Finite fibre components after an elementary etale change

First lift a monic coprime factorization $\bar{P}=\bar{I}\bar{H}$ over $k$, of positive degrees $r,s$, for a monic $P\in A[T]$. Introduce the $r+s$ coefficients of monic universal factors $I,H$ and impose the $r+s$ coefficient equations $IH=P$. Their square Jacobian acts by $(\delta I,\delta H)\mapsto H\delta I+I\delta H$, with $\mathrm{deg}\delta I<r$, $\mathrm{deg}\delta H<s$. At the chosen coprime factors its kernel is zero: the equation implies $\bar{H}$ divides $\delta H$, hence $\delta H=0$, and then $\delta I=0$. Invert the Jacobian determinant $d$. The resulting algebra $E$ is a square-Jacobian algebra, giving an elementary change by [F1] and [F4]. Substitution of the chosen coefficients gives a point $t$ with residue field $k$. The same invertible matrix solves $aI+bH=1$, so the factors are coprime over $E$. This construction works over any finite-type $k$-base and is compatible with restriction.

For one selected point choose affine charts $\mathrm{Spec}S\subseteq X$ and $\mathrm{Spec}A\subseteq Y$ around it and its image. Put $C={\mathrm{Int}}_A(S)$, finite by [F2]. At the selected maximal ideal CA-20 gives $g\in C$ with $C_g=S_g$. The selected point of the finite fibre of $C$ is isolated, and is the only source fibre point above it, because $g$ is nonzero there and identifies the two open fibre neighbourhoods. In the Artinian ring $C{\otimes }_{A}k$, take the idempotent equal to $1$ on that local factor and $0$ on every other factor, and lift it to $c\in C$; the residue field is $k$, so the quotient map onto this fibre is surjective. Choose monic $P$ with $P(c)=0$, multiply it by $T$ if necessary, and factor $\bar{P}=T^e\bar{H}$ with $e\ge 1$ and $\bar{H}(0)\mathrm{\ne }0$. The selected value of $c$ is $1$, so $\bar{H}(1)=0$ and $\mathrm{deg}\bar{H}\ge 1$.

Apply step 1.1 to $T^e\bar{H}$, obtaining $E$ and coprime monic factors $I,H$. In $E{\otimes }_{A}C$ and $E{\otimes }_{A}S$, the equations $I(c)H(c)=0$ and $a(c)I(c)+b(c)H(c)=1$ give compatible product decompositions. The factor where $I(c)$ is invertible, equivalently the quotient by $H(c)$, has just the selected point over $t$: on the fibre $c$ equals $1$ on the selected local factor and $0$ elsewhere, so $I(c)$ equals $1$ there and is nilpotent on the other factors. Let $C_1$ and $S_1$ be these factors. The ring $C_1$ is finite over $E$. Its closed locus $V(g)$ has closed image in $\mathrm{Spec}E$ by [F3], and that image omits $t$. Shrink to a principal neighbourhood of $t$ avoiding this image. Then $g$ is a unit in $C_1$, and also in $S_1$. The equality $C_g=S_g$ base changes and takes factors to give $(C_1{)}_g=(S_1{)}_g$, hence $C_1=S_1$. Thus $\mathrm{Spec}{S}_1$ is an open neighbourhood in the changed affine source, finite over the new base, with the required singleton chosen fibre.

This finite open neighbourhood $V$ is also closed in the entire changed source $X_T$. The finite map $V\to T$ is universally closed: tensoring its finite algebras by any base algebra stays finite and lying over on quotients proves closedness. The graph of $V\hookrightarrow X_T$ is closed in $V{\times }_T{X}_T$ by separatedness of $X_T\to T$, and projection of the graph to $X_T$ is closed by universal closedness of $V\to T$. Hence $V$ is open and closed. This argument uses separatedness exactly here.

Induct on the selected list. The first construction gives a clopen finite piece for $x_1$. In its clopen complement select the lift of $x_2$, apply the constructions of steps 1.2, 2.1 and 3.1 again, and base change the previous pieces. The lift of each $x_i$ is unique with unchanged residue field, because $k{\otimes }_{k}k=k$. By [F1] the composite base change is still elementary and the previous pieces remain clopen and finite. At each step the new piece has only its selected point in the chosen fibre, so it does not remove a later selected point. After finitely many steps the clopen complement $W$ has exactly the stated fibre exclusion. The empty list uses the identity change and $W=X$. If every fibre point was selected, the complement fibre is empty. No perfectness or source normality is used.