Lemma 37.23.6 (0572)—The Stacks project

Table of contents
Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.23: Slicing Cohen-Macaulay morphisms
Lemma 37.23.6 (cite)

previous lemma

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 Pieter Belmans on December 31, 2013 at 11:01

It's insanely nitpicky, but quasi-finite is written quasi finite in the proof. And $s\in g_s(T)_s$ should be $s\in g_s(T_s)$ .

Comment #419 by Pieter Belmans on December 31, 2013 at 14:05

The definition of the fppf topology is not contained in 34.5.1 as suggested in the statement of the lemma, but in 34.7.1.

Comment #420 by Johan on January 01, 2014 at 15:55

Fixed here. Thanks!

Comments (3)

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