired result.