The Stacks project

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. $\square$