Lemma 58.25.2. Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $f \in \mathfrak m$. Assume
$A$ is $f$-adically complete,
$f$ is a nonzerodivisor,
$H^1_\mathfrak m(A/fA)$ and $H^2_\mathfrak m(A/fA)$ are finite $A$-modules,
for every maximal ideal $\mathfrak p \subset A_ f$ purity holds for $(A_ f)_\mathfrak p$,
purity holds for $A$.
Then purity holds for $A/fA$.
Proof. The proof is identical to the proof of Lemma 58.25.1 using Lemma 58.24.2 in stead of Lemma 58.24.1. $\square$
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