Permalink
Find file
Fetching contributors…
Cannot retrieve contributors at this time
4562 lines (4080 sloc) 163 KB
\input{preamble}
% OK, start here.
%
\begin{document}
\title{Groupoid Schemes}
\maketitle
\phantomsection
\label{section-phantom}
\tableofcontents
\section{Introduction}
\label{section-introduction}
\noindent
This chapter is devoted to generalities concerning groupoid schemes.
See for example the beautiful paper \cite{K-M} by Keel and Mori.
\section{Notation}
\label{section-notation}
\noindent
Let $S$ be a scheme. If $U$, $T$ are schemes over $S$ we denote
$U(T)$ for the set of $T$-valued points of $U$ {\it over} $S$. In a formula:
$U(T) = \Mor_S(T, U)$. We try to reserve the letter $T$ to denote
a ``test scheme'' over $S$, as in the discussion that follows.
Suppose we are given schemes $X$, $Y$ over
$S$ and a morphism of schemes $f : X \to Y$ over $S$.
For any scheme $T$ over $S$ we get an induced map of sets
$$
f : X(T) \longrightarrow Y(T)
$$
which as indicated we denote by $f$ also. In fact this construction
is functorial in the scheme $T/S$. Yoneda's Lemma, see Categories,
Lemma \ref{categories-lemma-yoneda}, says that $f$ determines and is
determined by this transformation of functors $f : h_X \to h_Y$.
More generally, we use the same notation for maps between fibre
products. For example, if
$X$, $Y$, $Z$ are schemes over $S$, and if
$m : X \times_S Y \to Z \times_S Z$ is
a morphism of schemes over $S$, then we think of $m$ as corresponding
to a collection of maps between $T$-valued points
$$
X(T) \times Y(T) \longrightarrow Z(T) \times Z(T).
$$
And so on and so forth.
\medskip\noindent
We continue our convention to label projection maps starting with
index $0$, so we have $\text{pr}_0 : X \times_S Y \to X$ and
$\text{pr}_1 : X \times_S Y \to Y$.
\section{Equivalence relations}
\label{section-equivalence-relations}
\noindent
Recall that a {\it relation} $R$ on a set $A$ is just a subset
of $R \subset A \times A$. We usually write $a R b$ to indicate
$(a, b) \in R$. We say the relation is {\it transitive} if
$a R b, b R c \Rightarrow a R c$. We say the relation is
{\it reflexive} if $a R a$ for all $a \in A$. We say the relation is
{\it symmetric} if $a R b \Rightarrow b R a$.
A relation is called an {\it equivalence relation} if
it is transitive, reflexive and symmetric.
\medskip\noindent
In the setting of schemes we are going to relax the notion of a
relation a little bit and just require $R \to A \times A$ to
be a map. Here is the definition.
\begin{definition}
\label{definition-equivalence-relation}
Let $S$ be a scheme. Let $U$ be a scheme over $S$.
\begin{enumerate}
\item A {\it pre-relation} on $U$ over $S$ is any morphism
of schemes $j : R \to U \times_S U$. In this case we set
$t = \text{pr}_0 \circ j$ and $s = \text{pr}_1 \circ j$, so
that $j = (t, s)$.
\item A {\it relation} on $U$ over $S$ is a monomorphism
of schemes $j : R \to U \times_S U$.
\item A {\it pre-equivalence relation} is a pre-relation
$j : R \to U \times_S U$ such that the image of
$j : R(T) \to U(T) \times U(T)$ is an equivalence relation for
all $T/S$.
\item We say a morphism $R \to U \times_S U$ of schemes is
an {\it equivalence relation on $U$ over $S$}
if and only if for every scheme $T$ over $S$ the $T$-valued
points of $R$ define an equivalence relation
on the set of $T$-valued points of $U$.
\end{enumerate}
\end{definition}
\noindent
In other words, an equivalence relation is a pre-equivalence relation
such that $j$ is a relation.
\begin{lemma}
\label{lemma-restrict-relation}
Let $S$ be a scheme.
Let $U$ be a scheme over $S$.
Let $j : R \to U \times_S U$ be a pre-relation.
Let $g : U' \to U$ be a morphism of schemes.
Finally, set
$$
R' = (U' \times_S U')\times_{U \times_S U} R
\xrightarrow{j'}
U' \times_S U'
$$
Then $j'$ is a pre-relation on $U'$ over $S$.
If $j$ is a relation, then $j'$ is a relation.
If $j$ is a pre-equivalence relation, then $j'$ is a pre-equivalence relation.
If $j$ is an equivalence relation, then $j'$ is an equivalence relation.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{definition}
\label{definition-restrict-relation}
Let $S$ be a scheme.
Let $U$ be a scheme over $S$.
Let $j : R \to U \times_S U$ be a pre-relation.
Let $g : U' \to U$ be a morphism of schemes.
The pre-relation $j' : R' \to U' \times_S U'$ is called
the {\it restriction}, or {\it pullback} of the pre-relation $j$ to $U'$.
In this situation we sometimes write $R' = R|_{U'}$.
\end{definition}
\begin{lemma}
\label{lemma-pre-equivalence-equivalence-relation-points}
Let $j : R \to U \times_S U$ be a pre-relation.
Consider the relation on points of the scheme $U$ defined by
the rule
$$
x \sim y
\Leftrightarrow
\exists\ r \in R :
t(r) = x,
s(r) = y.
$$
If $j$ is a pre-equivalence relation then this is an
equivalence relation.
\end{lemma}
\begin{proof}
Suppose that $x \sim y$ and $y \sim z$.
Pick $r \in R$ with $t(r) = x$, $s(r) = y$ and
pick $r' \in R$ with $t(r') = y$, $s(r') = z$.
Pick a field $K$ fitting into the following commutative
diagram
$$
\xymatrix{
\kappa(r) \ar[r] & K \\
\kappa(y) \ar[u] \ar[r] & \kappa(r') \ar[u]
}
$$
Denote $x_K, y_K, z_K : \Spec(K) \to U$
the morphisms
$$
\begin{matrix}
\Spec(K) \to \Spec(\kappa(r))
\to
\Spec(\kappa(x)) \to U \\
\Spec(K) \to \Spec(\kappa(r))
\to
\Spec(\kappa(y)) \to U \\
\Spec(K) \to \Spec(\kappa(r'))
\to
\Spec(\kappa(z)) \to U
\end{matrix}
$$
By construction $(x_K, y_K) \in j(R(K))$ and
$(y_K, z_K) \in j(R(K))$. Since $j$ is a pre-equivalence relation
we see that also $(x_K, z_K) \in j(R(K))$.
This clearly implies that $x \sim z$.
\medskip\noindent
The proof that $\sim$ is reflexive and symmetric is omitted.
\end{proof}
\section{Group schemes}
\label{section-group-schemes}
\noindent
Let us recall that a {\it group} is a pair
$(G, m)$ where $G$ is a set, and $m : G \times G \to G$ is
a map of sets with the following properties:
\begin{enumerate}
\item (associativity) $m(g, m(g', g'')) = m(m(g, g'), g'')$
for all $g, g', g'' \in G$,
\item (identity) there exists a unique element $e \in G$
(called the {\it identity}, {\it unit}, or $1$ of $G$) such that
$m(g, e) = m(e, g) = g$ for all $g \in G$, and
\item (inverse) for all $g \in G$ there exists a $i(g) \in G$
such that $m(g, i(g)) = m(i(g), g) = e$, where $e$ is the
identity.
\end{enumerate}
Thus we obtain a map $e : \{*\} \to G$ and a map
$i : G \to G$ so that the quadruple $(G, m, e, i)$
satisfies the axioms listed above.
\medskip\noindent
A {\it homomorphism of groups} $\psi : (G, m) \to (G', m')$
is a map of sets $\psi : G \to G'$ such that
$m'(\psi(g), \psi(g')) = \psi(m(g, g'))$. This automatically
insures that $\psi(e) = e'$ and $i'(\psi(g)) = \psi(i(g))$.
(Obvious notation.) We will use this below.
\begin{definition}
\label{definition-group-scheme}
Let $S$ be a scheme.
\begin{enumerate}
\item A {\it group scheme over $S$} is a pair $(G, m)$, where
$G$ is a scheme over $S$ and $m : G \times_S G \to G$ is
a morphism of schemes over $S$ with the following property:
For every scheme $T$ over $S$ the pair $(G(T), m)$
is a group.
\item A {\it morphism $\psi : (G, m) \to (G', m')$ of group schemes over $S$}
is a morphism $\psi : G \to G'$ of schemes over $S$ such that for
every $T/S$ the induced map $\psi : G(T) \to G'(T)$ is a homomorphism
of groups.
\end{enumerate}
\end{definition}
\noindent
Let $(G, m)$ be a group scheme over the scheme $S$.
By the discussion above (and the discussion in Section \ref{section-notation})
we obtain morphisms of schemes over $S$:
(identity) $e : S \to G$ and (inverse) $i : G \to G$ such that
for every $T$ the quadruple $(G(T), m, e, i)$ satisfies the
axioms of a group listed above.
\medskip\noindent
Let $(G, m)$, $(G', m')$ be group schemes over $S$.
Let $f : G \to G'$ be a morphism of schemes over $S$.
It follows from the definition that $f$ is a morphism
of group schemes over $S$ if and only if the following diagram
is commutative:
$$
\xymatrix{
G \times_S G \ar[r]_-{f \times f} \ar[d]_m &
G' \times_S G' \ar[d]^m \\
G \ar[r]^f & G'
}
$$
\begin{lemma}
\label{lemma-base-change-group-scheme}
Let $(G, m)$ be a group scheme over $S$.
Let $S' \to S$ be a morphism of schemes.
The pullback $(G_{S'}, m_{S'})$ is a group scheme over $S'$.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{definition}
\label{definition-closed-subgroup-scheme}
Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$.
\begin{enumerate}
\item A {\it closed subgroup scheme} of $G$ is a closed subscheme
$H \subset G$ such that $m|_{H \times_S H}$ factors through $H$ and induces a
group scheme structure on $H$ over $S$.
\item An {\it open subgroup scheme} of $G$ is an open subscheme
$G' \subset G$ such that $m|_{G' \times_S G'}$ factors through $G'$
and induces a group scheme structure on $G'$ over $S$.
\end{enumerate}
\end{definition}
\noindent
Alternatively, we could say that $H$ is a closed subgroup scheme of $G$
if it is a group scheme over $S$ endowed with a morphism of group schemes
$i : H \to G$ over $S$ which identifies $H$ with a closed subscheme of $G$.
\begin{definition}
\label{definition-smooth-group-scheme}
Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$.
\begin{enumerate}
\item We say $G$ is a {\it smooth group scheme} if the structure
morphism $G \to S$ is smooth.
\item We say $G$ is a {\it flat group scheme} if the structure
morphism $G \to S$ is flat.
