I'm working on exercise II.5.15 in Hartshorne's book. I need to prove the following bit.
Let $ X $ be a noetherian scheme. Let $\mathscr {F }$ be a quasi coherent sheaf on $ X$. Then the subsheaf $ \mathscr{G}\subset\mathscr{F}$ generated by $s\in \mathscr{F}(X)$ is coherent.
One idea I have is to show that $ \mathscr{G}(V)=\mathcal{O}_{X}(V)s$. I'm unable to prove that the gluing axiom of sheaves in this case.
This answer explains the idea of a subsheaf generated by sections. However, the definition is nonconstructive and I don't see a way to use it:
Some question of sheaf generated by sections
As a bonus, can we generalise this question to finitely many sections?
0) To a section $s\in \mathscr F(X)$ of an $\mathscr O$-module (quasi-coherent or not) we can associate a morphism of $\mathscr O$-modules $h=h_s:\mathscr O\to \mathscr F$ given by $\mathscr O(U)\to \mathscr F(U):f\mapsto f\cdot (s\mid U) $.
The subsheaf of $\mathscr F$ generated by $s$ is then the image $Im(s)=s(\mathscr O)\subset \mathscr F$ of $h$.
1) If now $\mathscr F$ happens to be quasi-coherent then that image is quasi-coherent, like any image of a morphism between quasi-coherent $\mathscr O$-modules.
2) The same technique shows that a section of a coherent sheaf generates a coherent subsheaf as soon as $\mathscr O$ is coherent (which strangely is not always the case if $X$ is not noetherian).
NB
Hartshorne defines "coherent sheaf" in an idiosyncratic way , different from the definition in FAC, EGA or essentially all the literature in algebraic or analytic geometry. This is an unfortunate choice.