Im not sure if Ive got this right:
Let X be an integral scheme and $\mathcal{F}$ a coherent sheaf. Then $\mathcal{F}$ is torsion if and only if it is not supported at the generic point. It is is easy to see that if the stalk vanishes, any open affine section must be torsion,but Im not sure as to why every section over any open should be torsion.
Any help?
Thanks in advance
Best Answer
Recall the definition of a torsion sheaf. It is a sheaf associated to a torsion presheaf. Equivalently, every stalk is torsion. In particular it suffices to prove this locally. Therefore, open affines suffice.
(It is not true that every torsion sheaf has torsion global sections; we need quasi-compactness for this)