\item We say $G$ is a {\it separated group scheme} if the structure
morphism $G \to S$ is separated.
\end{enumerate}
Add more as needed.
\end{definition}
\section{Examples of group schemes}
\label{section-examples-group-schemes}
\begin{example}[Multiplicative group scheme]
\label{example-multiplicative-group}
Consider the functor which associates
to any scheme $T$ the group $\Gamma(T, \mathcal{O}_T^*)$
of units in the global sections of the structure sheaf.
This is representable by the scheme
$$
\mathbf{G}_m = \Spec(\mathbf{Z}[x, x^{-1}])
$$
The morphism giving the group structure is the morphism
\begin{eqnarray*}
\mathbf{G}_m \times \mathbf{G}_m & \to & \mathbf{G}_m \\
\Spec(\mathbf{Z}[x, x^{-1}] \otimes_{\mathbf{Z}} \mathbf{Z}[x, x^{-1}])
& \to &
\Spec(\mathbf{Z}[x, x^{-1}]) \\
\mathbf{Z}[x, x^{-1}] \otimes_{\mathbf{Z}} \mathbf{Z}[x, x^{-1}]
& \leftarrow &
\mathbf{Z}[x, x^{-1}] \\
x \otimes x & \leftarrow & x
\end{eqnarray*}
Hence we see that $\mathbf{G}_m$ is a group scheme over $\mathbf{Z}$.
For any scheme $S$ the base change $\mathbf{G}_{m, S}$ is a
group scheme over $S$ whose functor of points is
$$
T/S
\longmapsto
\mathbf{G}_{m, S}(T) = \mathbf{G}_m(T) = \Gamma(T, \mathcal{O}_T^*)
$$
as before.
\end{example}
\begin{example}[Roots of unity]
\label{example-roots-of-unity}
Let $n \in \mathbf{N}$.
Consider the functor which associates
to any scheme $T$ the subgroup of $\Gamma(T, \mathcal{O}_T^*)$
consisting of $n$th roots of unity.
This is representable by the scheme
$$
\mu_n = \Spec(\mathbf{Z}[x]/(x^n - 1)).
$$
The morphism giving the group structure is the morphism
\begin{eqnarray*}
\mu_n \times \mu_n & \to & \mu_n \\
\Spec(
\mathbf{Z}[x]/(x^n - 1)
\otimes_{\mathbf{Z}}
\mathbf{Z}[x]/(x^n - 1))
& \to &
\Spec(\mathbf{Z}[x]/(x^n - 1)) \\
\mathbf{Z}[x]/(x^n - 1) \otimes_{\mathbf{Z}} \mathbf{Z}[x]/(x^n - 1)
& \leftarrow &
\mathbf{Z}[x]/(x^n - 1) \\
x \otimes x & \leftarrow & x
\end{eqnarray*}
Hence we see that $\mu_n$ is a group scheme over $\mathbf{Z}$.
For any scheme $S$ the base change $\mu_{n, S}$ is a
group scheme over $S$ whose functor of points is
$$
T/S
\longmapsto
\mu_{n, S}(T) = \mu_n(T) = \{f \in \Gamma(T, \mathcal{O}_T^*) \mid f^n = 1\}
$$
as before.
\end{example}
\begin{example}[Additive group scheme]
\label{example-additive-group}
Consider the functor which associates
to any scheme $T$ the group $\Gamma(T, \mathcal{O}_T)$
of global sections of the structure sheaf.
This is representable by the scheme
$$
\mathbf{G}_a = \Spec(\mathbf{Z}[x])
$$
The morphism giving the group structure is the morphism
\begin{eqnarray*}
\mathbf{G}_a \times \mathbf{G}_a & \to & \mathbf{G}_a \\
\Spec(\mathbf{Z}[x] \otimes_{\mathbf{Z}} \mathbf{Z}[x])
& \to &
\Spec(\mathbf{Z}[x]) \\
\mathbf{Z}[x] \otimes_{\mathbf{Z}} \mathbf{Z}[x]
& \leftarrow &
\mathbf{Z}[x] \\
x \otimes 1 + 1 \otimes x & \leftarrow & x
\end{eqnarray*}
Hence we see that $\mathbf{G}_a$ is a group scheme over $\mathbf{Z}$.
For any scheme $S$ the base change $\mathbf{G}_{a, S}$ is a
group scheme over $S$ whose functor of points is
$$
T/S
\longmapsto
\mathbf{G}_{a, S}(T) = \mathbf{G}_a(T) = \Gamma(T, \mathcal{O}_T)
$$
as before.
\end{example}
\begin{example}[General linear group scheme]
\label{example-general-linear-group}
Let $n \geq 1$.
Consider the functor which associates
to any scheme $T$ the group
$$
\text{GL}_n(\Gamma(T, \mathcal{O}_T))
$$
of invertible $n \times n$ matrices over
the global sections of the structure sheaf.
This is representable by the scheme
$$
\text{GL}_n = \Spec(\mathbf{Z}[\{x_{ij}\}_{1 \leq i, j \leq n}][1/d])
$$
where $d = \det((x_{ij}))$ with $(x_{ij})$ the $n \times n$ matrix
with entry $x_{ij}$ in the $(i, j)$-spot.
The morphism giving the group structure is the morphism
\begin{eqnarray*}
\text{GL}_n \times \text{GL}_n & \to & \text{GL}_n \\
\Spec(\mathbf{Z}[x_{ij}, 1/d] \otimes_{\mathbf{Z}}
\mathbf{Z}[x_{ij}, 1/d])
& \to &
\Spec(\mathbf{Z}[x_{ij}, 1/d]) \\
\mathbf{Z}[x_{ij}, 1/d] \otimes_{\mathbf{Z}} \mathbf{Z}[x_{ij}, 1/d]
& \leftarrow &
\mathbf{Z}[x_{ij}, 1/d] \\
\sum x_{ik} \otimes x_{kj} & \leftarrow & x_{ij}
\end{eqnarray*}
Hence we see that $\text{GL}_n$ is a group scheme over $\mathbf{Z}$.
For any scheme $S$ the base change $\text{GL}_{n, S}$ is a
group scheme over $S$ whose functor of points is
$$
T/S
\longmapsto
\text{GL}_{n, S}(T) = \text{GL}_n(T) =\text{GL}_n(\Gamma(T, \mathcal{O}_T))
$$
as before.
\end{example}
\begin{example}
\label{example-determinant}
The determinant defines a morphism of group schemes
$$
\det : \text{GL}_n \longrightarrow \mathbf{G}_m
$$
over $\mathbf{Z}$. By base change it gives a morphism
of group schemes $\text{GL}_{n, S} \to \mathbf{G}_{m, S}$
over any base scheme $S$.
\end{example}
\begin{example}[Constant group]
\label{example-constant-group}
Let $G$ be an abstract group. Consider the functor
which associates to any scheme $T$ the group
of locally constant maps $T \to G$ (where $T$ has the Zariski topology
and $G$ the discrete topology). This is representable by the scheme
$$
G_{\Spec(\mathbf{Z})} =
\coprod\nolimits_{g \in G} \Spec(\mathbf{Z}).
$$
The morphism giving the group structure is the morphism
$$
G_{\Spec(\mathbf{Z})}
\times_{\Spec(\mathbf{Z})}
G_{\Spec(\mathbf{Z})}
\longrightarrow
G_{\Spec(\mathbf{Z})}
$$
which maps the component corresponding to the pair $(g, g')$ to the
component corresponding to $gg'$. For any scheme $S$ the base change
$G_S$ is a group scheme over $S$ whose functor of points is
$$
T/S
\longmapsto
G_S(T) = \{f : T \to G \text{ locally constant}\}
$$
as before.
\end{example}
\section{Properties of group schemes}
\label{section-properties-group-schemes}
\noindent
In this section we collect some simple properties of group schemes which
hold over any base.
\begin{lemma}
\label{lemma-group-scheme-separated}
Let $S$ be a scheme.
Let $G$ be a group scheme over $S$.
Then $G \to S$ is separated (resp.\ quasi-separated) if and only if
the identity morphism $e : S \to G$ is a closed immersion
(resp.\ quasi-compact).
\end{lemma}
\begin{proof}
We recall that by
Schemes, Lemma \ref{schemes-lemma-section-immersion}
we have that $e$ is an immersion which is a closed immersion
(resp.\ quasi-compact) if $G \to S$ is separated (resp.\ quasi-separated).
For the converse, consider the diagram
$$
\xymatrix{
G \ar[r]_-{\Delta_{G/S}} \ar[d] &
G \times_S G \ar[d]^{(g, g') \mapsto m(i(g), g')} \\
S \ar[r]^e & G
}
$$
It is an exercise in the functorial point of view in algebraic geometry
to show that this diagram is cartesian. In other words, we see that
$\Delta_{G/S}$ is a base change of $e$. Hence if $e$ is a
closed immersion (resp.\ quasi-compact) so is $\Delta_{G/S}$, see
Schemes, Lemma \ref{schemes-lemma-base-change-immersion}
(resp.\ Schemes, Lemma
\ref{schemes-lemma-quasi-compact-preserved-base-change}).
\end{proof}
\begin{lemma}
\label{lemma-flat-action-on-group-scheme}
Let $S$ be a scheme.
Let $G$ be a group scheme over $S$.
Let $T$ be a scheme over $S$ and let $\psi : T \to G$ be a morphism over $S$.
If $T$ is flat over $S$, then the morphism
$$
T \times_S G \longrightarrow G, \quad
(t, g) \longmapsto m(\psi(t), g)
$$
is flat. In particular, if $G$ is flat over $S$, then
$m : G \times_S G \to G$ is flat.
\end{lemma}
\begin{proof}
Consider the diagram
$$
\xymatrix{
T \times_S G \ar[rrr]_{(t, g) \mapsto (t, m(\psi(t), g))} & & &
T \times_S G \ar[r]_{\text{pr}} \ar[d] &
G \ar[d] \\
& & &
T \ar[r] &
S
}
$$
The left top horizontal arrow is an isomorphism and the
square is cartesian. Hence the lemma follows from
Morphisms, Lemma \ref{morphisms-lemma-base-change-flat}.
\end{proof}
\begin{lemma}
\label{lemma-group-scheme-module-differentials}
Let $(G, m, e, i)$ be a group scheme over the scheme $S$.
Denote $f : G \to S$ the structure morphism. Assume $f$ is flat.
Then there exist canonical isomorphisms
$$
\Omega_{G/S} \cong f^*\mathcal{C}_{S/G} \cong f^*e^*\Omega_{G/S}
$$
where $\mathcal{C}_{S/G}$ denotes the conormal sheaf of the
immersion $e$. In particular, if $S$ is the spectrum of a field, then
$\Omega_{G/S}$ is a free $\mathcal{O}_G$-module.
