= Coherence reflected by finite direct image
{title2=$\mathcal F\text{ coherent}\iff f_*\mathcal F\text{ coherent}$}
For a <finite morphism> between <Noetherian schemes>, a <quasi-coherent sheaf> is coherent exactly when its <direct image sheaf> is coherent. Over an affine target chart, let $B$ be the finite algebra over the target ring $A$, and let $M$ represent the sheaf on $\operatorname{Spec}B$. Finite generation of $M$ over $A$ implies finite generation over $B$, using the same generators; finite generation over $B$ implies finite generation over $A$ because $B$ is finite over $A$.
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