# Coherent schemes generated by global sections on a Noetherian Scheme

This question arises from the proof of proposition 7.10 in Hartshorne chapter II. Here, we have a Noetherian scheme $$X$$ and a coherent sheaf $$\mathcal{F}$$ of $$\mathcal{O}_X$$-modules which is generated by global sections. Hartshorne claims that $$\mathcal{F}$$ can in fact be generated by finitely many global sections.

How do we see that? Since $$X$$ is Noetherian and $$\mathcal{F}$$ we have a finite affine opening such that $$\mathcal{F}|_U \cong \tilde{M}$$ for some finitely generatered module $$M$$ over a Noetherian ring $$A_i$$. How does this lead us to conclude $$\mathcal{F}$$ is globally generated by finitely many elements? This is the furthest I can go. Any help given would be greatly appreciated!

Pick a finite number of generators of $$M_i$$ as an $$A_i$$ module. Since the restrictions of global sections generate $$M_i$$, we can write each generator of $$M_i$$ as a finite $$A_i$$-linear sum of the global generators (this works because we can only consider finite sums!). Thus we get a finite list of global generators which suffice to generate $$M_i$$. Now take the union of the lists for each $$M_i$$ across all elements of your finite cover, and you get a finite list of global generators for your global sheaf.