\end{lemma}
\begin{proof}
In
Morphisms, Lemma \ref{morphisms-lemma-differentials-diagonal}
we identified $\Omega_{G/S}$ with the conormal sheaf of the
diagonal morphism $\Delta_{G/S}$. In the proof of
Lemma \ref{lemma-group-scheme-separated}
we showed that $\Delta_{G/S}$ is a base change of the immersion $e$
by the morphism $(g, g') \mapsto m(i(g), g')$. This morphism
is isomorphic to the morphism $(g, g') \mapsto m(g, g')$
hence is flat by
Lemma \ref{lemma-flat-action-on-group-scheme}.
Hence we get the first isomorphism by
Morphisms, Lemma \ref{morphisms-lemma-conormal-functorial-flat}.
By
Morphisms, Lemma \ref{morphisms-lemma-differentials-relative-immersion-section}
we have $\mathcal{C}_{S/G} \cong e^*\Omega_{G/S}$.
\medskip\noindent
If $S$ is the spectrum of a field, then $G \to S$ is flat, and
any $\mathcal{O}_S$-module on $S$ is free.
\end{proof}
\begin{lemma}
\label{lemma-group-scheme-addition-tangent-vectors}
Let $S$ be a scheme. Let $G$ be a group scheme over $S$.
Let $s \in S$. Then the composition
$$
T_{G/S, e(s)} \oplus T_{G/S, e(s)} = T_{G \times_S G/S, (e(s), e(s))}
\rightarrow T_{G/S, e(s)}
$$
is addition of tangent vectors. Here the $=$ comes from
Varieties, Lemma \ref{varieties-lemma-tangent-space-product}
and the right arrow is induced from $m : G \times_S G \to G$ via
Varieties, Lemma \ref{varieties-lemma-map-tangent-spaces}.
\end{lemma}
\begin{proof}
We will use Varieties, Equation (\ref{varieties-equation-tangent-space-fibre})
and work with tangent vectors in fibres.
An element $\theta$ in the first factor $T_{G_s/s, e(s)}$
is the image of $\theta$ via the map
$T_{G_s/s, e(s)} \to T_{G_s \times G_s/s, (e(s), e(s))}$
coming from $(1, e) : G_s \to G_s \times G_s$.
Since $m \circ (1, e) = 1$ we see that $\theta$ maps to $\theta$
by functoriality. Since the map is linear we see that
$(\theta_1, \theta_2)$ maps to $\theta_1 + \theta_2$.
\end{proof}
\section{Properties of group schemes over a field}
\label{section-properties-group-schemes-field}
\noindent
In this section we collect some properties of group schemes over a
field. In the case of group schemes which are (locally) algebraic
over a field we can say a lot more, see
Section \ref{section-algebraic-group-schemes}.
\begin{lemma}
\label{lemma-group-scheme-over-field-open-multiplication}
If $(G, m)$ is a group scheme over a field $k$, then the
multiplication map $m : G \times_k G \to G$ is open.
\end{lemma}
\begin{proof}
The multiplication map is isomorphic to the projection map
$\text{pr}_0 : G \times_k G \to G$
because the diagram
$$
\xymatrix{
G \times_k G \ar[d]^m \ar[rrr]_{(g, g') \mapsto (m(g, g'), g')} & & &
G \times_k G \ar[d]^{(g, g') \mapsto g} \\
G \ar[rrr]^{\text{id}} & & & G
}
$$
is commutative with isomorphisms as horizontal arrows. The projection
is open by
Morphisms, Lemma \ref{morphisms-lemma-scheme-over-field-universally-open}.
\end{proof}
\begin{lemma}
\label{lemma-group-scheme-over-field-translate-open}
If $(G, m)$ is a group scheme over a field $k$. Let $U \subset G$
open and $T \to G$ a morphism of schemes. Then the image of the
composition $T \times_k U \to G \times_k G \to G$ is open.
\end{lemma}
\begin{proof}
For any field extension $k \subset K$ the morphism $G_K \to G$ is open
(Morphisms, Lemma \ref{morphisms-lemma-scheme-over-field-universally-open}).
Every point $\xi$ of $T \times_k U$ is the image of a morphism
$(t, u) : \Spec(K) \to T \times_k U$ for some $K$. Then the image of
$T_K \times_K U_K = (T \times_k U)_K \to G_K$ contains the translate
$t \cdot U_K$ which is open. Combining these facts we see that the
image of $T \times_k U \to G$ contains an open neighbourhood of
the image of $\xi$. Since $\xi$ was arbitrary we win.
\end{proof}
\begin{lemma}
\label{lemma-group-scheme-over-field-separated}
Let $G$ be a group scheme over a field.
Then $G$ is a separated scheme.
\end{lemma}
\begin{proof}
Say $S = \Spec(k)$ with $k$ a field, and let $G$ be a group scheme
over $S$. By
Lemma \ref{lemma-group-scheme-separated}
we have to show that $e : S \to G$ is a closed immersion. By
Morphisms, Lemma
\ref{morphisms-lemma-algebraic-residue-field-extension-closed-point-fibre}
the image of $e : S \to G$ is a closed point of $G$.
It is clear that $\mathcal{O}_G \to e_*\mathcal{O}_S$ is surjective,
since $e_*\mathcal{O}_S$ is a skyscraper sheaf supported at the neutral
element of $G$ with value $k$. We conclude that $e$ is a closed immersion by
Schemes, Lemma \ref{schemes-lemma-characterize-closed-immersions}.
\end{proof}
\begin{lemma}
\label{lemma-group-scheme-field-geometrically-irreducible}
Let $G$ be a group scheme over a field $k$.
Then
\begin{enumerate}
\item every local ring $\mathcal{O}_{G, g}$ of $G$ has a unique
minimal prime ideal,
\item there is exactly one irreducible component $Z$ of $G$
passing through $e$, and
\item $Z$ is geometrically irreducible over $k$.
\end{enumerate}
\end{lemma}
\begin{proof}
For any point $g \in G$ there exists a field extension
$k \subset K$ and a $K$-valued point $g' \in G(K)$ mapping to
$g$. If we think of $g'$ as a $K$-rational point of the
group scheme $G_K$, then we see that
$\mathcal{O}_{G, g} \to \mathcal{O}_{G_K, g'}$ is a faithfully flat
local ring map (as $G_K \to G$ is flat, and a local flat ring map
is faithfully flat, see
Algebra, Lemma \ref{algebra-lemma-local-flat-ff}).
The result for $\mathcal{O}_{G_K, g'}$ implies the
result for $\mathcal{O}_{G, g}$, see
Algebra, Lemma \ref{algebra-lemma-injective-minimal-primes-in-image}.
Hence in order to prove (1) it suffices to
prove it for $k$-rational points $g$ of $G$. In this case
translation by $g$ defines an automorphism $G \to G$
which maps $e$ to $g$. Hence $\mathcal{O}_{G, g} \cong \mathcal{O}_{G, e}$.
In this way we see that (2) implies (1), since irreducible components
passing through $e$ correspond one to one with minimal prime ideals
of $\mathcal{O}_{G, e}$.
\medskip\noindent
In order to prove (2) and (3) it suffices to prove (2) when $k$
is algebraically closed. In this case, let $Z_1$, $Z_2$ be two
irreducible components of $G$ passing through $e$.
Since $k$ is algebraically closed the closed subscheme
$Z_1 \times_k Z_2 \subset G \times_k G$ is irreducible too, see
Varieties, Lemma \ref{varieties-lemma-bijection-irreducible-components}.
Hence $m(Z_1 \times_k Z_2)$ is contained in an irreducible
component of $G$. On the other hand it contains
$Z_1$ and $Z_2$ since $m|_{e \times G} = \text{id}_G$ and
$m|_{G \times e} = \text{id}_G$. We conclude $Z_1 = Z_2$ as desired.
\end{proof}
\begin{remark}
\label{remark-warning-group-scheme-geometrically-irreducible}
Warning: The result of
Lemma \ref{lemma-group-scheme-field-geometrically-irreducible}
does not mean that every irreducible component of $G/k$ is
geometrically irreducible. For example the group scheme
$\mu_{3, \mathbf{Q}} = \Spec(\mathbf{Q}[x]/(x^3 - 1))$
over $\mathbf{Q}$ has two irreducible components corresponding
to the factorization $x^3 - 1 = (x - 1)(x^2 + x + 1)$.
The first factor corresponds to the irreducible component
passing through the identity, and the second irreducible component
is not geometrically irreducible over $\Spec(\mathbf{Q})$.
\end{remark}
\begin{lemma}
\label{lemma-reduced-subgroup-scheme-perfect}
Let $G$ be a group scheme over a perfect field $k$.
Then the reduction $G_{red}$ of $G$ is a closed subgroup scheme of $G$.
\end{lemma}
\begin{proof}
Omitted. Hint: Use that $G_{red} \times_k G_{red}$ is reduced by
Varieties, Lemmas \ref{varieties-lemma-perfect-reduced} and
\ref{varieties-lemma-geometrically-reduced-any-base-change}.
\end{proof}
\begin{lemma}
\label{lemma-open-subgroup-closed-over-field}
Let $k$ be a field. Let $\psi: G' \to G$ be a morphism of group schemes
over $k$. If $\psi(G')$ is open in $G$, then $\psi(G')$ is closed in $G$.
\end{lemma}
\begin{proof}
Let $U = \psi(G') \subset G$. Let $Z = G \setminus \psi(G') = G \setminus U$
with the reduced induced closed subscheme structure. By
Lemma \ref{lemma-group-scheme-over-field-translate-open}
the image of
$$
Z \times_k G' \longrightarrow
Z \times_k U \longrightarrow G
$$
is open (the first arrow is surjective). On the other hand, since $\psi$
is a homomorphism of group schemes, the image of $Z \times_k G' \to G$
is contained in $Z$ (because translation by $\psi(g')$ preserves
$U$ for all points $g'$ of $G'$; small detail omitted).
Hence $Z \subset G$ is an open subset (although not
necessarily an open subscheme). Thus $U = \psi(G')$ is closed.
\end{proof}
\begin{lemma}
\label{lemma-immersion-group-schemes-closed-over-field}
Let $i : G' \to G$ be an immersion of group schemes over a field $k$.
Then $i$ is a closed immersion, i.e., $i(G')$ is a closed subgroup scheme
of $G$.
\end{lemma}
\begin{proof}
To show that $i$ is a closed immersion it suffices to show that
$i(G')$ is a closed subset of $G$. Let $k \subset k'$ be a perfect
extension of $k$. If $i(G'_{k'}) \subset G_{k'}$ is closed, then
$i(G') \subset G$ is closed by
Morphisms, Lemma \ref{morphisms-lemma-fpqc-quotient-topology}
(as $G_{k'} \to G$ is flat, quasi-compact and surjective).
Hence we may and do assume $k$ is perfect. We will use without further
mention that products of reduced schemes over $k$ are reduced.
