Lemma 35.20.11. The property $\mathcal{P}(f) =$“$f$ is locally of finite presentation” is fpqc local on the base.

Proof. Being locally of finite presentation is preserved under base change, see Morphisms, Lemma 29.21.4. Being locally of finite type is Zariski local on the base, see Morphisms, Lemma 29.21.2. Finally, let $S' \to S$ be a flat surjective morphism of affine schemes, and let $f : X \to S$ be a morphism. Assume that the base change $f' : X' \to S'$ is locally of finite presentation. Let $U \subset X$ be an affine open. Then $U' = S' \times _ S U$ is affine and of finite type over $S'$. Write $S = \mathop{\mathrm{Spec}}(R)$, $S' = \mathop{\mathrm{Spec}}(R')$, $U = \mathop{\mathrm{Spec}}(A)$, and $U' = \mathop{\mathrm{Spec}}(A')$. We know that $R \to R'$ is faithfully flat, $A' = R' \otimes _ R A$ and $R' \to A'$ is of finite presentation. We have to show that $R \to A$ is of finite presentation. This is the result of Algebra, Lemma 10.126.2. It follows that $f$ is locally of finite presentation. Therefore Lemma 35.19.4 applies and we win. $\square$

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