The Stacks project

58.24 Finite étale covers of punctured spectra, IV

Let $X, X_0, U, U_0$ be as in Situation 58.19.1. In this section we ask when the restriction functor

\[ \textit{FÉt}_ U \longrightarrow \textit{FÉt}_{U_0} \]

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

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

  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 32.19.1 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

  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,

  4. 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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