We may replace $G'$ and $G$ by their reductions, see
Lemma \ref{lemma-reduced-subgroup-scheme-perfect}.
Let $\overline{G'} \subset G$ be the closure of $i(G')$ viewed
as a reduced closed subscheme. By
Varieties, Lemma \ref{varieties-lemma-closure-of-product}
we conclude that $\overline{G'} \times_k \overline{G'}$
is the closure of the image of $G' \times_k G' \to G \times_k G$. Hence
$$
m\Big(\overline{G'} \times_k \overline{G'}\Big)
\subset \overline{G'}
$$
as $m$ is continuous. It follows that $\overline{G'} \subset G$
is a (reduced) closed subgroup scheme. By
Lemma \ref{lemma-open-subgroup-closed-over-field}
we see that $i(G') \subset \overline{G'}$ is also closed
which implies that $i(G') = \overline{G'}$ as desired.
\end{proof}
\begin{lemma}
\label{lemma-irreducible-group-scheme-over-field-qc}
Let $G$ be a group scheme over a field $k$. If $G$ is irreducible,
then $G$ is quasi-compact.
\end{lemma}
\begin{proof}
Suppose that $k \subset K$ is a field extension. If $G_K$
is quasi-compact, then $G$ is too as $G_K \to G$ is surjective.
By Lemma \ref{lemma-group-scheme-field-geometrically-irreducible}
we see that $G_K$ is irreducible. Hence it suffices to prove the lemma
after replacing $k$ by some extension. Choose $K$ to be an algebraically
closed field extension of very large cardinality. Then by
Varieties, Lemma \ref{varieties-lemma-make-Jacobson},
we see that $G_K$ is a Jacobson scheme all of whose closed points have residue
field equal to $K$. In other words we may assume $G$ is a Jacobson
scheme all of whose closed points have residue field $k$.
\medskip\noindent
Let $U \subset G$ be a nonempty affine open. Let $g \in G(k)$. Then
$gU \cap U \not = \emptyset$. Hence we see that $g$ is in the image
of the morphism
$$
U \times_{\Spec(k)} U \longrightarrow G, \quad
(u_1, u_2) \longmapsto u_1u_2^{-1}
$$
Since the image of this morphism is open
(Lemma \ref{lemma-group-scheme-over-field-open-multiplication})
we see that the image is all of $G$ (because $G$ is Jacobson
and closed points are $k$-rational).
Since $U$ is affine, so is $U \times_{\Spec(k)} U$. Hence $G$ is the
image of a quasi-compact scheme, hence quasi-compact.
\end{proof}
\begin{lemma}
\label{lemma-connected-group-scheme-over-field-irreducible}
Let $G$ be a group scheme over a field $k$. If $G$ is connected,
then $G$ is irreducible.
\end{lemma}
\begin{proof}
By Varieties, Lemma \ref{varieties-lemma-geometrically-connected-criterion}
we see that $G$ is geometrically connected. If we show that $G_K$
is irreducible for some field extension $k \subset K$, then
the lemma follows. Hence we may apply
Varieties, Lemma \ref{varieties-lemma-make-Jacobson}
to reduce to the case where $k$ is algebraically closed,
$G$ is a Jacobson scheme, and all the closed points are $k$-rational.
\medskip\noindent
Let $Z \subset G$ be the unique irreducible component of $G$ passing
through the neutral element, see
Lemma \ref{lemma-group-scheme-field-geometrically-irreducible}.
Endowing $Z$ with the reduced induced closed subscheme structure,
we see that $Z \times_k Z$ is reduced and irreducible
(Varieties, Lemmas
\ref{varieties-lemma-geometrically-reduced-any-base-change} and
\ref{varieties-lemma-bijection-irreducible-components}).
We conclude that $m|_{Z \times_k Z} : Z \times_k Z \to G$ factors
through $Z$. Hence $Z$ becomes a closed subgroup scheme of $G$.
\medskip\noindent
To get a contradiction, assume there exists another irreducible
component $Z' \subset G$. Then $Z \cap Z' = \emptyset$ by
Lemma \ref{lemma-group-scheme-field-geometrically-irreducible}.
By Lemma \ref{lemma-irreducible-group-scheme-over-field-qc}
we see that $Z$ is quasi-compact. Thus we may choose a quasi-compact open
$U \subset G$ with $Z \subset U$ and $U \cap Z' = \emptyset$.
The image $W$ of $Z \times_k U \to G$ is open in $G$ by
Lemma \ref{lemma-group-scheme-over-field-translate-open}.
On the other hand, $W$ is quasi-compact as the image of a
quasi-compact space. We claim that $W$ is closed.
If the claim is true, then $W \subset G \setminus Z'$ is a proper open
and closed subset of $G$, which contradicts the assumption that
$G$ is connected.
\medskip\noindent
Proof of the claim. Since $W$ is quasi-compact, we see that
points in the closure of $W$ are specializations of points of $W$
(Morphisms, Lemma \ref{morphisms-lemma-reach-points-scheme-theoretic-image}).
Thus we have to show that any irreducible
component $Z'' \subset G$ of $G$ which meets $W$ is contained in $W$.
As $G$ is Jacobson and closed points are rational,
$Z'' \cap W$ has a rational point
$g \in Z''(k) \cap W(k)$ and hence $Z'' = Zg$. But $W = m(Z \times_k W)$
by construction, so $Z'' \cap W \not = \emptyset$ implies
$Z'' \subset W$.
\end{proof}
\begin{proposition}
\label{proposition-connected-component}
Let $G$ be a group scheme over a field $k$. There exists a canonical closed
subgroup scheme $G^0 \subset G$ with the following properties
\begin{enumerate}
\item $G^0 \to G$ is a flat closed immersion,
\item $G^0 \subset G$ is the connected component of the identity,
\item $G^0$ is geometrically irreducible, and
\item $G^0$ is quasi-compact.
\end{enumerate}
\end{proposition}
\begin{proof}
Let $G^0$ be the connected component of the identity with its canonical
scheme structure (Morphisms, Definition
\ref{morphisms-definition-scheme-structure-connected-component}).
By Varieties, Lemma \ref{varieties-lemma-geometrically-connected-criterion}
we see that $G^0$ is geometrically connected. Thus
$G^0 \times_k G^0$ is connected
(Varieties, Lemma \ref{varieties-lemma-bijection-connected-components}).
Thus $m(G^0 \times_k G^0) \subset G^0$ set theoretically.
To see that this holds scheme theoretically, note that
$G^0 \times_k G^0 \to G \times_k G$ is a flat closed immersion.
By Morphisms, Lemma \ref{morphisms-lemma-characterize-flat-closed-immersions}
it follows that $G^0 \times_k G^0$ is a closed subscheme of
$(G \times_k G) \times_{m, G} G^0$. Thus we see that
$m|_{G^0 \times_k G^0} : G^0 \times_k G^0 \to G$ factors through
$G^0$. Hence $G^0$ becomes a closed subgroup scheme of $G$.
By Lemma \ref{lemma-connected-group-scheme-over-field-irreducible}
we see that $G^0$ is irreducible. By
Lemma \ref{lemma-group-scheme-field-geometrically-irreducible}
we see that $G^0$ is geometrically irreducible. By
Lemma \ref{lemma-irreducible-group-scheme-over-field-qc}
we see that $G^0$ is quasi-compact.
\end{proof}
\begin{lemma}
\label{lemma-profinite-product-over-field}
Let $k$ be a field. Let $T = \Spec(A)$ where $A$ is a directed colimit of
algebras which are finite products of copies of $k$. For any scheme $X$
over $k$ we have $|T \times_k X| = |T| \times |X|$ as topological spaces.
\end{lemma}
\begin{proof}
By taking an affine open covering we reduce to the case of an affine $X$.
Say $X = \Spec(B)$.
Write $A = \colim A_i$ with $A_i = \prod_{t \in T_i} k$ and $T_i$ finite.
Then $T_i = |\Spec(A_i)|$ with the discrete topology and the transition
morphisms $A_i \to A_{i'}$ are given by set maps $T_{i'} \to T_i$. Thus
$|T| = \lim T_i$ as a topological space, see
Limits, Lemma \ref{limits-lemma-topology-limit}. Similarly we have
\begin{align*}
|T \times_k X| & =
|\Spec(A \otimes_k B)| \\
& =
|\Spec(\colim A_i \otimes_k B)| \\
& =
\lim |\Spec(A_i \otimes_k B)| \\
& =
\lim |\Spec(\prod\nolimits_{t \in T_i} B)| \\
& =
\lim T_i \times |X| \\
& =
(\lim T_i) \times |X| \\
& =
|T| \times |X|
\end{align*}
by the lemma above and the fact that limits commute with limits.
\end{proof}
\noindent
The following lemma says that in fact we can put a
``algebraic profinite family of points'' in an affine open.
We urge the reader to read Lemma \ref{lemma-points-in-affine} first.
\begin{lemma}
\label{lemma-compact-set-in-affine}
Let $k$ be an algebraically closed field. Let $G$ be a group scheme over $k$.
Assume that $G$ is Jacobson and that all closed points are $k$-rational.
Let $T = \Spec(A)$ where $A$ is a directed colimit of algebras which
are finite products of copies of $k$. For any morphism $f : T \to G$
there exists an affine open $U \subset G$ containing $f(T)$.
\end{lemma}
\begin{proof}
Let $G^0 \subset G$ be the closed subgroup scheme found in
Proposition \ref{proposition-connected-component}. The first two paragraphs
serve to reduce to the case $G = G^0$.
\medskip\noindent
Observe that $T$ is a directed inverse limit of finite topological spaces
(Limits, Lemma \ref{limits-lemma-topology-limit}), hence profinite as a
topological space (Topology, Definition \ref{topology-definition-profinite}).
Let $W \subset G$ be a quasi-compact open containing the image of $T \to G$.
After replacing $W$ by the image of $G^0 \times W \to G \times G \to G$ we may
assume that $W$ is invariant under the action of left translation by $G^0$, see
Lemma \ref{lemma-group-scheme-over-field-translate-open}.
Consider the composition
$$
\psi = \pi \circ f : T \xrightarrow{f} W \xrightarrow{\pi} \pi_0(W)
$$
The space $\pi_0(W)$ is profinite
(Topology, Lemma \ref{topology-lemma-spectral-pi0} and
Properties, Lemma
\ref{properties-lemma-quasi-compact-quasi-separated-spectral}).
Let $F_\xi \subset T$ be the fibre of $T \to \pi_0(W)$ over $\xi \in \pi_0(W)$.
Assume that for all $\xi$ we can find an affine open $U_\xi \subset W$ with
$F \subset U$. Since $\psi : T \to \pi_0(W)$ is proper as a map of
topological spaces (Topology, Lemma \ref{topology-lemma-closed-map}),
we can find a quasi-compact open $V_\xi \subset \pi_0(W)$ such that
$\psi^{-1}(V_\xi) \subset f^{-1}(U_\xi)$ (easy topological argument omitted).
