Lemma 37.60.8. Let f : X \to S be a finite morphism of schemes. Then f is pseudo-coherent if and only if f_*\mathcal{O}_ X is pseudo-coherent as an \mathcal{O}_ S-module.
Proof. Translated into algebra this lemma says the following: If R \to A is a finite ring map, then R \to A is pseudo-coherent as a ring map (which means by definition that A as an A-module is pseudo-coherent relative to R) if and only if A is pseudo-coherent as an R-module. This follows from the more general More on Algebra, Lemma 15.81.5. \square
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