The Stacks project

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

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

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

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. $\square$