Lemma 58.25.1 (0DXZ)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 58: Fundamental Groups of Schemes
Section 58.25: Purity in local case, II
Lemma 58.25.1 (cite)

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Lemma 58.25.1. Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $f \in \mathfrak m$ . Assume

$A$ has a dualizing complex and is $f$ -adically complete,
one of the following is true
1. $A_ f$ is $(S_2)$ and every irreducible component of $X$ not contained in $X_0$ has dimension $\geq 4$ , or
2. if $\mathfrak p \not\in V(f)$ and $V(\mathfrak p) \cap V(f) \not= \{ \mathfrak m\}$ , then $\text{depth}(A_\mathfrak p) + \dim (A/\mathfrak p) > 3$ .
for every maximal ideal $\mathfrak p \subset A_ f$ purity holds for $(A_ f)_\mathfrak p$ , and
purity holds for $A$ .

Then purity holds for $A/fA$ .

Proof. Denote $X = \mathop{\mathrm{Spec}}(A)$ and $U = X \setminus \{ \mathfrak m\}$ the punctured spectrum. Similarly we have $X_0 = \mathop{\mathrm{Spec}}(A/fA)$ and $U_0 = X_0 \setminus \{ \mathfrak m\}$ . Let $V_0 \to U_0$ be a finite étale morphism. By Lemma 58.24.1 we find a finite étale morphism $V \to U$ whose base change to $U_0$ is isomorphic to $V_0 \to U_0$ . By assumption (5) we find that $V \to U$ extends to a finite étale morphism $Y \to X$ . Then the restriction of $Y$ to $X_0$ is the desired extension of $V_0 \to U_0$ . $\square$

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