The Stacks project

Lemma 37.73.4. Let $f : X \to Y$ be a separated, locally quasi-finite morphism with finite fibres. Let $w : X \to \mathbf{Z}_{> 0}$ be a positive weighting of $f$. Then $\int _ f w$ is lower semi-continuous.

Proof. Let $y \in Y$. Let $x_1, \ldots , x_ r \in X$ be the points lying over $y$. Apply Lemma 37.40.5 to get an ├ętale neighbourhood $(U, u) \to (Y, y)$ and a decomposition

\[ U \times _ Y X = W \amalg \ \coprod \nolimits _{i = 1, \ldots , n} \ \coprod \nolimits _{j = 1, \ldots , m_ i} V_{i, j} \]

as in locus citatus. Observe that $(\int _ f w)(y) = \sum w(v_{i, j})$ where $w(v_{i, j}) = w(x_ i)$. Since $\int _{V_{i, j} \to U} w|_{V_{i, j}}$ is locally constant by definition, we may after shrinking $U$ assume these functions are constant with value $w(v_{i, j})$. We conclude that

\[ \textstyle {\int }_{U \times _ Y X \to U} w|_{U \times _ Y X} = \textstyle {\int }_{W \to U} w|_ W + \sum \textstyle {\int }_{V_{i, j} \to U} w|_{V_{i, j}} = \textstyle {\int }_{W \to U} w|_ W + (\int _ f w)(y) \]

This is $\geq (\int _ f w)(y)$ and we conclude because $U \to Y$ is open and formation of the integral commutes with base change (Lemma 37.72.1). $\square$


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