Definition 39.20.1. Let $\tau $, $S$, and the pre-relation $j : R \to U \times _ S U$ be as above. In this setting the quotient sheaf $U/R$ associated to $j$ is the sheafification of the presheaf (39.20.0.1) in the $\tau $-topology. If $j : R \to U \times _ S U$ comes from the action of a group scheme $G/S$ on $U$ as in Lemma 39.16.1 then we sometimes denote the quotient sheaf $U/G$.
39.20 Quotient sheaves
Let $\tau \in \{ Zariski, {\acute{e}tale}, fppf, smooth, syntomic\} $. Let $S$ be a scheme. Let $j : R \to U \times _ S U$ be a pre-relation over $S$. Say $U, R, S$ are objects of a $\tau $-site $\mathit{Sch}_\tau $ (see Topologies, Section 34.2). Then we can consider the functors
\[ h_ U, h_ R : (\mathit{Sch}/S)_\tau ^{opp} \longrightarrow \textit{Sets}. \]
These are sheaves, see Descent, Lemma 35.13.7. The morphism $j$ induces a map $j : h_ R \to h_ U \times h_ U$. For each object $T \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_\tau )$ we can take the equivalence relation $\sim _ T$ generated by $j(T) : R(T) \to U(T) \times U(T)$ and consider the quotient. Hence we get a presheaf
39.20.0.1
\begin{equation} \label{groupoids-equation-quotient-presheaf} (\mathit{Sch}/S)_\tau ^{opp} \longrightarrow \textit{Sets}, \quad T \longmapsto U(T)/\sim _ T \end{equation}
This means exactly that the diagram
\[ \xymatrix{ h_ R \ar@<1ex>[r] \ar@<-1ex>[r] & h_ U \ar[r] & U/R } \]
is a coequalizer diagram in the category of sheaves of sets on $(\mathit{Sch}/S)_\tau $. Using the Yoneda embedding we may view $(\mathit{Sch}/S)_\tau $ as a full subcategory of sheaves on $(\mathit{Sch}/S)_\tau $ and hence identify schemes with representable functors. Using this abuse of notation we will often depict the diagram above simply
\[ \xymatrix{ R \ar@<1ex>[r]^ s \ar@<-1ex>[r]_ t & U \ar[r] & U/R } \]
We will mostly work with the fppf topology when considering quotient sheaves of groupoids/equivalence relations.
Definition 39.20.2. In the situation of Definition 39.20.1. We say that the pre-relation $j$ has a representable quotient if the sheaf $U/R$ is representable. We will say a groupoid $(U, R, s, t, c)$ has a representable quotient if the quotient $U/R$ with $j = (t, s)$ is representable.
The following lemma characterizes schemes $M$ representing the quotient. It applies for example if $\tau = fppf$, $U \to M$ is flat, of finite presentation and surjective, and $R \cong U \times _ M U$.
Lemma 39.20.3. In the situation of Definition 39.20.1. Assume there is a scheme $M$, and a morphism $U \to M$ such that
the morphism $U \to M$ equalizes $s, t$,
the morphism $U \to M$ induces a surjection of sheaves $h_ U \to h_ M$ in the $\tau $-topology, and
the induced map $(t, s) : R \to U \times _ M U$ induces a surjection of sheaves $h_ R \to h_{U \times _ M U}$ in the $\tau $-topology.
In this case $M$ represents the quotient sheaf $U/R$.
Proof. Condition (1) says that $h_ U \to h_ M$ factors through $U/R$. Condition (2) says that $U/R \to h_ M$ is surjective as a map of sheaves. Condition (3) says that $U/R \to h_ M$ 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 39.20.4. Let $\tau \in \{ Zariski, {\acute{e}tale}, fppf, smooth, syntomic\} $. Let $S$ be a scheme. Let $j : R \to U \times _ S U$ be a pre-equivalence relation over $S$. Assume $U, R, S$ are objects of a $\tau $-site $\mathit{Sch}_\tau $. For $T \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_\tau )$ and $a, b \in U(T)$ the following are equivalent:
$a$ and $b$ map to the same element of $(U/R)(T)$, and
there exists a $\tau $-covering $\{ f_ i : T_ i \to T\} $ of $T$ and morphisms $r_ i : T_ i \to R$ 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 $\tau $-sheaves
\[ h_ R \longrightarrow h_ U \times _{U/R} h_ U \]
is surjective.
