Can we prove that the section is globally 0?

Question

Suppose $X$ is an irreducible space (meaning every open nonempty subset is dense), of dimension 1 (meaning the only closed irreducible subspaces are singletons and $X$), such that points are closed in $X$. Suppose $F$ is a sheaf of abelian groups on $X$, and $s \in F(X)$ is a global section such that $s|_{X-\{p\}} = 0$ for some point $p \in X$. Can we conclude that $s =0$ ? What I tried so far is the following: if we can prove $\{x \in X \mid s_x = 0 \in F_x\}$ is closed we're done. But I cannot prove it. I'm also not sure whether the result holds, but it should at least hold if $X$ is a smooth curve and $F$ is the sheaf of $1$-forms on it (I need it in this context). Could someone help me out? I'm completely lost.

Answer 1

The statement in the full generality you've made is not true - consider a skyscraper sheaf located at $p$, for instance. The case you describe where $X$ is a smooth curve is true - when $X$ is smooth, $\Omega^1_X$ is locally free, which means it has no torsion. A nonzero section supported only at $p$ would be torsion, so such a section must be zero.