## 58.23 Finite étale covers of punctured spectra, III

In this section we study when in Situation 58.19.1. the restriction functor

\[ \mathop{\mathrm{colim}}\nolimits _{U_0 \subset U' \subset U\text{ open}} \textit{FÉt}_{U'} \longrightarrow \textit{FÉt}_{U_0} \]

is an equivalence of categories.

Lemma 58.23.1. In Situation 58.19.1 assume

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

one of the following is true

$A_ f$ is $(S_2)$ and every irreducible component of $X$ not contained in $X_0$ has dimension $\geq 4$, or

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

Then the restriction functor

\[ \mathop{\mathrm{colim}}\nolimits _{U_0 \subset U' \subset U\text{ open}} \textit{FÉt}_{U'} \longrightarrow \textit{FÉt}_{U_0} \]

is an equivalence.

**Proof.**
This follows from Lemma 58.17.4 and Algebraic and Formal Geometry, Lemma 52.24.1.
$\square$

Lemma 58.23.2. In Situation 58.19.1 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.

Then the restriction functor

\[ \mathop{\mathrm{colim}}\nolimits _{U_0 \subset U' \subset U\text{ open}} \textit{FÉt}_{U'} \longrightarrow \textit{FÉt}_{U_0} \]

is an equivalence.

**Proof.**
This follows from Lemma 58.17.4 and Algebraic and Formal Geometry, Lemma 52.24.2.
$\square$

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