The Stacks project

Lemma 37.58.10. The property $\mathcal{P}(f) =$“$f$ is perfect” is fpqc local on the base.

Proof. We will use the criterion of Descent, Lemma 35.21.4 to prove this. By Definition 37.58.2 being perfect is Zariski local on the base. By Lemma 37.58.3 being perfect is preserved under flat base change. The final hypothesis (3) of Descent, Lemma 35.21.4 translates into the following algebra statement: Let $A \to B$ be a faithfully flat ring map. Let $C = A[x_1, \ldots , x_ n]/I$ be an $A$-algebra. If $C \otimes _ A B$ is perfect as an $B[x_1, \ldots , x_ n]$-module, then $C$ is perfect as a $A[x_1, \ldots , x_ n]$-module. This is More on Algebra, Lemma 15.74.13. $\square$


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