The Stacks project

Lemma 15.82.5. Let $A \to B$ be a finite type ring map with $A$ a regular ring of finite dimension. Then $A \to B$ is perfect.

Proof. By Algebra, Lemma 10.110.8 the assumption on $A$ means that $A$ has finite global dimension. Hence every module has finite tor dimension, see Lemma 15.66.19, in particular $B$ does. By Lemma 15.82.3 the map is pseudo-coherent. $\square$

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