Remark 52.27.2. Let $(A, \mathfrak m)$ be a Noetherian local ring and $f \in \mathfrak m$. The conclusion of Lemma 52.27.1 holds if we assume

$A$ has a dualizing complex,

$A$ is $f$-adically complete,

$f$ is a nonzerodivisor,

one of the following is true

$A_ f$ is $(S_2)$ and for $\mathfrak p \subset A$, $f \not\in \mathfrak p$ minimal we have $\dim (A/\mathfrak p) \geq 4$, or

if $\mathfrak p \not\in V(f)$ and $V(\mathfrak p) \cap V(f) \not= \{ \mathfrak m\} $, then $\text{depth}(A_\mathfrak p) + \dim (A/\mathfrak p) > 3$.

$H^3_{\mathfrak m}(A/fA) = 0$.

The proof is exactly the same as the proof of Lemma 52.27.1 using Lemma 52.24.1 instead of Lemma 52.24.2. Two points need to be made here: (a) it seems hard to find examples where one knows $H^3_{\mathfrak m}(A/fA) = 0$ without assuming $\text{depth}(A/fA) \geq 4$, and (b) the proof of Lemma 52.24.1 is a good deal harder than the proof of Lemma 52.24.2.

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