Lemma 37.57.10. The property $\mathcal{P}(f) =$“$f$ is pseudo-coherent” is fpqc local on the base.

**Proof.**
We will use the criterion of Descent, Lemma 35.21.4 to prove this. By Definition 37.57.2 being pseudo-coherent is Zariski local on the base. By Lemma 37.57.3 being pseudo-coherent 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 pseudo-coherent as an $B[x_1, \ldots , x_ n]$-module, then $C$ is pseudo-coherent as a $A[x_1, \ldots , x_ n]$-module. This is More on Algebra, Lemma 15.64.15.
$\square$

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