Proof. Omitted. Hint: The reason this works is that the presheaf (39.20.0.1) 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 39.3.1. $\square$
Lemma 39.20.5. Let $\tau \in \{ Zariski, {\acute{e}tale}, fppf, smooth, syntomic\} $. Let $S$ be a scheme. Let $j : R \to U \times _ S U$ be a pre-equivalence relation over $S$ and $g : U' \to U$ a morphism of schemes over $S$. Let $j' : R' \to U' \times _ S U'$ be the restriction of $j$ to $U'$. Assume $U, U', R, S$ are objects of a $\tau $-site $\mathit{Sch}_\tau $. The map of quotient sheaves is injective. If $g$ defines a surjection $h_{U'} \to h_ U$ of sheaves in the $\tau $-topology (for example if $\{ g : U' \to U\} $ is a $\tau $-covering), then $U'/R' \to U/R$ is an isomorphism.
\[ U'/R' \longrightarrow U/R \]
Proof. Suppose $\xi , \xi ' \in (U'/R')(T)$ are sections which map to the same section of $U/R$. Then we can find a $\tau $-covering $\mathcal{T} = \{ T_ i \to T\} $ of $T$ such that $\xi |_{T_ i}, \xi '|_{T_ i}$ are given by $a_ i, a_ i' \in U'(T_ i)$. By Lemma 39.20.4 and the axioms of a site we may after refining $\mathcal{T}$ assume there exist morphisms $r_ i : T_ i \to R$ 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 $(r_ i, (a_ i, a_ i')) \in R'(T_ i)$ and this shows that $a_ i$ and $a_ i'$ define the same section of $U'/R'$ over $T_ i$. By the sheaf condition this implies $\xi = \xi '$.
If $h_{U'} \to h_ U$ is a surjection of sheaves, then of course $U'/R' \to U/R$ is surjective also. If $\{ g : U' \to U\} $ is a $\tau $-covering, then the map of sheaves $h_{U'} \to h_ U$ is surjective, see Sites, Lemma 7.12.4. Hence $U'/R' \to U/R$ is surjective also in this case. $\square$
Lemma 39.20.6. Let $\tau \in \{ Zariski, {\acute{e}tale}, fppf, smooth, syntomic\} $. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $g : U' \to U$ a morphism of schemes over $S$. Let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ to $U'$. Assume $U, U', R, S$ are objects of a $\tau $-site $\mathit{Sch}_\tau $. The map of quotient sheaves is injective. If the composition defines a surjection of sheaves in the $\tau $-topology then the map is bijective. This holds for example if $\{ h : U' \times _{g, U, t} R \to U\} $ is a $\tau $-covering, or if $U' \to U$ defines a surjection of sheaves in the $\tau $-topology, or if $\{ g : U' \to U\} $ is a covering in the $\tau $-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 39.13.2 and 39.20.5. 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 $ is the image of $f \in U(T)$. By the assumption that $h$ is a surjection of sheaves, we can find a $\tau $-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 (39.20.0.1). 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 a $\tau $-covering, then it induces a surjection of sheaves, see Sites, Lemma 7.12.4. 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). $\square$
Lemma 39.20.7. Let $S$ be a scheme. Let $f : (U, R, j) \to (U', R', j')$ be a morphism between equivalence relations over $S$. Assume that is cartesian. For any $\tau \in \{ Zariski, {\acute{e}tale}, fppf, smooth, syntomic\} $ the diagram is a fibre product square of $\tau $-sheaves.
\[ \xymatrix{ R \ar[d]_ s \ar[r]_ f & R' \ar[d]^{s'} \\ U \ar[r]^ f & U' } \]
\[ \xymatrix{ U \ar[d] \ar[r] & U/R \ar[d]^ f \\ U' \ar[r] & U'/R' } \]
Proof. By Lemma 39.20.4 the quotient sheaves have a simple description which we will use below without further mention. We first show that
\[ U \longrightarrow U' \times _{U'/R'} U/R \]
is injective. Namely, assume $a, b \in U(T)$ map to the same element on the right hand side. Then $f(a) = f(b)$. After replacing $T$ by the members of a $\tau $-covering we may assume that there exists an $r \in R(T)$ such that $a = s(r)$ and $b = t(r)$. Then $r' = f(r)$ is a $T$-valued point of $R'$ with $s'(r') = t'(r')$. Hence $r' = e'(f(a))$ (where $e'$ is the identity of the groupoid scheme associated to $j'$, see Lemma 39.13.3). Because the first diagram of the lemma is cartesian this implies that $r$ has to equal $e(a)$. Thus $a = b$.
Finally, we show that the displayed arrow is surjective. Let $T$ be a scheme over $S$ and let $(a', \overline{b})$ be a section of the sheaf $U' \times _{U'/R'} U/R$ over $T$. After replacing $T$ by the members of a $\tau $-covering we may assume that $\overline{b}$ is the class of an element $b \in U(T)$. After replacing $T$ by the members of a $\tau $-covering we may assume that there exists an $r' \in R'(T)$ such that $a' = t(r')$ and $s'(r') = f(b)$. Because the first diagram of the lemma is cartesian we can find $r \in R(T)$ such that $s(r) = b$ and $f(r) = r'$. Then it is clear that $a = t(r) \in U(T)$ is a section which maps to $(a', \overline{b})$. $\square$
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