Finite neighbourhood of an isolated fibre point after elementary etale change

Reduction to a finite-type affine chart. Choose an affine open $V_0=\mathrm{Spec}A\subseteq S$ with $s\in V_0$ and an affine open $U_0=\mathrm{Spec}B\subseteq X$ with $x\in U_0$ and $f(U_0)\subseteq V_0$; this is possible by [F3] because $f$ is locally of finite type. Then $A\to B$ is of finite type, and the prime $\mathfrak{q}\subseteq B$ of $x$ lies over the prime $\mathfrak{p}\subseteq A$ of $s$. Under the identification of [F3] the chart fibre $\mathrm{Spec}(B{\otimes }_A\kappa (\mathfrak{p}))$ is an open subscheme of $X_s$ containing the point $x$; since $x$ is isolated in $X_s$, the corresponding point $\bar{\mathfrak{q}}$ is isolated in $\mathrm{Spec}(B{\otimes }_A\kappa (\mathfrak{p}))$.

Quasi-finiteness at $\mathfrak{q}$. Put $K=\kappa (\mathfrak{p})$ and $F=B{\otimes }_{A}K$, a finite-type $K$-algebra. Since the fibre point $\bar{\mathfrak{q}}$ is isolated, a basic open $D_F(h)$ contains it and no other point by [F1]. The nonzero finite-type algebra $F_h$ therefore has a one-point spectrum and dimension $0$. Noether normalization [F1] makes it finite over a polynomial ring $K[z_1,\dots ,z_d]$, and dimension preservation plus the polynomial dimension formula in [F1] force $d=0$. Thus $F_h$ is a finite-dimensional $K$-algebra. As its spectrum has the single point $\bar{\mathfrak{q}}$, it is local and its localization at that point is itself; this local ring is $B_{\mathfrak{q}}\mathrm{/}\mathfrak{p}{B}_{\mathfrak{q}}$ by [F3]. Hence the fibre local algebra is finite over $K$, precisely the quasi-finiteness condition at $\mathfrak{q}$ (Quasi-finiteness at a prime of a finite-type algebra).

The local integral closure. Put $B^0={\mathrm{Int}}_A(B)$. By [F2] applied to step 2.1, choose $g\in B^0\setminus \mathfrak{q}$ with $(B^0{)}_g\cong B_g$. Write $K=\kappa (\mathfrak{p})$, $F_0=B^0{\otimes }_{A}K$ and $F=B{\otimes }_{A}K$. Every element of $F_0$ is integral over $K$, so every prime of $F_0$ is maximal: its quotient is an integral domain algebraic over a field, hence a field. Let ${\mathfrak{q}}_0$ be the contraction of the selected point of $F$. It is closed in $\mathrm{Spec}{F}_0$; it is also open, because the localization at $g$ identifies an open neighbourhood of it with the isolated singleton in $\mathrm{Spec}F$ from step 1.1. Thus $\{{\mathfrak{q}}_0\}$ is clopen. The same localization shows that the selected point of $F$ is the only point over ${\mathfrak{q}}_0$.

An element separating the selected fibre point. By [F2], a fibre idempotent $e\in F_0$ is $1$ at ${\mathfrak{q}}_0$ and $0$ at every other fibre prime. Write $e$ as a fraction of an element of $B^0$ with denominator from $A\setminus \mathfrak{p}$, and choose $t\in B^0$ whose image in $F_0$ is a nonzero scalar multiple of $e$. Thus $t$ is nonzero at ${\mathfrak{q}}_0$ and zero at every other fibre prime. Since $t$ is integral over $A$, choose a monic $P\in A[T]$ with $P(t)=0$. Over $K$, factor $\bar{P}=T^{e}H$ with $H(0)\mathrm{\ne }0$; if needed multiply $P$ by $T$ so $e\ge 1$. The selected nonzero value of $t$ is a root of $H$, so $\mathrm{deg}H\ge 1$, and $T^e$ and $H$ are coprime over $K$.

The étale factorization and finite component. Apply the coprime lifting part of [F2] to $\bar{P}=T^{e}H$ to obtain an everywhere étale $A$-algebra $A^{\mathrm{\prime }}$ and a prime ${\mathfrak{p}}^{\mathrm{\prime }}$ with $\kappa ({\mathfrak{p}}^{\mathrm{\prime }})=K$, together with monic coprime $I,H^{\mathrm{\prime }}\in A^{\mathrm{\prime }}[T]$ with $P=I{H}^{\mathrm{\prime }}$ and reductions $I<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>{\mathfrak{p}}^{\mathrm{\prime }}=T^e$, $H^{\mathrm{\prime }}<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>{\mathfrak{p}}^{\mathrm{\prime }}=H$. In $D_0=A^{\mathrm{\prime }}{\otimes }_A{B}^0$ and $D=A^{\mathrm{\prime }}{\otimes }_{A}B$, the equations $I(t)H^{\mathrm{\prime }}(t)=0$ and $a(t)I(t)+b(t)H^{\mathrm{\prime }}(t)=1$ yield product decompositions. Let $C_0=D_0\mathrm{/}(H^{\mathrm{\prime }}(t))$ and $C=D\mathrm{/}(H^{\mathrm{\prime }}(t))$ be the selected factors: on the fibre over ${\mathfrak{p}}^{\mathrm{\prime }}$, $H^{\mathrm{\prime }}(t)$ vanishes exactly at the selected point, while $I(t)$ vanishes at all the other points. Thus $C_0$ and $C$ each have exactly one prime over ${\mathfrak{p}}^{\mathrm{\prime }}$, and the prime of $C$ lies over the original $\mathfrak{q}$. Base changing $(B^0{)}_g\cong B_g$ and taking these factors gives $(C_0{)}_g\cong C_g$. The algebra $C_0$ is integral over $A^{\mathrm{\prime }}$, while $C$ is finite type over $A^{\mathrm{\prime }}$.

