Quotient sheaves and representable quotients for pre-relations and group actions

Assume the Axiom of Choice for the sheafification and represented-sheaf suppliers used below. Work on a fixed big fppf site of $k$-schemes as in Stacks Section 34.7. Let $k$ be a field, and let $s,t:R\to U$ be two morphisms of $k$-schemes with common target (Morphisms of schemes); such a pair is a pre-relation on $U$.

The naive quotient presheaf $P_{U\mathrm{/}R}$ is the functor on the category of $k$-schemes sending a $k$-scheme $T$ to the quotient of the set $U(T)$ (Fibre product of schemes) by the equivalence relation generated by $\{(s(x),t(x)):x\in R(T)\}$; its value at $T=\mathrm{Spec}k$ is the set of $k$-points of $U$ modulo the relation generated by $R(k)$. In general $P_{U\mathrm{/}R}$ is only a presheaf (Presheaves, covariantly and contravariantly representable functors, and representations).

An fppf sheaf on this category is a presheaf $F$ such that for every $k$-scheme $T$ and every fppf covering family $\{T_i\to T\}$ — a set-indexed family of flat morphisms locally of finite presentation which is jointly surjective onto $T$ (Faithfully flat scheme morphism, Locally finite presentation morphisms) — the diagram $F(T)\to {\prod }_{i}F(T_i)\rightrightarrows {\prod }_{i,j}F(T_i{\times }_T{T}_j)$ is an equalizer. The fppf quotient sheaf $U\mathrm{/}R$ is the sheafification of $P_{U\mathrm{/}R}$ for this topology: a morphism $P_{U\mathrm{/}R}\to U\mathrm{/}R$ of presheaves with $U\mathrm{/}R$ an fppf sheaf, initial among morphisms from $P_{U\mathrm{/}R}$ to fppf sheaves. Every representable presheaf is an fppf sheaf (Scheme morphisms satisfy fppf descent); a $k$-scheme $M$ represents the quotient sheaf $U\mathrm{/}R$ when there is a natural isomorphism $h_M\cong U\mathrm{/}R$ (Presheaves, covariantly and contravariantly representable functors, and representations), and then $M$ is unique up to unique isomorphism.

For a group scheme $G$ of finite type over $k$ (Group schemes of finite type over a field) acting on $U$ one takes $R=G{\times }_{k}U$ with $s$ the second projection and $t$ the action morphism, and writes $U\mathrm{/}G$; for $U=G$ and a closed subgroup scheme $H\subseteq G$ (Morphisms and closed subgroup schemes of group schemes) acting by right translation one takes $R=G{\times }_{k}H$, $s(g,h)=g$, $t(g,h)=gh$, and writes $G\mathrm{/}H$. The natural map from the naive quotient presheaf to its quotient sheaf is generally not an isomorphism; sheafification can both identify sections that agree locally and introduce sections that only have local representatives. The orbit set is only $P_{U\mathrm{/}R}(k)$.