The kernel of a morphism of quasi-coherent sheaves on a scheme $(X,\mathcal{O}_X)$ is quasi-coherent.

Question

Let $(X,\mathcal{O}_X)$ be a scheme. I know that an $\mathcal{O}_X$-module $\mathcal{F}$ is quasi-coherent if for each $x \in X$ there exists an open neighborhood $U$ of $x$ and an exact sequence of $\mathcal{O}_X$-modules

$ \left. \mathcal{O}_X^{(J)} \right\vert_{U} \to \left. \mathcal{O}_X^{(I)} \right\vert_{U} \to \mathcal{F} \to 0$.

How can i use that definition to prove that given a morphism $\alpha \colon \mathcal{F} \to \mathcal{G}$ of quasi-coherent $\mathcal{O}_X$-modules, then $\mathrm{Ker} \mbox{ } \alpha$ is a quasi-coherent $\mathcal{O}_X$-module?

Answer 1

At some point you will have to use that quasicoherent sheaves of modules have a nice form on affine opens. For general (locally) ringed spaces, qc sheaves of modules can be a bit pathological. I'm sure there's a counterexample somewhere in the stacks project... Your definition is the general one for ringed spaces, so at some point you will have to add extra ingredients!

In any case, the proof outline goes roughly like this:

1. Reduce to the affine case. (quasicoherence being a local property)
2. Apply the tilde functor (the functor that associated a sheaf of modules to a module) to the kernel-image short exact sequence.
3. Use the fact that the tilde functor is exact, to see that the kernel is qc.

For reference, this is Cor I.1.3.9 in EGA I or Prop. 7.14 in Görtz + Wedhorn.