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

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

This means exactly that the diagram

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

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

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.

**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.

**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.

**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

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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