The Stacks project

Lemma 37.72.5. Let $f : X \to Y$ and $g : Y \to Z$ be locally quasi-finite morphisms. Let $w_ f : X \to \mathbf{Z}$ be a weighting of $f$ and let $w_ g : Y \to \mathbf{Z}$ be a weighting of $g$. Then the function

\[ X \longrightarrow \mathbf{Z},\quad x \longmapsto w_ f(x) w_ g(f(x)) \]

is a weighting of $g \circ f$.

Proof. Let us set $w_{g \circ f}(x) = w_ f(x) w_ g(f(x))$ for $x \in X$. Consider a diagram

\[ \xymatrix{ X \ar[d]_{g \circ f} & U \ar[l] \ar[d]^\pi \\ Z & W \ar[l] } \]

where $W \to Z$ is étale, $U \subset X_ W$ is open, and $U \to W$ finite. We have to show that $\int _\pi w_{g \circ f}|_ U$ is locally constant. Choose a point $w \in W$. By Lemma 37.72.1 (and the fact that étale morphisms are open) it suffices to show that $\int _\pi w_{g \circ f}|_ U$ is constant after replacing $(W, w)$ by an étale neighbourhood. After replacing $(W, w)$ by an étale neighbourhood we may assume $U = U_1 \amalg \ldots \amalg U_ n$ where each $U_ i$ has a unique point $u_ i$ lying over $w$ such that $\kappa (u_ i)/\kappa (w)$ is purely inseparable, see Lemma 37.40.5. Clearly, it suffices to show that $\int _{U_ i \to W} w_{g \circ f}|_{U_ i}$ is constant in an étale neighbourhood of $w$. This reduces us to the case discussed in the next paragraph.

We have $w \in W$ and there is a unique point $u \in U$ lying over $w$ with $\kappa (u)/\kappa (w)$ purely inseparable. Consider the point $v = f(u) \in Y$. After replacing $(W, w)$ by an elementary étale neighbourhood we may assume there is an open neighbourhood $V \subset Y_ W$ of $v$ such that $V \to W$ is finite, see Lemma 37.40.1. Then $f_ W^{-1}(V) \cap U$ is an open neighbourhood of $u$ where $f_ W : X_ W \to Y_ W$ is the base change of $f$ to $W$. Hence after Zariski shrinking $W$, we may assume $f_ W(U) \subset V$. Thus we obtain morphisms

\[ U \xrightarrow {a} V \xrightarrow {b} W \]

and $U \to V$ is finite as $V \to W$ is separated (because finite). Since $w_ f$ and $w_ g$ are weightings of $f$ and $g$ we see that $\int _ a w_ f|_ U$ is locally constant on $V$ and $\int _ b w_ g|_ V$ is locally constant on $W$. Thus after shrinking $W$ one more time we may assume these functions are constant say with values $n$ and $m$. It follows immediately that $\int _\pi w_{g \circ f}|_ U = \int _{b \circ a} w_{g \circ f}|_ U$ is constant with value $nm$ as desired. $\square$


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