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. $\square$

Comment #2258 by Katharina on

The point $x$ has not been defined: $x$ is the point corresponding to the maximal ideal $\mathfrak{m}$.

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