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

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

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

1. $A$ is $f$-adically complete,

2. $f$ is a nonzerodivisor,

3. $H^1_\mathfrak m(A/fA)$ 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. $\square$

Remark 58.23.3. Let $(A, \mathfrak m)$ be a complete local Noetherian ring and $f \in \mathfrak m$ nonzero. Suppose that $A_ f$ is $(S_2)$ and every irreducible component of $\mathop{\mathrm{Spec}}(A)$ has dimension $\geq 4$. Then Lemma 58.23.1 tells us that the category

$\mathop{\mathrm{colim}}\nolimits \nolimits _{U' \subset U\text{ open, }U_0 \subset U} \text{ category of schemes finite étale over }U'$

is equivalent to the category of schemes finite étale over $U_0$. For example this holds if $A$ is a normal domain of dimension $\geq 4$!

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