# II.1.14

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This problem has Hartshorne Height 1.

### HAPPY

• Show the complement of $\mbox{Supp} s$ is open. If $s \in F(U)$ then since the stalk is a direct limit, for s to map to 0 in the stalk means it maps to 0 in some small enough open neighborhood of the point. So the any point in the complement is an interior point.
• One example for $\mbox{Supp} F$ not being closed. Take $X = \mathbb{R}$ with the usual topology. The for any open set $U$ consider the set of real valued functions which are 0 if $0 \in U$. Then $\mbox{Supp} F$ is $\mathbb{R} \backslash 0$