The fppf quotient sheaf of a pre-relation

Let $S$ be a scheme and let $s,t\text{\hspace{0.17em}⁣}:R\to U$ be morphisms of $S$-schemes (Morphisms of schemes). For an $S$-scheme $T$, let ${\sim }_T$ be the equivalence relation on the set $U(T)={\mathrm{Mor}}_S(T,U)$ generated by the pairs $(s(\xi ),t(\xi ))$ for $\xi \in R(T)$ (Equivalence relation, equivalence class, and the quotient set $A\mathrm{/}\sim$, Fibre product of schemes). The relations ${\sim }_T$ are compatible with restriction along $T^{\mathrm{\prime }}\to T$, because a point $\xi \in R(T)$ restricts to $R(T^{\mathrm{\prime }})$ and $s,t$ are natural, so $T⟼U(T)\mathrm{/}{\sim }_T$ is a presheaf of sets, the naive quotient presheaf $P_{U\mathrm{/}R}$, and the quotient maps $U(T)\to U(T)\mathrm{/}{\sim }_T$ are natural.

The fppf quotient sheaf $U\mathrm{/}R$ is the sheafification of $P_{U\mathrm{/}R}$ (Fppf sheaves of sets and sheafification, Sheafification exists for the fppf site): the initial fppf sheaf receiving $P_{U\mathrm{/}R}$, with its universal property. Its construction uses the Axiom of Choice (The Axiom of Choice), inherited from the sheafification lemma. The naive quotient presheaf and the quotient sheaf differ in general, and no representability of $U\mathrm{/}R$ by a scheme or algebraic space is asserted.

Two standard special cases are used below. If $S=\mathrm{Spec}k$ is a field and $G$ is a group scheme of finite type over $k$ (Group schemes of finite type over a field) acting on a $k$-scheme $U$, one takes $R=G{\times }_{k}U$ with $s$ the second projection $G{\times }_{k}U\to U$ and $t$ the action morphism $G{\times }_{k}U\to U$, and writes $U\mathrm{/}G$ for the quotient sheaf. If $U=G$ and $H\subseteq G$ is a closed subgroup scheme (Morphisms and closed subgroup schemes of group schemes) acting by right translation, one takes $R=G{\times }_{k}H$ with $s(g,h)=g$ and $t(g,h)=gh$, and writes $G\mathrm{/}H$. In both cases the quotient sheaf is the fppf sheafification of the corresponding naive quotient.