A flat finite-type equivalence relation has generic saturated quasi-sections

Choose a closed point $z\in X$ and an affine neighbourhood $E$ of the source of an arrow targeting $z$ (the identity arrow suffices). Apply [F1] to $s:t^{-1}(z)\cap s^{-1}(E)\to E$ and the flat map $t:s^{-1}(E)\to X$. It gives a closed $F\subset E$ with nonempty fibre and finite source image over $z$, while $a=t:s^{-1}(F)\to X$ is flat along that fibre. This fibre has finitely many points: over a fixed source point $f$ and target $z$, the possible product points lie in $\mathrm{Spec}(\kappa (f){\otimes }_k\kappa (z))$, which is finite over $\kappa (f)$ because $\kappa (z)\mathrm{/}k$ is finite. A monomorphism $R\to X\times X$ has at most one point over each such product point. The fibre of $a$ is therefore a finite-type $\kappa (z)$-scheme with finitely many points, hence zero-dimensional with finite residue fields. Thus $a$ is also quasi-finite at those fibre points.

Let $P\subset s^{-1}(F)$ be the locus where $a$ is flat and quasi-finite. Composition gives the following invariance: for arrows $f\to x$ and $x\to y$, composing identifies the space of arrows $f\to x$ with that of arrows $f\to y$ after base change to the arrow scheme parametrizing $x\to y$. The two target base changes use $s,t:R\to X$, both faithfully flat and of finite presentation. Flatness descends by [F2], and quasi-finiteness is detected by geometric fibre dimension, which is unchanged by residue-field extension. Consequently the two inverse images of $P$ on $s^{-1}(F){\times }_F{s}^{-1}(F)$ coincide. The map $s:s^{-1}(F)\to F$ is open and onto, so $P=s^{-1}(F^{\mathrm{\prime }})$ for an open $F^{\mathrm{\prime }}\subset F$. It contains the entire fibre over $z$. Replace $F$ by $F^{\mathrm{\prime }}$; now $a$ is flat, quasi-finite and separated everywhere. Its open image $D$ contains $z$ and is saturated by composition.

Inside $D$ take the union $W_z$ of all opens over which $a$ is finite. This open contains the generic points of every irreducible component of $X$ through $z$. Indeed those points lie in the open image $D$; over their Artinian local rings a quasi-finite finite-type separated scheme is finite. To see this, its reduced closed fibre is a finite discrete scheme, so its finitely many affine point neighbourhoods are disjoint and cover the scheme, and lifting finite module generators through the nilpotent maximal ideal proves module finiteness. A finite compactification from [F2], replaced by the schematic closure of its source, then has no boundary over that local scheme; the finite image of the closed boundary can be removed from a neighbourhood of the generic point, making $a$ finite there.

The finite locus just defined is invariant along $R$: its two inverse images are the finite loci of the two isomorphic base changes of $a$ given by composition. Here finiteness descends under our faithfully flat open covers. An explicit verification is as follows. If a separated quasi-finite finite-type map becomes finite after such a cover, it becomes universally closed. For every further base change and closed source subset, its image pulls back to a closed set on the covering target; an open surjective map detects closed sets, so that image is closed downstairs. The original map is therefore proper, and [F2] makes it finite. This also proves equality of the maximal finite loci, by descending each covering open's saturated image. Hence $W_z$ is saturated. Set $U_z=F^{\mathrm{\prime }}\cap W_z$. Saturation identifies $s^{-1}(U_z)$ with $a^{-1}(W_z)$, so its target map is finite, flat and onto. Its base change by $U_z\subset W_z$ is one projection of $R_{U_z}$, and inversion gives the other.

If $W_z$ is not dense, repeat in the interior of $X\setminus W_z$. This interior is saturated: openness of the relation projections implies that the closure of a saturated subset is saturated, since the inverse image of its closure equals the closure of its inverse image for an open map. Each repetition meets a previously missed irreducible component at its generic point, and there are only finitely many components. We obtain finitely many disjoint $W_i$ with dense union. Finally $U_i$ is open in the affine $F$: for any finite subset, the ideal defining the complement of $U_i$ avoids its point primes; prime avoidance gives a principal open in $F$ containing the subset and contained in $U_i$. This proves the affine-neighbourhood assertion. AC is inherited from [F1]–[F2].