Lemma 61.5.2 (096Y)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 61: Pro-étale Cohomology
Section 61.5: Constructing w-local affine schemes
Lemma 61.5.2 (cite)

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Lemma 61.5.2. Let $X = \mathop{\mathrm{Spec}}(A)$ as above. Given any finite stratification $X = \coprod T_ i$ by constructible subsets, there exists a finite subset $E \subset A$ such that the stratification (61.5.1.2) refines $X = \coprod T_ i$ .

Proof. We may write $T_ i = \bigcup _ j U_{i, j} \cap V_{i, j}^ c$ as a finite union for some $U_{i, j}$ and $V_{i, j}$ quasi-compact open in $X$ . Then we may write $U_{i, j} = \bigcup D(f_{i, j, k})$ and $V_{i, j} = \bigcup D(g_{i, j, l})$ . Then we set $E = \{ f_{i, j, k}\} \cup \{ g_{i, j, l}\}$ . This does the job, because the stratification (61.5.1.2) is the one whose strata are labeled by the vanishing pattern of the elements of $E$ which clearly refines the given stratification. $\square$

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