The Stacks project

Lemma 34.20.30. The property $\mathcal{P}(f) =$“$f$ is finite locally free” is fpqc local on the base. Let $d \geq 0$. The property $\mathcal{P}(f) =$“$f$ is finite locally free of degree $d$” is fpqc local on the base.

Proof. Being finite locally free is equivalent to being finite, flat and locally of finite presentation (Morphisms, Lemma 28.46.2). Hence this follows from Lemmas 34.20.23, 34.20.15, and 34.20.11. If $f : Z \to U$ is finite locally free, and $\{ U_ i \to U\} $ is a surjective family of morphisms such that each pullback $Z \times _ U U_ i \to U_ i$ has degree $d$, then $Z \to U$ has degree $d$, for example because we can read off the degree in a point $u \in U$ from the fibre $(f_*\mathcal{O}_ Z)_ u \otimes _{\mathcal{O}_{U, u}} \kappa (u)$. $\square$


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