# II.1.20

219,534pages on
this wiki

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

### HAPPY

• Part a) just requires the zero and gluing axiom; and important point is $0 \in \Gamma_{V\cap Z}(V,F|_V)$ is equivalent to a section whose support is empty. The gluing axiom from the fact that $F$ is already a sheaf, the only thing to be careful is to make sure the elements you get by gluing have support contained in $Z$.
• 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 $U$, a sheaf map $F(U) \to G(U)$ is surjective, implies that the map $F \to G$ is surjective.