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
U'/R' \longrightarrow U/R
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.
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
U'/R' \longrightarrow U/R
is injective. If the composition
\xymatrix{ U' \times _{g, U, t} R \ar[r]_-{\text{pr}_1} \ar@/^3ex/[rr]^ h & R \ar[r]_ s & U }
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.
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
\xymatrix{ R \ar[d]_ s \ar[r]_ f & R' \ar[d]^{s'} \\ U \ar[r]^ f & U' }
is cartesian. For any \tau \in \{ Zariski, {\acute{e}tale}, fppf, smooth, syntomic\} the diagram
\xymatrix{ U \ar[d] \ar[r] & U/R \ar[d]^ f \\ U' \ar[r] & U'/R' }
is a fibre product square of \tau -sheaves.
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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