After replacing $U_\xi$ by $U_\xi \cap \pi^{-1}(V_\xi)$, which is open and
closed in $U_\xi$ hence affine, we see that $U_\xi \subset \pi^{-1}(V_\xi)$
and $U_\xi \cap T = \psi^{-1}(V_\xi)$.
By Topology, Lemma \ref{topology-lemma-profinite-refine-open-covering}
we can find a finite disjoint union decomposition
$\pi_0(W) = \bigcup_{i = 1, \ldots, n} V_i$ by quasi-compact opens such that
$V_i \subset V_{\xi_i}$ for some $i$. Then we see that
$$
f(T) \subset \bigcup\nolimits_{i = 1, \ldots, n} U_{\xi_i} \cap \pi^{-1}(V_i)
$$
the right hand side of which is a finite disjoint union of affines, therefore
affine.
\medskip\noindent
Let $Z$ be a connected component of $G$ which meets $f(T)$. Then $Z$
has a $k$-rational point $z$ (because all residue fields of the scheme $T$
are isomorphic to $k$). Hence $Z = G^0 z$. By our choice of $W$, we see
that $Z \subset W$. The argument in the preceding paragraph reduces us to
the problem of finding an affine open neighbourhood of $f(T) \cap Z$ in $W$.
After translation by a rational point we may assume that $Z = G^0$
(details omitted). Observe that the scheme theoretic inverse image
$T' = f^{-1}(G^0) \subset T$ is a closed subscheme, which has the same type.
After replacing $T$ by $T'$ we may assume that $f(T) \subset G^0$.
Choose an affine open neighbourhood $U \subset G$
of $e \in G$, so that in particular $U \cap G^0$ is nonempty. We will show
there exists a $g \in G^0(k)$ such that $f(T) \subset g^{-1}U$.
This will finish the proof as $g^{-1}U \subset W$ by the left
$G^0$-invariance of $W$.
\medskip\noindent
The arguments in the preceding two paragraphs allow us to pass to $G^0$
and reduce the problem to the following:
Assume $G$ is irreducible and $U \subset G$ an affine
open neighbourhood of $e$. Show that $f(T) \subset g^{-1}U$
for some $g \in G(k)$. Consider the morphism
$$
U \times_k T \longrightarrow G \times_k T,\quad
(t, u) \longrightarrow (uf(t)^{-1}, t)
$$
which is an open immersion (because the extension of this morphism to
$G \times_k T \to G \times_k T$ is an isomorphism).
By our assumption on $T$ we see that we have $|U \times_k T| = |U| \times |T|$
and similarly for $G \times_k T$, see
Lemma \ref{lemma-profinite-product-over-field}.
Hence the image of the displayed open immersion is a finite union
of boxes $\bigcup_{i = 1, \ldots, n} U_i \times V_i$ with
$V_i \subset T$ and $U_i \subset G$ quasi-compact open. This means that
the possible opens $Uf(t)^{-1}$, $t \in T$ are finite in number, say
$Uf(t_1)^{-1}, \ldots, Uf(t_r)^{-1}$. Since $G$ is irreducible the
intersection
$$
Uf(t_1)^{-1} \cap \ldots \cap Uf(t_r)^{-1}
$$
is nonempty and since $G$ is Jacobson with closed points $k$-rational,
we can choose a $k$-valued point $g \in G(k)$ of this intersection.
Then we see that $g \in Uf(t)^{-1}$ for all $t \in T$ which
means that $f(t) \in g^{-1}U$ as desired.
\end{proof}
\begin{remark}
\label{remark-easy}
If $G$ is a group scheme over a field, is there always a quasi-compact
open and closed subgroup scheme? By
Proposition \ref{proposition-connected-component}
this question is only interesting if $G$ has infinitely many connected
components (geometrically).
\end{remark}
\begin{lemma}
\label{lemma-group-scheme-field-countable-affine}
Let $G$ be a group scheme over a field.
There exists an open and closed subscheme $G' \subset G$
which is a countable union of affines.
\end{lemma}
\begin{proof}
Let $e \in U(k)$ be a quasi-compact open neighbourhood of the identity
element. By replacing $U$ by $U \cap i(U)$ we may assume that
$U$ is invariant under the inverse map. As $G$ is separated this is
still a quasi-compact set. Set
$$
G' = \bigcup\nolimits_{n \geq 1} m_n(U \times_k \ldots \times_k U)
$$
where $m_n : G \times_k \ldots \times_k G \to G$ is the $n$-slot
multiplication map
$(g_1, \ldots, g_n) \mapsto m(m(\ldots (m(g_1, g_2), g_3), \ldots ), g_n)$.
Each of these maps are open (see
Lemma \ref{lemma-group-scheme-over-field-open-multiplication})
hence $G'$ is an open subgroup scheme. By
Lemma \ref{lemma-open-subgroup-closed-over-field}
it is also a closed subgroup scheme.
\end{proof}
\section{Properties of algebraic group schemes}
\label{section-algebraic-group-schemes}
\noindent
Recall that a scheme over a field $k$ is (locally) algebraic if it is
(locally) of finite type over $\Spec(k)$, see
Varieties, Definition \ref{varieties-definition-algebraic-scheme}.
This is the sense of algebraic we are using in the title of this section.
\begin{lemma}
\label{lemma-group-scheme-finite-type-field}
Let $k$ be a field. Let $G$ be a locally algebraic group scheme over $k$.
Then $G$ is equidimensional and $\dim(G) = \dim_g(G)$ for all $g \in G$.
For any closed point $g \in G$ we have $\dim(G) = \dim(\mathcal{O}_{G, g})$.
\end{lemma}
\begin{proof}
Let us first prove that $\dim_g(G) = \dim_{g'}(G)$ for any
pair of points $g, g' \in G$. By
Morphisms, Lemma \ref{morphisms-lemma-dimension-fibre-after-base-change}
we may extend the ground field at will. Hence we may assume that
both $g$ and $g'$ are defined over $k$. Hence there exists an
automorphism of $G$ mapping $g$ to $g'$, whence the equality.
By
Morphisms, Lemma \ref{morphisms-lemma-dimension-fibre-at-a-point}
we have
$\dim_g(G) = \dim(\mathcal{O}_{G, g}) +
\text{trdeg}_k(\kappa(g))$.
On the other hand, the dimension of $G$ (or any open subset of $G$)
is the supremum of the dimensions of the local rings of of $G$, see
Properties, Lemma \ref{properties-lemma-codimension-local-ring}.
Clearly this is maximal for closed points $g$ in which case
$\text{trdeg}_k(\kappa(g)) = 0$ (by the Hilbert Nullstellensatz, see
Morphisms, Section \ref{morphisms-section-points-finite-type}).
Hence the lemma follows.
\end{proof}
\noindent
The following result is sometimes referred to as Cartier's theorem.
\begin{lemma}
\label{lemma-group-scheme-characteristic-zero-smooth}
Let $k$ be a field of characteristic $0$. Let $G$ be a
locally algebraic group scheme over $k$. Then the structure
morphism $G \to \Spec(k)$ is smooth, i.e., $G$ is a smooth
group scheme.
\end{lemma}
\begin{proof}
By
Lemma \ref{lemma-group-scheme-module-differentials}
the module of differentials of $G$ over $k$ is free.
Hence smoothness follows from
Varieties, Lemma \ref{varieties-lemma-char-zero-differentials-free-smooth}.
\end{proof}
\begin{remark}
\label{remark-when-reduced}
Any group scheme over a field of characteristic $0$ is reduced, see
\cite[I, Theorem 1.1 and I, Corollary 3.9, and II, Theorem 2.4]{Perrin-thesis}
and also
\cite[Proposition 4.2.8]{Perrin}.
This was a question raised in
\cite[page 80]{Oort}.
We have seen in
Lemma \ref{lemma-group-scheme-characteristic-zero-smooth}
that this holds when the group scheme is locally of finite type.
\end{remark}
\begin{lemma}
\label{lemma-reduced-group-scheme-prefect-field-characteristic-p-smooth}
Let $k$ be a perfect field of characteristic $p > 0$ (see
Lemma \ref{lemma-group-scheme-characteristic-zero-smooth}
for the characteristic zero case).
Let $G$ be a locally algebraic group scheme over $k$.
If $G$ is reduced then the structure
morphism $G \to \Spec(k)$ is smooth, i.e., $G$ is a smooth
group scheme.
\end{lemma}
\begin{proof}
By
Lemma \ref{lemma-group-scheme-module-differentials}
the sheaf $\Omega_{G/k}$ is free. Hence the lemma follows from
Varieties, Lemma \ref{varieties-lemma-char-p-differentials-free-smooth}.
\end{proof}
\begin{remark}
\label{remark-reduced-smooth-not-true-general}
Let $k$ be a field of characteristic $p > 0$.
Let $\alpha \in k$ be an element which is not a $p$th power.
The closed subgroup scheme
$$
G = V(x^p + \alpha y^p) \subset \mathbf{G}_{a, k}^2
$$
is reduced and irreducible but not smooth (not even normal).
\end{remark}
\noindent
The following lemma is a special case of
Lemma \ref{lemma-compact-set-in-affine}
with a somewhat easier proof.
\begin{lemma}
\label{lemma-points-in-affine}
Let $k$ be an algebraically closed field.
Let $G$ be a locally algebraic group scheme over $k$.
Let $g_1, \ldots, g_n \in G(k)$ be $k$-rational points.
Then there exists an affine open $U \subset G$ containing $g_1, \ldots, g_n$.
\end{lemma}
\begin{proof}
We first argue by induction on $n$ that we may assume all $g_i$ are
on the same connected component of $G$. Namely, if not, then we can
find a decomposition $G = W_1 \amalg W_2$ with $W_i$ open in $G$ and
(after possibly renumbering) $g_1, \ldots, g_r \in W_1$ and
$g_{r + 1}, \ldots, g_n \in W_2$ for some $0 < r < n$. By
induction we can find affine opens $U_1$ and $U_2$ of $G$ with
$g_1, \ldots, g_r \in U_1$ and $g_{r + 1}, \ldots, g_n \in U_2$.
Then
$$
g_1, \ldots, g_n \in (U_1 \cap W_1) \cup (U_2 \cap W_2)
$$
is a solution to the problem. Thus we may assume $g_1, \ldots, g_n$
are all on the same connected component of $G$. Translating by $g_1^{-1}$
we may assume $g_1, \ldots, g_n \in G^0$ where $G^0 \subset G$ is as in
Proposition \ref{proposition-connected-component}. Choose an affine
open neighbourhood $U$ of $e$, in particular $U \cap G^0$ is nonempty.
