# II.1.6

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• 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.