Criterion for a scheme to represent an fppf quotient sheaf

For every $k$-scheme $T$ the map $U(T)\to M(T)$, $u\mapsto q(u)$, is constant on each generating pair $(s(x),t(x))$ of the relation on $U(T)$ by (1), hence constant on the equivalence relation it generates, so it descends to a map $P_{U\mathrm{/}R}(T)\to M(T)$; these maps are natural in $T$ and define a morphism of presheaves $P_{U\mathrm{/}R}\to h_M$. By [F2] the functor $h_M$ is an fppf sheaf, so the universal property in [F1] factors this morphism uniquely through a morphism $\phi :U\mathrm{/}R\to h_M$.

Every section of $U\mathrm{/}R$ is locally represented by a section of $U$. Let $V(T)\subseteq (U\mathrm{/}R)(T)$ be the set of sections $s$ for which there are an fppf covering $\{T_i\to T\}$ and elements $u_i\in U(T_i)$ whose images in $(U\mathrm{/}R)(T_i)$ equal the restrictions of $s$. Restrictions of such sections lie in $V$, so $V$ is a subpresheaf of $U\mathrm{/}R$; and $V$ is a sheaf, because an fppf covering of each member of an fppf covering of $T$ composes to an fppf covering of $T$. Since $U(T)\to P_{U\mathrm{/}R}(T)$ is surjective for every $T$, the canonical morphism $P_{U\mathrm{/}R}\to U\mathrm{/}R$ takes values in $V$, so there is a factorization $P_{U\mathrm{/}R}\to V$ through the inclusion $V\hookrightarrow U\mathrm{/}R$. Applying the initiality in [F1] to the target sheaf $V$ and to the target sheaf $U\mathrm{/}R$ shows that the composite $U\mathrm{/}R\to V\hookrightarrow U\mathrm{/}R$ is the identity: both maps make the triangle from $P_{U\mathrm{/}R}$ commute, and such a map is unique. Therefore $V=U\mathrm{/}R$, which is the claim.

The morphism $\phi$ is surjective as a map of fppf sheaves. Let $T$ be a $k$-scheme and $m\in M(T)=h_M(T)$. By (2) there are an fppf covering $\{T_i\to T\}$ and $u_i\in U(T_i)$ with $q(u_i)=m{\mathrm{\mid }}_{T_i}$. For every pair $i,j$, the restrictions $u_i{\mathrm{\mid }}_{T_{ij}}$ and $u_j{\mathrm{\mid }}_{T_{ij}}$ have the same image in $M(T_{ij})$, so by [F3] they define a section of $h_{U{\times }_{M}U}(T_{ij})$. Hypothesis (3) lifts this section, after an fppf refinement of $T_{ij}$, to $r$ with $(t(r),s(r))=(u_j,u_i)$. Thus $[u_i]$ and $[u_j]$ agree in the quotient presheaf locally on $T_{ij}$ and hence agree in its sheafification $U\mathrm{/}R$ on $T_{ij}$. They are therefore a matching family in the sheaf $U\mathrm{/}R$, so glue to $s\in (U\mathrm{/}R)(T)$. Step 1.1 gives $\phi ([u_i])=q(u_i)=m{\mathrm{\mid }}_{T_i}$; since $h_M$ is a sheaf, $\phi (s)=m$. Thus every section of $h_M$ lies in the image of $\phi$.

The morphism $\phi$ is injective as a map of presheaves. Let $\xi ,{\xi }^{\mathrm{\prime }}\in (U\mathrm{/}R)(T)$ satisfy $\phi (\xi )=\phi ({\xi }^{\mathrm{\prime }})=m$. By step 1.2 there are an fppf covering $\{T_i\to T\}$ of $T$ and elements $u_i,u_i^{\mathrm{\prime }}\in U(T_i)$ with $\xi {\mathrm{\mid }}_{T_i}=[u_i]$ and ${\xi }^{\mathrm{\prime }}{\mathrm{\mid }}_{T_i}=[u_i^{\mathrm{\prime }}]$. Then $q(u_i)=\phi ([u_i])=m{\mathrm{\mid }}_{T_i}=\phi ([u_i^{\mathrm{\prime }}])=q(u_i^{\mathrm{\prime }})$, so $(u_i^{\mathrm{\prime }},u_i)$ is a $T_i$-point of $U{\times }_{M}U$ by [F3]. Hypothesis (3) provides, after refining the covering, an element $r_i\in R(T_i)$ with $(t,s)(r_i)=(u_i^{\mathrm{\prime }},u_i)$, that is $t(r_i)=u_i^{\mathrm{\prime }}$ and $s(r_i)=u_i$. Then $(s(r_i),t(r_i))=(u_i,u_i^{\mathrm{\prime }})$ is a generating pair of the relation defining $P_{U\mathrm{/}R}(T_i)$, so $[u_i]=[u_i^{\mathrm{\prime }}]$ in $(U\mathrm{/}R)(T_i)$. Hence $\xi$ and ${\xi }^{\mathrm{\prime }}$ agree on the members of an fppf covering, and separatedness of the sheaf $U\mathrm{/}R$ forces $\xi ={\xi }^{\mathrm{\prime }}$.

A morphism of fppf sheaves that is surjective and injective on sections is an isomorphism, so steps 2.1 and 2.2 show that $\phi :U\mathrm{/}R\to h_M$ is an isomorphism; hence $h_M\cong U\mathrm{/}R$ naturally and $M$ represents the quotient sheaf $U\mathrm{/}R$. For the parenthetical instances, if $q$ is faithfully flat and locally of finite presentation then $h_U\to h_M$ is a surjection of fppf sheaves by [F2], and the same argument with $X=R$ and $Y=U{\times }_{M}U$ gives the second instance; this finishes the proof.