Since $G^0$ is irreducible we see that
$$
G^0 \cap (Ug_1^{-1} \cap \ldots \cap Ug_n^{-1})
$$
is nonempty. Since $G \to \Spec(k)$ is locally of finite type, also
$G^0 \to \Spec(k)$ is locally of finite type, hence any nonempty
open has a $k$-rational point. Thus we can pick $g \in G^0(k)$ with
$g \in Ug_i^{-1}$ for all $i$. Then $g_i \in g^{-1}U$ for all $i$
and $g^{-1}U$ is the affine open we were looking for.
\end{proof}
\begin{lemma}
\label{lemma-algebraic-quasi-projective}
Let $k$ be a field. Let $G$ be an algebraic group scheme over $k$.
Then $G$ is quasi-projective over $k$.
\end{lemma}
\begin{proof}
By Varieties, Lemma \ref{varieties-lemma-ample-after-field-extension}
we may assume that $k$ is algebraically closed. Let $G^0 \subset G$
be the connected component of $G$ as in
Proposition \ref{proposition-connected-component}.
Then every other connected component of $G$ has a $k$-rational
point and hence is isomorphic to $G^0$ as a scheme.
Since $G$ is quasi-compact and Noetherian, there are finitely many of these
connected components. Thus we reduce to the case discussed in
the next paragraph.
\medskip\noindent
Let $G$ be a connected algebraic group scheme over an algebraically closed
field $k$. If the characteristic of $k$ is zero, then $G$ is smooth over
$k$ by Lemma \ref{lemma-group-scheme-characteristic-zero-smooth}.
If the characteristic of $k$ is $p > 0$, then we let $H = G_{red}$
be the reduction of $G$. By
Divisors, Proposition \ref{divisors-proposition-push-down-ample}
it suffices to show that $H$ has an ample invertible sheaf.
(For an algebraic scheme over $k$ having an ample invertible
sheaf is equivalent to being quasi-projective over $k$, see
for example the very general
More on Morphisms, Lemma \ref{more-morphisms-lemma-quasi-projective}.)
By Lemma \ref{lemma-reduced-subgroup-scheme-perfect}
we see that $H$ is a group scheme over $k$.
By Lemma \ref{lemma-reduced-group-scheme-prefect-field-characteristic-p-smooth}
we see that $H$ is smooth over $k$.
This reduces us to the situation discussed in the next
paragraph.
\medskip\noindent
Let $G$ be a quasi-compact irreducible smooth group scheme over an
algebraically closed field $k$. Observe that the local rings of $G$
are regular and hence UFDs
(Varieties, Lemma \ref{varieties-lemma-smooth-regular} and
More on Algebra, Lemma \ref{more-algebra-lemma-regular-local-UFD}).
The complement of a nonempty affine open of $G$
is the support of an effective Cartier divisor $D$.
This follows from Divisors, Lemma
\ref{divisors-lemma-complement-open-affine-effective-cartier-divisor}.
(Observe that $G$ is separated by
Lemma \ref{lemma-group-scheme-over-field-separated}.)
We conclude there exists an effective Cartier divisor $D \subset G$
such that $G \setminus D$ is affine. We will use below that
for any $n \geq 1$ and $g_1, \ldots, g_n \in G(k)$ the complement
$G \setminus \bigcup D g_i$ is affine. Namely, it is the intersection
of the affine opens $G \setminus Dg_i \cong G \setminus D$
in the separated scheme $G$.
\medskip\noindent
We may choose the top row of the diagram
$$
\xymatrix{
G & U \ar[l]_j \ar[r]^\pi & \mathbf{A}^d_k \\
& W \ar[r]^{\pi'} \ar[u] & V \ar[u]
}
$$
such that $U \not = \emptyset$, $j : U \to G$ is an open immersion, and
$\pi$ is \'etale, see
Morphisms, Lemma \ref{morphisms-lemma-smooth-etale-over-affine-space}.
There is a nonempty affine open $V \subset \mathbf{A}^d_k$ such that
with $W = \pi^{-1}(V)$ the morphism $\pi' = \pi|_W : W \to V$ is finite \'etale.
In particular $\pi'$ is finite locally free, say of degree $n$.
Consider the effective Cartier divisor
$$
\mathcal{D} = \{(g, w) \mid m(g, j(w)) \in D\} \subset G \times W
$$
(This is the restriction to $G \times W$ of the pullback of $D \subset G$
under the flat morphism $m : G \times G \to G$.)
Consider the closed subset\footnote{Using the material
in Divisors, Section \ref{divisors-section-norms}
we could take as effective Cartier
divisor $E$ the norm of the effective Cartier divisor $\mathcal{D}$
along the finite locally free morphism $1 \times \pi'$ bypassing
some of the arguments.}
$T = (1 \times \pi')(\mathcal{D}) \subset G \times V$.
Since $\pi'$ is finite locally free, every irreducible component
of $T$ has codimension $1$ in $G \times V$. Since $G \times V$
is smooth over $k$ we conclude these components are effective Cartier
divisors (Divisors, Lemma \ref{divisors-lemma-weil-divisor-is-cartier-UFD}
and lemmas cited above)
and hence $T$ is the support of an effective Cartier divisor
$E$ in $G \times V$. If $v \in V(k)$, then
$(\pi')^{-1}(v) = \{w_1, \ldots, w_n\} \subset W(k)$ and we see that
$$
E_v = \bigcup\nolimits_{i = 1, \ldots, n} D j(w_i)^{-1}
$$
in $G$ set theoretically. In particular we see that $G \setminus E_v$
is affine open (see above).
Moreover, if $g \in G(k)$, then there exists a $v \in V$
such that $g \not \in E_v$. Namely, the set $W'$ of $w \in W$ such that
$g \not \in Dj(w)^{-1}$ is nonempty open and it suffices to pick $v$
such that the fibre of $W' \to V$ over $v$ has $n$ elements.
\medskip\noindent
Consider the invertible sheaf
$\mathcal{M} = \mathcal{O}_{G \times V}(E)$ on $G \times V$.
By Varieties, Lemma \ref{varieties-lemma-rational-equivalence-for-Pic}
the isomorphism class $\mathcal{L}$ of the restriction
$\mathcal{M}_v = \mathcal{O}_G(E_v)$ is independent of $v \in V(k)$.
On the other hand, for every $g \in G(k)$ we can find a $v$
such that $g \not \in E_v$ and such that $G \setminus E_v$
is affine. Thus the canonical section
(Divisors, Definition
\ref{divisors-definition-invertible-sheaf-effective-Cartier-divisor})
of $\mathcal{O}_G(E_v)$
corresponds to a section $s_v$ of $\mathcal{L}$ which does not
vanish at $g$ and such that $G_{s_v}$ is affine.
This means that $\mathcal{L}$ is ample by definition
(Properties, Definition \ref{properties-definition-ample}).
\end{proof}
\begin{lemma}
\label{lemma-algebraic-center}
Let $k$ be a field. Let $G$ be a locally algebraic group scheme over $k$.
Then the center of $G$ is a closed subgroup scheme of $G$.
\end{lemma}
\begin{proof}
Let $\text{Aut}(G)$ denote the contravariant functor on the category of
schemes over $k$ which associates to $S/k$ the set of automorphisms
of the base change $G_S$ as a group scheme over $S$. There is a natural
transformation
$$
G \longrightarrow \text{Aut}(G),\quad
g \longmapsto \text{inn}_g
$$
sending an $S$-valued point $g$ of $G$ to the inner automorphism of $G$
determined by $g$. The center $C$ of $G$ is by definition the kernel of
this transformation, i.e., the functor which to $S$ associates those
$g \in G(S)$ whose associated inner automorphism is trivial. The statement
of the lemma is that this functor is representable by a closed subgroup
scheme of $G$.
\medskip\noindent
Choose an integer $n \geq 1$. Let $G_n \subset G$ be the $n$th infinitesimal
neighbourhood of the identity element $e$ of $G$. For every scheme $S/k$
the base change $G_{n, S}$ is the $n$th infinitesimal neighbourhood of
$e_S : S \to G_S$. Thus we see that there is a natural transformation
$\text{Aut}(G) \to \text{Aut}(G_n)$ where the right hand side is the
functor of automorphisms of $G_n$ as a scheme ($G_n$ isn't in general
a group scheme). Observe that $G_n$ is the spectrum of an artinian
local ring $A_n$ with residue field $k$ which has finite dimension
as a $k$-vector space
(Varieties, Lemma \ref{varieties-lemma-algebraic-scheme-dim-0}).
Since every automorphism of $G_n$ induces in particular an invertible
linear map $A_n \to A_n$, we obtain transformations of functors
$$
G \to \text{Aut}(G) \to \text{Aut}(G_n) \to \text{GL}(A_n)
$$
The final group valued functor is representable, see
Example \ref{example-general-linear-group}, and the
last arrow is visibly injective.
Thus for every $n$ we obtain a closed subgroup scheme
$$
H_n = \Ker(G \to \text{Aut}(G_n)) = \Ker(G \to \text{GL}(A_n)).
$$
As a first approximation we set $H = \bigcap_{n \geq 1} H_n$
(scheme theoretic intersection). This is a closed subgroup scheme
which contains the center $C$.
\medskip\noindent
Let $h$ be an $S$-valued point of $H$ with $S$ locally Noetherian.
Then the automorphism $\text{inn}_h$ induces the identity on all
the closed subschemes $G_{n, S}$. Consider the kernel
$K = \Ker(\text{inn}_h : G_S \to G_S)$.
This is a closed subgroup scheme of $G_S$ over $S$ containing the
closed subschemes $G_{n, S}$ for $n \geq 1$.
This implies that $K$ contains an open neighbourhood of
$e(S) \subset G_S$, see
Algebra, Remark \ref{algebra-remark-intersection-powers-ideal}.
Let $G^0 \subset G$ be as in Proposition \ref{proposition-connected-component}.
Since $G^0$ is geometrically irreducible, we conclude that $K$ contains
$G^0_S$ (for any nonempty open $U \subset G^0_{k'}$ and any field extension
$k'/k$ we have $U \cdot U^{-1} = G^0_{k'}$, see proof of
Lemma \ref{lemma-irreducible-group-scheme-over-field-qc}).
Applying this with $S = H$ we find that $G^0$ and $H$
are subgroup schemes of $G$ whose points commute: for any scheme $S$
and any $S$-valued points $g \in G^0(S)$, $h \in H(S)$ we have
$gh = hg$ in $G(S)$.
\medskip\noindent
Assume that $k$ is algebraically closed. Then we can pick a $k$-valued
point $g_i$ in each irreducible component $G_i$ of $G$. Observe that in
this case the connected components of $G$ are the irreducible components
of $G$ are the translates of $G^0$ by our $g_i$. We claim that
$$
C = H \cap \bigcap\nolimits_i \Ker(\text{inn}_{g_i} : G \to G)
\quad (\text{scheme theoretic intersection})
$$
Namely, $C$ is contained in the right hand side. On the other hand, every
$S$-valued point $h$ of the right hand side commutes with $G^0$
and with $g_i$ hence with everything in $G = \bigcup G^0g_i$.
