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The Stacks project

Lemma 58.20.2. Let (A, \mathfrak m) be a Noetherian local ring. Set X = \mathop{\mathrm{Spec}}(A) and let U = X \setminus \{ \mathfrak m\} . Let V be finite étale over U. Assume A has depth \geq 2. The following are equivalent

  1. V = Y \times _ X U for some Y \to X finite étale,

  2. B = \Gamma (V, \mathcal{O}_ V) is finite étale over A.

Proof. Denote \pi : V \to U the given finite étale morphism. Assume Y as in (1) exists. Let x \in X be the point corresponding to \mathfrak m. Let y \in Y be a point mapping to x. We claim that \text{depth}(\mathcal{O}_{Y, y}) \geq 2. This is true because Y \to X is étale and hence A = \mathcal{O}_{X, x} and \mathcal{O}_{Y, y} have the same depth (Algebra, Lemma 10.163.2). Hence Lemma 58.20.1 applies and Y = \mathop{\mathrm{Spec}}(B).

The implication (2) \Rightarrow (1) is easier and the details are omitted. \square


Comments (2)

Comment #2258 by Katharina on

The point has not been defined: is the point corresponding to the maximal ideal .


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