= Coherence of an affine module sheaf
{title2=$\widetilde M\text{ coherent}\iff M\text{ finitely generated}$}
For a <Noetherian ring> $A$, a <finitely generated module> $M$ has a <finite presentation of a module>, and kernels of maps from <finite free modules> to $M$ are finitely generated. Its <affine module sheaf> is therefore a <coherent sheaf>. Conversely, if $\widetilde M$ is coherent, choose a finite <principal open subset> cover on which its section <modules> are finitely generated. Lift their finitely many generators to numerators in $M$. The <submodule> generated by these numerators becomes all of $M$ on every chart, and hence is all of $M$: its quotient has zero <localization> at every point. This proves finite generation without assuming that <global sections> preserve arbitrary <sheaf> <surjections>.
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