\medskip\noindent
The case of a general base field $k$ follows from the result for the
algebraic closure $\overline{k}$ by descent. Namely, let
$A \subset G_{\overline{k}}$ the closed subgroup scheme representing
the center of $G_{\overline{k}}$. Then we have
$$
A \times_{\Spec(k)} \Spec(\overline{k}) =
\Spec(\overline{k}) \times_{\Spec(k)} A
$$
as closed subschemes of $G_{\overline{k} \otimes_k \overline{k}}$ by
the functorial nature of the center. Hence we see that $A$ descends
to a closed subgroup scheme $Z \subset G$ by
Descent, Lemma \ref{descent-lemma-closed-immersion}
(and Descent, Lemma \ref{descent-lemma-descending-property-closed-immersion}).
Then $Z$ represents $C$ (small argument omitted) and the proof is complete.
\end{proof}
\section{Abelian varieties}
\label{section-abelian-varieties}
\noindent
An excellent reference for this material is Mumford's book on
abelian varieties, see \cite{AVar}. We encourage the reader to
look there. There are many equivalent definitions; here is one.
\begin{definition}
\label{definition-abelian-variety}
Let $k$ be a field. An {\it abelian variety} is a group scheme over
$k$ which is also a proper, geometrically integral variety over $k$.
\end{definition}
\noindent
We prove a few lemmas about this notion and then we collect
all the results together in
Proposition \ref{proposition-review-abelian-varieties}.
\begin{lemma}
\label{lemma-abelian-variety-projective}
Let $k$ be a field. Let $A$ be an abelian variety over $k$.
Then $A$ is projective.
\end{lemma}
\begin{proof}
This follows from
Lemma \ref{lemma-algebraic-quasi-projective} and
More on Morphisms, Lemma \ref{more-morphisms-lemma-projective}.
\end{proof}
\begin{lemma}
\label{lemma-abelian-variety-change-field}
Let $k$ be a field. Let $A$ be an abelian variety over $k$.
For any field extension $K/k$ the base change $A_K$ is an
abelian variety over $K$.
\end{lemma}
\begin{proof}
Omitted. Note that this is why we insisted on $A$ being
geometrically integral; without that condition this lemma
(and many others below) would be wrong.
\end{proof}
\begin{lemma}
\label{lemma-abelian-variety-smooth}
Let $k$ be a field. Let $A$ be an abelian variety over $k$.
Then $A$ is smooth over $k$.
\end{lemma}
\begin{proof}
If $k$ is perfect then this follows from
Lemma \ref{lemma-group-scheme-characteristic-zero-smooth}
(characteristic zero) and
Lemma \ref{lemma-reduced-group-scheme-prefect-field-characteristic-p-smooth}
(positive characteristic).
We can reduce the general case to this case by descent for smoothness
(Descent, Lemma \ref{descent-lemma-descending-property-smooth})
and going to the perfect closure using
Lemma \ref{lemma-abelian-variety-change-field}.
\end{proof}
\begin{lemma}
\label{lemma-abelian-variety-abelian}
An abelian variety is an abelian group scheme, i.e., the group
law is commutative.
\end{lemma}
\begin{proof}
Let $k$ be a field. Let $A$ be an abelian variety over $k$.
By Lemma \ref{lemma-abelian-variety-change-field} we may replace
$k$ by its algebraic closure. Consider the morphism
$$
h : A \times_k A \longrightarrow A \times_k A,\quad
(x, y) \longmapsto (x, xyx^{-1}y^{-1})
$$
This is a morphism over $A$ via the first projection on either side.
Let $e \in A(k)$ be the unit. Then we see that $h|_{e \times A}$ is
constant with value $(e, e)$. By More on Morphisms, Lemma
\ref{more-morphisms-lemma-flat-proper-family-cannot-collapse-fibre}
there exists an open neighbourhood $U \subset A$ of $e$
such that $h|_{U \times A}$ factors through some $Z \subset U \times A$
finite over $U$. This means that for $x \in U(k)$ the morphism
$A \to A$, $y \mapsto xyx^{-1}y^{-1}$ takes finitely many values.
Of course this means it is constant with value $e$. Thus
$(x, y) \mapsto xyx^{-1}y^{-1}$ is
constant with value $e$ on $U \times A$ which implies
that the group law on $A$ is abelian.
\end{proof}
\begin{lemma}
\label{lemma-apply-cube}
Let $k$ be a field. Let $A$ be an abelian variety over $k$.
Let $\mathcal{L}$ be an invertible $\mathcal{O}_A$-module.
Then there is an isomorphism
$$
m_{1, 2, 3}^*\mathcal{L} \otimes
m_1^*\mathcal{L} \otimes
m_2^*\mathcal{L} \otimes
m_3^*\mathcal{L} \cong
m_{1, 2}^*\mathcal{L} \otimes
m_{1, 3}^*\mathcal{L} \otimes
m_{2, 3}^*\mathcal{L}
$$
of invertible modules on $A \times_k A \times_k A$
where $m_{i_1, \ldots, i_t} : A \times_k A \times_k A \to A$
is the morphism $(x_1, x_2, x_3) \mapsto \sum x_{i_j}$.
\end{lemma}
\begin{proof}
Apply the theorem of the cube
(More on Morphisms, Theorem \ref{more-morphisms-theorem-of-the-cube})
to the difference
$$
\mathcal{M} =
m_{1, 2, 3}^*\mathcal{L} \otimes
m_1^*\mathcal{L} \otimes
m_2^*\mathcal{L} \otimes
m_3^*\mathcal{L} \otimes
m_{1, 2}^*\mathcal{L}^{\otimes -1} \otimes
m_{1, 3}^*\mathcal{L}^{\otimes -1} \otimes
m_{2, 3}^*\mathcal{L}^{\otimes -1}
$$
This works because the restriction of $\mathcal{M}$
to $A \times A \times e = A \times A$ is equal to
$$
n_{1, 2}^*\mathcal{L} \otimes
n_1^*\mathcal{L} \otimes
n_2^*\mathcal{L} \otimes
n_{1, 2}^*\mathcal{L}^{\otimes -1} \otimes
n_1^*\mathcal{L}^{\otimes -1} \otimes
n_2^*\mathcal{L}^{\otimes -1} \cong \mathcal{O}_{A \times_k A}
$$
where $n_{i_1, \ldots, i_t} : A \times_k A \to A$
is the morphism $(x_1, x_2) \mapsto \sum x_{i_j}$.
Similarly for $A \times e \times A$ and $e \times A \times A$.
\end{proof}
\begin{lemma}
\label{lemma-pullbacks-by-n}
Let $k$ be a field. Let $A$ be an abelian variety over $k$.
Let $\mathcal{L}$ be an invertible $\mathcal{O}_A$-module.
Then
$$
[n]^*\mathcal{L} \cong
\mathcal{L}^{\otimes n(n + 1)/2} \otimes
([-1]^*\mathcal{L})^{\otimes n(n - 1)/2}
$$
where $[n] : A \to A$ sends $x$ to $x + x + \ldots + x$ with $n$ summands
and where $[-1] : A \to A$ is the inverse of $A$.
\end{lemma}
\begin{proof}
Consider the morphism $A \to A \times_k A \times_k A$,
$x \mapsto (x, x, -x)$ where $-x = [-1](x)$. Pulling back
the relation of Lemma \ref{lemma-apply-cube} we obtain
$$
\mathcal{L} \otimes
\mathcal{L} \otimes
\mathcal{L} \otimes
[-1]^*\mathcal{L} \cong
[2]^*\mathcal{L}
$$
which proves the result for $n = 2$. By induction assume the result holds
for $1, 2, \ldots, n$. Then consider the morphism
$A \to A \times_k A \times_k A$, $x \mapsto (x, x, [n - 1]x)$.
Pulling back
the relation of Lemma \ref{lemma-apply-cube} we obtain
$$
[n + 1]^*\mathcal{L} \otimes
\mathcal{L} \otimes
\mathcal{L} \otimes
[n - 1]^*\mathcal{L} \cong
[2]^*\mathcal{L} \otimes
[n]^*\mathcal{L} \otimes
[n]^*\mathcal{L}
$$
and the result follows by elementary arithmetic.
\end{proof}
\begin{lemma}
\label{lemma-degree-multiplication-by-d}
Let $k$ be a field. Let $A$ be an abelian variety over $k$.
Let $[d] : A \to A$ be the multiplication by $d$.
Then $[d]$ is finite locally free of degree $d^{2\dim(A)}$.
\end{lemma}
\begin{proof}
By Lemma \ref{lemma-abelian-variety-projective}
(and More on Morphisms, Lemma \ref{more-morphisms-lemma-projective})
we see that $A$ has an ample invertible module $\mathcal{L}$.
Since $[-1] : A \to A$ is an automorphism, we see that
$[-1]^*\mathcal{L}$ is an ample invertible $\mathcal{O}_X$-module
as well. Thus $\mathcal{N} = \mathcal{L} \otimes [-1]^*\mathcal{L}$
is ample, see
Properties, Lemma \ref{properties-lemma-ample-tensor-globally-generated}.
Since $\mathcal{N} \cong [-1]^*\mathcal{N}$ we see that
$[d]^*\mathcal{N} \cong \mathcal{N}^{\otimes n^2}$ by
Lemma \ref{lemma-pullbacks-by-n}.
\medskip\noindent
To get a contradiction $C \subset X$ be a proper curve contained in a
fibre of $[d]$. Then $\mathcal{N}^{\otimes d^2}|_C \cong \mathcal{O}_C$
is an ample invertible $\mathcal{O}_C$-module of degree $0$ which
contradicts Varieties, Lemma \ref{varieties-lemma-ample-curve} for example.
(You can also use Varieties, Lemma \ref{varieties-lemma-ample-positive}.)
Thus every fibre of $[d]$ has dimension $0$ and hence $[d]$ is finite
for example by Cohomology of Schemes, Lemma
\ref{coherent-lemma-characterize-finite}.
Moreover, since $A$ is smooth over $k$ by
Lemma \ref{lemma-abelian-variety-smooth}
we see that $[d] : A \to A$ is flat by
Algebra, Lemma \ref{algebra-lemma-CM-over-regular-flat}
(we also use that schemes smooth over fields are regular and that
regular rings are Cohen-Macaulay, see
Varieties, Lemma \ref{varieties-lemma-smooth-regular} and
Algebra, Lemma \ref{algebra-lemma-regular-ring-CM}).
Thus $[d]$ is finite flat hence finite locally free by
Morphisms, Lemma \ref{morphisms-lemma-finite-flat}.
