Definition 78.19.1. Let $B \to S$ and the pre-relation $j : R \to U \times _ B U$ be as above. In this setting the quotient sheaf $U/R$ associated to $j$ is the sheafification of the presheaf (78.19.0.1) on $(\mathit{Sch}/S)_{fppf}$. If $j : R \to U \times _ B U$ comes from the action of a group algebraic space $G$ over $B$ on $U$ as in Lemma 78.15.1 then we denote the quotient sheaf $U/G$.
78.19 Quotient sheaves
Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j : R \to U \times _ B U$ be a pre-relation over $B$. For each scheme $S'$ over $S$ we can take the equivalence relation $\sim _{S'}$ generated by the image of $j(S') : R(S') \to U(S') \times U(S')$. Hence we get a presheaf
78.19.0.1
\begin{equation} \label{spaces-groupoids-equation-quotient-presheaf} \begin{matrix} (\mathit{Sch}/S)^{opp}_{fppf}
& \longrightarrow
& \textit{Sets},
\\ S'
& \longmapsto
& U(S')/\sim _{S'}
\end{matrix} \end{equation}
Note that since $j$ is a morphism of algebraic spaces over $B$ and into $U \times _ B U$ there is a canonical transformation of presheaves from the presheaf (78.19.0.1) to $B$.
This means exactly that the diagram
\[ \xymatrix{ R \ar@<1ex>[r] \ar@<-1ex>[r] & U \ar[r] & U/R } \]
is a coequalizer diagram in the category of sheaves of sets on $(\mathit{Sch}/S)_{fppf}$. Again there is a canonical map of sheaves $U/R \to B$ as $j$ is a morphism of algebraic spaces over $B$ into $U \times _ B U$.
Remark 78.19.2. A variant of the construction above would have been to sheafify the functor where now $\sim _ X \subset U(X) \times U(X)$ is the equivalence relation generated by the image of $j : R(X) \to U(X) \times U(X)$. Here of course $U(X) = \mathop{\mathrm{Mor}}\nolimits _ B(X, U)$ and $R(X) = \mathop{\mathrm{Mor}}\nolimits _ B(X, R)$. In fact, the result would have been the same, via the identifications of (insert future reference in Topologies of Spaces here).
\[ \begin{matrix} (\textit{Spaces}/B)^{opp}_{fppf}
& \longrightarrow
& \textit{Sets},
\\ X
& \longmapsto
& U(X)/\sim _ X
\end{matrix} \]
Definition 78.19.3. In the situation of Definition 78.19.1. We say that the pre-relation $j$ has a quotient representable by an algebraic space if the sheaf $U/R$ is an algebraic space. We say that the pre-relation $j$ has a representable quotient if the sheaf $U/R$ is representable by a scheme. We will say a groupoid in algebraic spaces $(U, R, s, t, c)$ over $B$ has a representable quotient (resp. quotient representable by an algebraic space) if the quotient $U/R$ with $j = (t, s)$ is representable (resp. an algebraic space).
If the quotient $U/R$ is representable by $M$ (either a scheme or an algebraic space over $S$), then it comes equipped with a canonical structure morphism $M \to B$ as we've seen above.
The following lemma characterizes $M$ representing the quotient. It applies for example if $U \to M$ is flat, of finite presentation and surjective, and $R \cong U \times _ M U$.
Lemma 78.19.4. In the situation of Definition 78.19.1. Assume there is an algebraic space $M$ over $S$, and a morphism $U \to M$ such that
the morphism $U \to M$ equalizes $s, t$,
the map $U \to M$ is a surjection of sheaves, and
the induced map $(t, s) : R \to U \times _ M U$ is a surjection of sheaves.
In this case $M$ represents the quotient sheaf $U/R$.
Proof. Condition (1) says that $U \to M$ factors through $U/R$. Condition (2) says that $U/R \to M$ is surjective as a map of sheaves. Condition (3) says that $U/R \to M$ is injective as a map of sheaves. Hence the lemma follows. $\square$
The following lemma is wrong if we do not require $j$ to be a pre-equivalence relation (but just a pre-relation say).
Lemma 78.19.5. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $j : R \to U \times _ B U$ be a pre-equivalence relation over $B$. For a scheme $S'$ over $S$ and $a, b \in U(S')$ the following are equivalent:
$a$ and $b$ map to the same element of $(U/R)(S')$, and
there exists an fppf covering $\{ f_ i : S_ i \to S'\} $ of $S'$ and morphisms $r_ i : S_ i \to R$ such that $a \circ f_ i = s \circ r_ i$ and $b \circ f_ i = t \circ r_ i$.
