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The Stacks project

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 32.20.1 applied n times we see that V' \to U' extends to a finite étale morphism V \to U. \square


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