The Stacks project

Lemma 37.74.3. Let $f : X \to Y$ be a locally quasi-finite morphism. Let $w : X \to \mathbf{Z}_{> 0}$ be a positive weighting of $f$. Then $w$ is upper semi-continuous.

Proof. Let $x \in X$ with image $y \in Y$. Choose an ├ętale neighbourhood $(V, v) \to (Y, y)$ and an open $U \subset X_ V$ such that $\pi : U \to V$ is finite and there is a unique point $u \in U$ mapping to $v$ with $\kappa (u)/\kappa (v)$ purely inseparable. See Lemma 37.41.3. Then $(\int _\pi w|_ U)(v) = w(u)$. It follows from Definition 37.73.2 that after replacing $V$ by a neighbourhood of $v$ we we have $w|_ U(u') \leq w|_ U(u) = w(x)$ for all $u' \in U$. Namely, $w|_ U(u')$ occurs as a summand in the expression for $(\int _\pi w|_ U)(\pi (u'))$. This proves the lemma because the ├ętale morphism $U \to X$ is open. $\square$

Comments (0)

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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0F3I. Beware of the difference between the letter 'O' and the digit '0'.