Let $X, X_0, U, U_0$ be as in Situation 58.19.1. In this section we ask when the restriction functor
is essentially surjective. We will do this by taking results from Section 58.23 and then filling in the gaps using purity. Recall that we say purity holds for a Noetherian local ring $(A, \mathfrak m)$ if the restriction functor $\textit{FÉt}_ X \to \textit{FÉt}_ U$ is essentially surjective where $X = \mathop{\mathrm{Spec}}(A)$ and $U = X \setminus \{ \mathfrak m\} $.
Lemma 58.24.1. 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 $\text{depth}(A_\mathfrak p) + \dim (A/\mathfrak p) > 3$.
for every maximal ideal $\mathfrak p \subset A_ f$ purity holds for $(A_ f)_\mathfrak p$.
Then the restriction functor $\textit{FÉt}_ U \to \textit{FÉt}_{U_0}$ is essentially surjective.
Proof. Let $V_0 \to U_0$ be a finite étale morphism. By Lemma 58.23.1 there exists an open $U' \subset U$ containing $U_0$ and a finite étale morphism $V' \to U$ whose base change to $U_0$ is isomorphic to $V_0 \to U_0$. Since $U' \supset U_0$ we see that $U \setminus U'$ consists of points corresponding to prime ideals $\mathfrak p_1, \ldots , \mathfrak p_ n$ as in (3). By assumption we can find finite étale morphisms $V'_ i \to \mathop{\mathrm{Spec}}(A_{\mathfrak p_ i})$ agreeing with $V' \to U'$ over $U' \times _ U \mathop{\mathrm{Spec}}(A_{\mathfrak p_ i})$. By Limits, Lemma 32.20.1 applied $n$ times we see that $V' \to U'$ extends to a finite étale morphism $V \to U$. $\square$
Lemma 58.24.2. 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 $(A_ f)_\mathfrak p$.
Then the restriction functor $\textit{FÉt}_ U \to \textit{FÉt}_{U_0}$ is essentially surjective.
Proof. The proof is identical to the proof of Lemma 58.24.1 using Lemma 58.23.2 in stead of Lemma 58.23.1. $\square$
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