The Stacks project

Lemma 37.23.6. Let $S$ be a scheme. Let $\mathcal{U} = \{ S_ i \to S\} _{i \in I}$ be an fppf covering of $S$, see Topologies, Definition 34.7.1. Then there exists an fppf covering $\mathcal{V} = \{ T_ j \to S\} _{j \in J}$ which refines (see Sites, Definition 7.8.1) $\mathcal{U}$ such that each $T_ j \to S$ is locally quasi-finite.

Proof. For every $s \in S$ there exists an $i \in I$ such that $s$ is in the image of $S_ i \to S$. By Lemma 37.23.5 we can find a morphism $g_ s : T_ s \to S$ such that $s \in g_ s(T_ s)$ which is flat, locally of finite presentation and locally quasi-finite and such that $g_ s$ factors through $S_ i \to S$. Hence $\{ T_ s \to S\} $ is the desired covering of $S$ that refines $\mathcal{U}$. $\square$


Comments (3)

Comment #417 by on

It's insanely nitpicky, but quasi-finite is written quasi finite in the proof. And should be .


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Your email address will not be published.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 0572. Beware of the difference between the letter 'O' and the digit '0'.