This problem has Hartshorne Height 2. II.1.2 can be used to solve this.
- Part a) just requires the zero and gluing axiom; and important point is is equivalent to a section whose support is empty. The gluing axiom from the fact that is already a sheaf, the only thing to be careful is to make sure the elements you get by gluing have support contained in .
- Part b) can be done by using II.1.2 and showing exactness on stalks.
- Finally for the last statement, note that the stronger condition that for every , a sheaf map is surjective, implies that the map is surjective.