Ramanujam-Samuel for finite étale covers

Lemma 58.26.2. Let $(A, \mathfrak m)$ be a Noetherian local ring of depth $\geq 2$. Let $B = A[[x_1, \ldots , x_ d]]$ with $d \geq 1$. For any open $V \subset Y = \mathop{\mathrm{Spec}}(B)$ which contains

1. any prime $\mathfrak q \subset B$ such that $\mathfrak q \cap A \not= \mathfrak m$,

2. the prime $\mathfrak m B$

the functor $\textit{FÉt}_ Y \to \textit{FÉt}_ V$ is an equivalence. In particular purity holds for $B$.

Proof. A prime $\mathfrak q \subset B$ which is not contained in $V$ lies over $\mathfrak m$. In this case $A \to B_\mathfrak q$ is a flat local homomorphism and hence $\text{depth}(B_\mathfrak q) \geq 2$ (Algebra, Lemma 10.163.2). Thus the functor is fully faithful by Lemma 58.10.3 combined with Local Cohomology, Lemma 51.3.1.

Let $W \to V$ be a finite étale morphism. Let $B \to C$ be the unique finite ring map such that $\mathop{\mathrm{Spec}}(C) \to Y$ is the finite morphism extending $W \to V$ constructed in Lemma 58.21.5. Observe that $C = \Gamma (W, \mathcal{O}_ W)$.

Set $Y_0 = V(x_1, \ldots , x_ d)$ and $V_0 = V \cap Y_0$. Set $X = \mathop{\mathrm{Spec}}(A)$. If we use the map $Y \to X$ to identify $Y_0$ with $X$, then $V_0$ is identified with the punctured spectrum $U$ of $A$. Thus we may view $W_0 = W \times _ Y Y_0$ as a finite étale scheme over $U$. Then

$W_0 \times _ U (U \times _ X Y) \quad \text{and}\quad W \times _ V (U \times _ X Y)$

are schemes finite étale over $U \times _ X Y$ which restrict to isomorphic finite étale schemes over $V_0$. By Lemma 58.26.1 applied to the open $U \times _ X Y$ we obtain an isomorphism

$W_0 \times _ U (U \times _ X Y) \longrightarrow W \times _ V (U \times _ X Y)$

over $U \times _ X Y$.

Observe that $C_0 = \Gamma (W_0, \mathcal{O}_{W_0})$ is a finite $A$-algebra by Lemma 58.21.5 applied to $W_0 \to U \subset X$ (exactly as we did for $B \to C$ above). Since the construction in Lemma 58.21.5 is compatible with flat base change and with change of opens, the isomorphism above induces an isomorphism

$\Psi : C \longrightarrow C_0 \otimes _ A B$

of finite $B$-algebras. However, we know that $\mathop{\mathrm{Spec}}(C) \to Y$ is étale at all points above at least one point of $Y$ lying over $\mathfrak m \in X$. Since $\Psi$ is an isomorphism, we conclude that $\mathop{\mathrm{Spec}}(C_0) \to X$ is étale above $\mathfrak m$ (small detail omitted). Of course this means that $A \to C_0$ is finite étale and hence $B \to C$ is finite étale. $\square$

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