Finite decomposition around isolated fibre points after an elementary étale change

For $n=0$ take $U=S$, $u=s$, the identity étale morphism, and $W=X$; the empty list of finite pieces satisfies every condition. Suppose inductively that the assertion holds for $x_1,\dots ,x_{i-1}$, over an elementary étale neighbourhood $(U,u)\to (S,s)$ with decomposition $X_U=W\bigsqcup V_1\bigsqcup \cdots \bigsqcup V_{i-1}$. By [F4] the point $x_i$ has a unique lift ${\tilde{x}}_i\in (X_U{)}_u$ with residue field $\kappa (x_i)$; distinctness of the original points and the fibre conditions put it in $W_u$. It remains isolated in $W_u$, because $W_u$ is open in $(X_U{)}_u$ and base change by the residue-field isomorphism identifies this fibre with $X_s$.

Apply [F1] to the separated, locally finite type morphism $W\to U$ at ${\tilde{x}}_i$. It gives an elementary étale neighbourhood $(U^{\mathrm{\prime }},u^{\mathrm{\prime }})\to (U,u)$ and an open $V_i^{\mathrm{\prime }}\subseteq W_{U^{\mathrm{\prime }}}$ containing the selected point, finite over $U^{\mathrm{\prime }}$, with singleton fibre over $u^{\mathrm{\prime }}$ and the same residue field. The composite $(U^{\mathrm{\prime }},u^{\mathrm{\prime }})\to (S,s)$ is elementary étale by [F2] and $\kappa (u^{\mathrm{\prime }})=\kappa (u)=\kappa (s)$. Base change all prior pieces to $U^{\mathrm{\prime }}$; they remain disjoint open-and-closed pieces, finite over $U^{\mathrm{\prime }}$, with their singleton chosen fibres and residue fields by [F4] and finite-module base change.

The new open immersion $V_i^{\mathrm{\prime }}\hookrightarrow W_{U^{\mathrm{\prime }}}$ is also closed. Indeed $V_i^{\mathrm{\prime }}\to U^{\mathrm{\prime }}$ is finite and hence proper by [F3], and $W_{U^{\mathrm{\prime }}}\to U^{\mathrm{\prime }}$ is separated as an open-and-closed subscheme of the separated base change $X_{U^{\mathrm{\prime }}}\to U^{\mathrm{\prime }}$; [F3] makes $V_i^{\mathrm{\prime }}\to W_{U^{\mathrm{\prime }}}$ proper and therefore closed. Thus $V_i^{\mathrm{\prime }}$ is open and closed in $W_{U^{\mathrm{\prime }}}$ and in $X_{U^{\mathrm{\prime }}}$. Let $W^{\mathrm{\prime }}=W_{U^{\mathrm{\prime }}}\setminus V_i^{\mathrm{\prime }}$, an open-and-closed complement. It has no point over $u^{\mathrm{\prime }}$ mapping to any $x_j$ with $j\le i$: the earlier points lie in their preserved $V_j$ and the new point lies in $V_i^{\mathrm{\prime }}$. This completes the induction step.

Finite induction through $i=n$ gives the displayed decomposition and all fibre conditions. The case of an empty fibre is covered by $n=0$. No quasi-compactness of $X$ or $S$ is used, and nonreduced singleton fibres are allowed. AC is spent only through [F1] and the finite/proper suppliers; the single-point étale neighbourhood is supplied by [F1].