Definition 37.72.2. Let $f : X \to Y$ be a locally quasi-finite morphism. A *weighting* or a *pondération* of $f$ is a map $w : X \to \mathbf{Z}$ such that for any diagram

\[ \xymatrix{ X \ar[d]_ f & U \ar[l]^ h \ar[d]^\pi \\ Y & V \ar[l]_ g } \]

where $V \to Y$ is étale, $U \subset X_ V$ is open, and $U \to V$ finite, the function $\int _\pi (w \circ h)$ is locally constant.

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