A flat equivalence relation with a suitable quasi-section has a scheme quotient

The source and target projections of $R_U$ are finite locally free: its target projection is the base change of $V\to X$ by $U\subset X$, and inversion exchanges the two. By [F1] it has a scheme quotient $Y$, with $U\to Y$ finite locally free and onto. The morphism $V\stackrel{s}{\to }U\to Y$ has equal pullbacks along $V{\times }_{X}V$: two arrows with equal target determine, by composition and inverse, a relation arrow between their source points in $U$. Therefore [F2] descends it to a morphism $q:X\to Y$.

As fppf sheaves, the quotient of $X$ by $R$ equals that of $U$ by $R_U$. Indeed every point of $X$ lifts locally along $V\to X$ and is then related to its source point in $U$, proving local surjectivity of the latter quotient into the former. Two points of $U$ are identified exactly when related by $R_U$, by restriction of the original equivalence relation. Thus [F2] gives $Y\cong X\mathrm{/}R$. Since $R$ is an equivalence relation, its arrows are unique when source and target are specified, so this equality of sheaves identifies $R$ with the representable kernel pair $X{\times }_{Y}X$. In particular $X{\times }_{Y}U\cong V$, by the arrow/source description.

Base change of $q$ by the finite locally free cover $U\to Y$ is $V\to U$, which is flat as a base change of the original relation's source map. Thus [F2] makes $q$ flat. It is onto since the composite $U\to X\to Y$ is onto, and is of finite presentation: both $X$ and $Y$ are finite-type $k$-schemes, so any $k$-morphism between them is finite type, and over their Noetherian affine charts finite type implies finite presentation. Therefore $q$ is faithfully flat of finite presentation with the asserted kernel pair and quotient. AC is inherited from [F1]–[F2].