Finite after one base shrink. The closed subset $V_{C_0}(g)$ has closed image in $\mathrm{Spec}{A}^{\mathrm{\prime }}$: apply the lying-over assertion of [F2] to the integral quotient map $A^{\mathrm{\prime }}\to C_0\mathrm{/}g{C}_0$ to identify its image with $V(\ker (A^{\mathrm{\prime }}\to C_0\mathrm{/}g{C}_0))$. The selected prime ${\mathfrak{p}}^{\mathrm{\prime }}$ is outside this image, since the unique prime of $C_0$ over it avoids $g$. Choose $a\in A^{\mathrm{\prime }}\setminus {\mathfrak{p}}^{\mathrm{\prime }}$ such that $D(a)$ misses the image. Then $g$ is a unit in $(C_0{)}_a$, so $(C_0{)}_a\cong C_a$ by step 5.1. The ring $C_a$ is both integral and finite type over $A_a^{\mathrm{\prime }}$, hence module-finite by [F2]. Replacing $A^{\mathrm{\prime }},C$ by these localizations, we obtain the finite selected factor with exactly one point over ${\mathfrak{p}}^{\mathrm{\prime }}$.

The elementary étale neighbourhood. Put $U=\mathrm{Spec}{A}_a^{\mathrm{\prime }}$ and $u={\mathfrak{p}}^{\mathrm{\prime }}{A}_a^{\mathrm{\prime }}$ as in step 6.1. The ring map $A\to A^{\mathrm{\prime }}$ is étale everywhere by [F2], and principal localization preserves étaleness by [F6], so $U\to \mathrm{Spec}A$ is étale. Composing with the open immersion $\mathrm{Spec}A\subseteq S$ gives an étale map $\phi :U\to S$. Moreover $\kappa (u)=\kappa ({\mathfrak{p}}^{\mathrm{\prime }})=\kappa (\mathfrak{p})=\kappa (s)$, so $(U,u)\to (S,s)$ is an elementary étale neighbourhood.

The open finite piece. Put $U=\mathrm{Spec}{A}_a^{\mathrm{\prime }}$ and $u={\mathfrak{p}}^{\mathrm{\prime }}{A}_a^{\mathrm{\prime }}$. The affine chart base change $\mathrm{Spec}(A_a^{\mathrm{\prime }}{\otimes }_{A}B)$ is open in $X_U$ by [F3]. Its product decomposition from step 5.1 has the selected factor $V=\mathrm{Spec}{C}_a$, which is open and closed in that chart, hence open in $X_U$. The unique point of $V_u$ lies over $x$ by step 5.1 and is the canonical fibre-product point $(x,u)$ because $\kappa (u)=\kappa (s)$.

$V\to U$ is finite. The base $U=\mathrm{Spec}{A}_a^{\mathrm{\prime }}$ is affine and $A_a^{\mathrm{\prime }}\to C_a$ is module-finite by step 6.1, so $V\to U$ is finite by [F5].

The fibre $V_u$ and its residue field. The fibre $V_u=V{\times }_U\mathrm{Spec}\kappa (u)$ is $\mathrm{Spec}(C_a{\otimes }_{A_a^{\mathrm{\prime }}}\kappa ({\mathfrak{p}}^{\mathrm{\prime }}))$, and by step 5.1 the ring $C_a$ has exactly one prime over ${\mathfrak{p}}^{\mathrm{\prime }}$; hence $V_u$ consists of exactly one point, necessarily the image of $x_U$ by step 7.2. For its residue field, apply [F4] to the fibre product $X{\times }_{S}U$ at the point $x_U$ over $x\in X$ and $u\in U$: the residue field is $\kappa (\mathfrak{r})$ for a prime $\mathfrak{r}$ of $\kappa (x){\otimes }_{\kappa (s)}\kappa (u)$, and since $\kappa (u)=\kappa (s)$ by step 7.1 this tensor product is $\kappa (x){\otimes }_{\kappa (s)}\kappa (s)=\kappa (x)$, whose spectrum is a single point with residue field $\kappa (x)$. Hence the unique point of $V_u$ has residue field $\kappa (x)$, the original residue field of $x$.

Choice accounting and conclusion. The Axiom of Choice [F7] is assumed in the Statement. It is used exactly through [F1], [F2] and the cited Zariski Main and coprime-factorisation proofs and through the 'etale stability statement [F6]; the chart choices of step 1.1 are finitely many selections, and later steps use only finite choices. Steps 3.1--7.2 produce the elementary 'etale neighbourhood $(U,u)\to (S,s)$ of claim 1, the open subscheme $V\subseteq X_U$ of claim 2 with $V\to U$ finite, and the one-point fibre $V_u$ with residue field $\kappa (x)$; the empty-source case of the Statement is vacuous because there is no point $x$ to treat. [F1, F2, F6, F7, step 8.1, step 8.2] $\mathrm{\square }$