The Stacks project

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

  1. $A$ has a dualizing complex and is $f$-adically complete,

  2. 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 $\text{depth}(A_\mathfrak p) + \dim (A/\mathfrak p) > 3$.

  3. for every maximal ideal $\mathfrak p \subset A_ f$ purity holds for $(A_ f)_\mathfrak p$, and

  4. 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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