Proof. This translates into the following algebra result: Let $A \to B$ be a perfect ring map. Let $A \to A'$ be flat. Then $A' \to B \otimes _ A A'$ is perfect. This result for pseudo-coherent ring maps we have seen in Lemma 37.57.3. The corresponding fact for finite tor dimension follows from More on Algebra, Lemma 15.66.14. $\square$

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