The Stacks project

Lemma 29.48.5. Let $f : X \to S$ be a finite locally free morphism of schemes. There exists a disjoint union decomposition $S = \coprod _{d \geq 0} S_ d$ by open and closed subschemes such that setting $X_ d = f^{-1}(S_ d)$ the restrictions $f|_{X_ d}$ are finite locally free morphisms $X_ d \to S_ d$ of degree $d$.

Proof. This is true because a finite locally free sheaf locally has a well defined rank. Details omitted. $\square$

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