The Stacks project

Lemma 37.58.9. Let $f : X \to S$ be a morphism of schemes. If $S$ is locally Noetherian, then $f$ is pseudo-coherent if and only if $f$ is locally of finite type.

Proof. This translates into the following algebra result: If $R \to A$ is a finite type ring map with $R$ Noetherian, then $R \to A$ is pseudo-coherent if and only if $R \to A$ is of finite type. To see this, note that a pseudo-coherent ring map is of finite type by definition. Conversely, if $R \to A$ is of finite type, then we can write $A = R[x_1, \ldots , x_ n]/I$ and it follows from More on Algebra, Lemma 15.64.17 that $A$ is pseudo-coherent as an $R[x_1, \ldots , x_ n]$-module, i.e., $R \to A$ is a pseudo-coherent ring map. $\square$

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