The Stacks project

Lemma 29.57.8. Let $f : X \to Y$ be an ├ętale morphism of schemes. Let $n \geq 0$. The following are equivalent

  1. the integer $n$ bounds the degrees of the fibres,

  2. for every field $k$ and morphism $\mathop{\mathrm{Spec}}(k) \to Y$ the base change $X_ k = \mathop{\mathrm{Spec}}(k) \times _ Y X$ has at most $n$ points, and

  3. for every $y \in Y$ and every separable algebraic closure $\kappa (y) \subset \kappa (y)^{sep}$ the scheme $X_{\kappa (y)^{sep}}$ has at most $n$ points.

Proof. This follows from Lemma 29.57.2 and the fact that the fibres $X_ y$ are disjoint unions of spectra of finite separable field extensions of $\kappa (y)$, see Lemma 29.36.7. $\square$

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