In other words, in this case the map of sheaves
\[ R \longrightarrow U \times _{U/R} U \]
is surjective.
Proof. Omitted. Hint: The reason this works is that the presheaf (78.19.0.1) in this case is really given by $T \mapsto U(T)/j(R(T))$ as $j(R(T)) \subset U(T) \times U(T)$ is an equivalence relation, see Definition 78.4.1. $\square$
Lemma 78.19.6. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $j : R \to U \times _ B U$ be a pre-equivalence relation over $B$ and $g : U' \to U$ a morphism of algebraic spaces over $B$. Let $j' : R' \to U' \times _ B U'$ be the restriction of $j$ to $U'$. The map of quotient sheaves is injective. If $U' \to U$ is surjective as a map of sheaves, for example if $\{ g : U' \to U\} $ is an fppf covering (see Topologies on Spaces, Definition 73.7.1), then $U'/R' \to U/R$ is an isomorphism of sheaves.
\[ U'/R' \longrightarrow U/R \]
Proof. Suppose $\xi , \xi ' \in (U'/R')(S')$ are sections which map to the same section of $U/R$. Then we can find an fppf covering $\mathcal{S} = \{ S_ i \to S'\} $ of $S'$ such that $\xi |_{S_ i}, \xi '|_{S_ i}$ are given by $a_ i, a_ i' \in U'(S_ i)$. By Lemma 78.19.5 and the axioms of a site we may after refining $\mathcal{T}$ assume there exist morphisms $r_ i : S_ i \to R$ such that $g \circ a_ i = s \circ r_ i$, $g \circ a_ i' = t \circ r_ i$. Since by construction $R' = R \times _{U \times _ S U} (U' \times _ S U')$ we see that $(r_ i, (a_ i, a_ i')) \in R'(S_ i)$ and this shows that $a_ i$ and $a_ i'$ define the same section of $U'/R'$ over $S_ i$. By the sheaf condition this implies $\xi = \xi '$.
If $U' \to U$ is a surjective map of sheaves, then $U'/R' \to U/R$ is surjective also. Finally, if $\{ g : U' \to U\} $ is a fppf covering, then the map of sheaves $U' \to U$ is surjective, see Topologies on Spaces, Lemma 73.7.5. $\square$
Lemma 78.19.7. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $g : U' \to U$ a morphism of algebraic spaces over $B$. Let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ to $U'$. The map of quotient sheaves is injective. If the composition is a surjection of fppf sheaves then the map is bijective. This holds for example if $\{ h : U' \times _{g, U, t} R \to U\} $ is an $fppf$-covering, or if $U' \to U$ is a surjection of sheaves, or if $\{ g : U' \to U\} $ is a covering in the fppf topology.
\[ U'/R' \longrightarrow U/R \]
\[ \xymatrix{ U' \times _{g, U, t} R \ar[r]_-{\text{pr}_1} \ar@/^3ex/[rr]^ h & R \ar[r]_ s & U } \]
Proof. Injectivity follows on combining Lemmas 78.11.2 and 78.19.6. To see surjectivity (see Sites, Section 7.11 for a characterization of surjective maps of sheaves) we argue as follows. Suppose that $T$ is a scheme and $\sigma \in U/R(T)$. There exists a covering $\{ T_ i \to T\} $ such that $\sigma |_{T_ i}$ is the image of some element $f_ i \in U(T_ i)$. Hence we may assume that $\sigma $ if the image of $f \in U(T)$. By the assumption that $h$ is a surjection of sheaves, we can find an fppf covering $\{ \varphi _ i : T_ i \to T\} $ and morphisms $f_ i : T_ i \to U' \times _{g, U, t} R$ such that $f \circ \varphi _ i = h \circ f_ i$. Denote $f'_ i = \text{pr}_0 \circ f_ i : T_ i \to U'$. Then we see that $f'_ i \in U'(T_ i)$ maps to $g \circ f'_ i \in U(T_ i)$ and that $g \circ f'_ i \sim _{T_ i} h \circ f_ i = f \circ \varphi _ i$ notation as in (78.19.0.1). Namely, the element of $R(T_ i)$ giving the relation is $\text{pr}_1 \circ f_ i$. This means that the restriction of $\sigma $ to $T_ i$ is in the image of $U'/R'(T_ i) \to U/R(T_ i)$ as desired.
If $\{ h\} $ is an fppf covering, then it induces a surjection of sheaves, see Topologies on Spaces, Lemma 73.7.5. If $U' \to U$ is surjective, then also $h$ is surjective as $s$ has a section (namely the neutral element $e$ of the groupoid scheme). $\square$
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