Lemma 35.23.30. 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 29.48.2). Hence this follows from Lemmas 35.23.23, 35.23.15, and 35.23.11. 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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