The Stacks project

Lemma 37.58.14. The property $\mathcal{P}(f) =$“$f$ is perfect” is fppf local on the source.

Proof. Let $\{ g_ i : X_ i \to X\} _{i \in I}$ be an fppf covering of schemes and let $f : X \to S$ be a morphism such that each $f \circ g_ i$ is perfect. By Lemma 37.57.13 we conclude that $f$ is pseudo-coherent. Hence by Lemma 37.58.11 it suffices to check that $\mathcal{O}_{X, x}$ is an $\mathcal{O}_{S, f(x)}$-module of finite tor dimension for all $x \in X$. Pick $i \in I$ and $x_ i \in X_ i$ mapping to $x$. Then we see that $\mathcal{O}_{X_ i, x_ i}$ has finite tor dimension over $\mathcal{O}_{S, f(x)}$ and that $\mathcal{O}_{X, x} \to \mathcal{O}_{X_ i, x_ i}$ is faithfully flat. The desired conclusion follows from More on Algebra, Lemma 15.66.17. $\square$


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