Lemma 42.13.2 (02R9)—The Stacks project

Processing math: 100%

Table of contents
Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.13: Preparation for flat pullback
Lemma 42.13.2 (cite)

previous lemma

numberstags

Lemma 42.13.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ , $Y$ be locally of finite type over $S$ . Let $f : X \to Y$ be a morphism. Assume $\{ Z_ i\} _{i \in I}$ is a locally finite collection of closed subsets of $Y$ . Then $\{ f^{-1}(Z_ i)\} _{i \in I}$ is a locally finite collection of closed subsets of $X$ .

Proof. Let $U \subset X$ be a quasi-compact open subset. Since the image $f(U) \subset Y$ is a quasi-compact subset there exists a quasi-compact open $V \subset Y$ such that $f(U) \subset V$ . Note that

$\{ i \in I \mid f^{-1}(Z_ i) \cap U \not= \emptyset \} \subset \{ i \in I \mid Z_ i \cap V \not= \emptyset \} .$

Since the right hand side is finite by assumption we win. $\square$

Comments (2)

Comment #2250 by Daniel Heiß on October 09, 2016 at 10:08

There is a typo in the last sentence. It should be "[...] of closed subsets of $X$ ."

Comment #2284 by Johan on November 03, 2016 at 19:57

Thanks. Fixed here.

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).

Comments (2)

Post a comment