Sections of a torsion sheaf

Question

I am trying to get a better understanding of torsion sheaves over projective schemes.

Is it true that if $\mathcal{F}$ is torsion over a projective scheme $X$, then $$\Gamma(X,\mathcal{F}) \cong \bigoplus_{p\in X} \mathcal{F}_p?$$

Answer 1

No. For instance, let $X = \mathbb{P}^2$, let $i \colon \mathbb{P}^1 \to \mathbb{P}^2$ be the embedding of a line, and let $$ F = i_*\mathcal{O}(-1). $$ Then $F$ is a nonzero torsion sheaf, but $\Gamma(X,F) = 0$.

EDIT. However, if $X$ is a curve, the result is true. Indeed, the question reduces to the case where the support of $F$ is connected, and if $X$ is a curve, this means that $F$ is supported at a point $p$. Finally, in this case the isomorphism $$ \Gamma(X,F) = F_p $$ is obvious --- just note that both sides (as functors from the category of modules over the local ring of $X$ at $p$ to the category of vector spaces) are just the forgetful functors.