Lemma 37.73.6. Let $f : X \to Y$ be a separated, locally quasi-finite morphism. Let $w : X \to \mathbf{Z}_{> 0}$ be a positive weighting of $f$. Assume $\int _ w f$ attains its maximum $d$ and let $Y_ d \subset Y$ be the open set of points $y$ with $(\int _ f w)(y) = d$. Then the morphism $f^{-1}(Y_ d) \to Y_ d$ is finite.

**Proof.**
Observe that $Y_ d$ is open by Lemma 37.73.4. Let $y \in Y_ d$. Say $x_1, \ldots , x_ n$ are the points of $X$ lying over $y$. Apply Lemma 37.40.5 to get an étale neighbourhood $(U, u) \to (Y, y)$ and a decomposition

as in locus citatus. Observe that $d = \sum w(v_{i, j})$ 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

This is $\geq (\int _ f w)(y) = d$ and we conclude that $W$ must be the emptyset. Thus $U \times _ Y X \to U$ is finite. By Descent, Lemma 35.22.23 this implies that $X \to Y$ is finite over the image of the open morphism $U \to Y$. In other words, we see that $f$ is finite over an open neighbourhood of $y$ as desired. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)