II.1.6

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

Hints

 * 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.