The Stacks project

Lemma 37.75.8. Let $f : X \to Y$ be a morphism of schemes. Assume

  1. $f$ is locally quasi-finite, and

  2. $Y$ is geometrically unibranch and locally Noetherian.

Then there is a weighting $w : X \to \mathbf{Z}_{\geq 0}$ given by the rule that sends $x \in X$ lying over $y \in Y$ to the “generic separable degree” of $\mathcal{O}_{X, x}^{sh}$ over $\mathcal{O}_{Y, y}^{sh}$.

Proof. It follows from Algebra, Lemma 10.156.3 that $\mathcal{O}_{Y, y}^{sh} \to \mathcal{O}_{X, x}^{sh}$ is finite. Since $Y$ is geometrically unibranch there is a unique minimal prime $\mathfrak p$ in $\mathcal{O}_{Y, y}^{sh}$, see More on Algebra, Lemma 15.106.5. Write

\[ (\kappa (\mathfrak p) \otimes _{\mathcal{O}_{Y, y}^{sh}} \mathcal{O}_{X, x}^{sh})_{red} = \prod K_ i \]

as a finite product of fields. We set $w(x) = \sum [K_ i : \kappa (\mathfrak p)]_ s$.

Since this definition is clearly insensitive to étale localization, in order to show that $w$ is a weighting we reduce to showing that if $f$ is a finite morphism, then $\int _ f w$ is locally constant. Observe that the value of $\int _ f w$ in a generic point $\eta $ of $Y$ is just the number of points of the geometric fibre $X_{\overline{\eta }}$ of $X \to Y$ over $\eta $. Moreover, since $Y$ is unibranch a point $y$ of $Y$ is the specialization of a unique generic point $\eta $. Hence it suffices to show that $(\int _ f w)(y)$ is equal to the number of points of $X_{\overline{\eta }}$. After passing to an affine neighbourhood of $y$ we may assume $X \to Y$ is given by a finite ring map $A \to B$. Suppose $\mathcal{O}_{Y, y}^{sh}$ is constructed using a map $\kappa (y) \to k$ into an algebraically closed field $k$. Then

\[ \mathcal{O}_{Y, y}^{sh} \otimes _ A B = \prod \nolimits _{f(x) = y} \prod \nolimits _{\varphi \in \mathop{\mathrm{Mor}}\nolimits _{\kappa (y)}(\kappa (x), k)} \mathcal{O}_{X, x}^{sh} \]

by Algebra, Lemma 10.153.4 and the lemma used above. Observe that the minimal prime $\mathfrak p$ of $\mathcal{O}_{Y, y}^{sh}$ maps to the prime of $A$ corresponding to $\eta $. Hence we see that the desired equality holds because the number of points of a geometric fibre is unchanged by a field extension. $\square$

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