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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