\medskip\noindent
Finally, we come to the formula for the degree. By
Varieties, Lemma \ref{varieties-lemma-degree-finite-morphism-in-terms-degrees}
we see that
$$
\deg_{\mathcal{N}^{\otimes d^2}}(A) = \deg([d]) \deg_\mathcal{N}(A)
$$
Since the degree of $A$ with respect to
$\mathcal{N}^{\otimes d^2}$, respectively $\mathcal{N}$
is the coefficient of $n^{\dim(A)}$ in the polynomial
$$
n \longmapsto \chi(A, \mathcal{N}^{\otimes nd^2}),\quad
\text{respectively}\quad n \longmapsto \chi(A, \mathcal{N}^{\otimes n})
$$
we see that $\deg([d]) = d^{2 \dim(A)}$.
\end{proof}
\begin{lemma}
\label{lemma-abelian-variety-multiplication-by-d-etale}
Let $k$ be a field. Let $A$ be an abelian variety over $k$.
Then $[d] : A \to A$ is \'etale if and only if $d$ is invertible in $k$.
\end{lemma}
\begin{proof}
Observe that $[d](x + y) = [d](x) + [d](y)$. Since translation by a
point is an automorphism of $A$, we see that the set of points where
$[d] : A \to A$ is \'etale is either empty or equal to $A$ (some details
omitted). Thus it suffices to check whether $[d]$ is \'etale at
the unit $e \in A(k)$. Since we know that $[d]$ is finite locally free
(Lemma \ref{lemma-degree-multiplication-by-d})
to see that it is \'etale at $e$ is equivalent to
proving that $\text{d}[d] : T_{A/k, e} \to T_{A/k, e}$ is injective. See
Varieties, Lemma \ref{varieties-lemma-injective-tangent-spaces-unramified} and
Morphisms, Lemma \ref{morphisms-lemma-flat-unramified-etale}.
By Lemma \ref{lemma-group-scheme-addition-tangent-vectors} we see that
$\text{d}[d]$ is given by multiplication by $d$ on $T_{A/k, e}$.
\end{proof}
\begin{lemma}
\label{lemma-abelian-variety-multiplication-by-p}
Let $k$ be a field of characteristic $p > 0$. Let $A$ be an abelian variety
over $k$. The fibre of $[p] : A \to A$ over $0$ has at most
$p^g$ distinct points.
\end{lemma}
\begin{proof}
To prove this, we may and do replace $k$ by the algebraic closure.
By Lemma \ref{lemma-group-scheme-addition-tangent-vectors}
the derivative of $[p]$ is multiplication by $p$ as a map
$T_{A/k, e} \to T_{A/k, e}$ and hence is zero (compare
with proof of Lemma \ref{lemma-abelian-variety-multiplication-by-d-etale}).
Since $[p]$ commutes with translation we conclude that the derivative of $[p]$
is everywhere zero, i.e., that the induced map
$[p]^*\Omega_{A/k} \to \Omega_{A/k}$ is zero.
Looking at generic points, we find that
the corresponding map $[p]^* : k(A) \to k(A)$
of function fields induces the zero map on $\Omega_{k(A)/k}$.
Let $t_1, \ldots, t_g$ be a p-basis of $k(A)$ over $k$
(More on Algebra, Definition \ref{more-algebra-definition-p-basis} and
Lemma \ref{more-algebra-lemma-p-basis}). Then $[p]^*(t_i)$
has a $p$th root by
Algebra, Lemma \ref{algebra-lemma-derivative-zero-pth-power}.
We conclude that
$k(A)[x_1, \ldots, x_g]/(x_1^p - t_1, \ldots, x_g^p - t_g)$ is a subextension
of $[p]^* : k(A) \to k(A)$.
Thus we can find an affine open $U \subset A$ such that
$t_i \in \mathcal{O}_A(U)$ and $x_i \in \mathcal{O}_A([p]^{-1}(U))$.
We obtain a factorization
$$
[p]^{-1}(U)
\xrightarrow{\pi_1}
\Spec(\mathcal{O}(U)[x_1, \ldots, x_g]/(x_1^p - t_1, \ldots, x_g^p - t_g))
\xrightarrow{\pi_2}
U
$$
of $[p]$ over $U$. After shrinking $U$ we may assume that $\pi_1$
is finite locally free (for example by generic flatness -- actually it is
already finite locally free in our case).
By Lemma \ref{lemma-degree-multiplication-by-d} we see that
$[p]$ has degree $p^{2g}$. Since $\pi_2$
has degree $p^g$ we see that $\pi_1$ has degree $p^g$ as well.
The morphism $\pi_2$ is a universal homeomorphism hence the fibres are
singletons. We conclude that the (set theoretic) fibres of $[p]^{-1}(U) \to U$
are the fibres of $\pi_1$. Hence they
have at most $p^g$ elements. Since $[p]$ is a homomorphism of group
schemes over $k$, the fibre of $[p] : A(k) \to A(k)$ has the
same cardinality for every $a \in A(k)$ and the proof is complete.
\end{proof}
\begin{proposition}
\label{proposition-review-abelian-varieties}
\begin{reference}
Wonderfully explained in \cite{AVar}.
\end{reference}
Let $A$ be an abelian variety over a field $k$. Then
\begin{enumerate}
\item $A$ is projective over $k$,
\item $A$ is a commutative group scheme,
\item the morphism $[n] : A \to A$ is surjective for all $n \geq 1$,
\item if $k$ is algebraically closed, then $A(k)$ is a divisible abelian group,
\item $A[n] = \Ker([n] : A \to A)$ is a finite group scheme of degree
$n^{2\dim A}$ over $k$,
\item $A[n]$ is \'etale over $k$ if and only if $n \in k^*$,
\item if $n \in k^*$ and $k$ is algebraically closed,
then $A(k)[n] \cong (\mathbf{Z}/n\mathbf{Z})^{\oplus 2\dim(A)}$,
\item if $k$ is algebraically closed of characteristic $p > 0$, then
there exists an integer $0 \leq f \leq \dim(A)$ such that
$A(k)[p^m] \cong (\mathbf{Z}/p^m\mathbf{Z})^{\oplus f}$
for all $m \geq 1$.
\end{enumerate}
\end{proposition}
\begin{proof}
Part (1) follows from Lemma \ref{lemma-abelian-variety-projective}.
Part (2) follows from Lemma \ref{lemma-abelian-variety-abelian}.
Part (3) follows from Lemma \ref{lemma-degree-multiplication-by-d}.
If $k$ is algebraically closed then surjective morphisms of varieties
over $k$ induce surjective maps on $k$-rational points, hence
(4) follows from (3).
Part (5) follows from Lemma \ref{lemma-degree-multiplication-by-d}
and the fact that a base change of a finite locally free morphism
of degree $N$ is a finite locally free morphism of degree $N$.
Part (6) follows from
Lemma \ref{lemma-abelian-variety-multiplication-by-d-etale}.
Namely, if $n$ is invertible in $k$, then $[n]$ is \'etale
and hence $A[n]$ is \'etale over $k$.
On the other hand, if $n$ is not invertible in $k$, then
$[n]$ is not \'etale at $e$ and it follows that $A[n]$
is not \'etale over $k$ at $e$ (use
Morphisms, Lemmas \ref{morphisms-lemma-flat-unramified-etale} and
\ref{morphisms-lemma-set-points-where-fibres-unramified}).
\medskip\noindent
Assume $k$ is algebraically closed. Set $g = \dim(A)$. Proof of (7).
Let $\ell$ be a prime number which is invertible in $k$. Then we see that
$$
A[\ell](k) = A(k)[\ell]
$$
is a finite abelian group, annihilated by $\ell$, of order $\ell^{2g}$.
It follows that it is isomorphic to $(\mathbf{Z}/\ell\mathbf{Z})^{2g}$
by the structure theory for finite abelian groups. Next, we consider
the short exact sequence
$$
0 \to A(k)[\ell] \to A(k)[\ell^2] \xrightarrow{\ell} A(k)[\ell] \to 0
$$
Arguing similarly as above we conclude that
$A(k)[\ell^2] \cong (\mathbf{Z}/\ell^2\mathbf{Z})^{2g}$.
By induction on the exponent we find that
$A(k)[\ell^m] \cong (\mathbf{Z}/\ell^m\mathbf{Z})^{2g}$.
For composite integers $n$ prime to the characteristic of $k$
we take primary parts and we find the correct shape of the
$n$-torsion in $A(k)$.
The proof of (8) proceeds in exactly the same way, using that
Lemma \ref{lemma-abelian-variety-multiplication-by-p} gives
$A(k)[p] \cong (\mathbf{Z}/p\mathbf{Z})^{\oplus f}$ for some $0 \leq f \leq g$.
\end{proof}
\section{Actions of group schemes}
\label{section-action-group-scheme}
\noindent
Let $(G, m)$ be a group and let $V$ be a set.
Recall that a {\it (left) action} of $G$ on $V$ is given
by a map $a : G \times V \to V$ such that
\begin{enumerate}
\item (associativity) $a(m(g, g'), v) = a(g, a(g', v))$ for all
$g, g' \in G$ and $v \in V$, and
\item (identity) $a(e, v) = v$ for all $v \in V$.
\end{enumerate}
We also say that $V$ is a {\it $G$-set} (this usually means we
drop the $a$ from the notation -- which is abuse of notation).
A {\it map of $G$-sets} $\psi : V \to V'$ is any set map
such that $\psi(a(g, v)) = a(g, \psi(v))$ for all $v \in V$.
\begin{definition}
\label{definition-action-group-scheme}
Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$.
\begin{enumerate}
\item An {\it action of $G$ on the scheme $X/S$} is
a morphism $a : G \times_S X \to X$ over $S$ such that
for every $T/S$ the map $a : G(T) \times X(T) \to X(T)$
defines the structure of a $G(T)$-set on $X(T)$.
\item Suppose that $X$, $Y$ are schemes over $S$ each endowed
with an action of $G$. An {\it equivariant} or more precisely
a {\it $G$-equivariant} morphism $\psi : X \to Y$
is a morphism of schemes over $S$ such
that for every $T/S$ the map $\psi : X(T) \to Y(T)$ is
a morphism of $G(T)$-sets.
\end{enumerate}
\end{definition}
\noindent
In situation (1) this means that the diagrams
\begin{equation}
\label{equation-action}
\xymatrix{
G \times_S G \times_S X \ar[r]_-{1_G \times a} \ar[d]_{m \times 1_X} &
G \times_S X \ar[d]^a \\
G \times_S X \ar[r]^a & X
}
\quad
\xymatrix{
G \times_S X \ar[r]_-a & X \\
X\ar[u]^{e \times 1_X} \ar[ru]_{1_X}
}
\end{equation}
are commutative. In situation (2) this just means that the diagram
$$
\xymatrix{
G \times_S X \ar[r]_-{\text{id} \times f} \ar[d]_a &
G \times_S Y \ar[d]^a \\
X \ar[r]^f & Y
}
$$
commutes.
\begin{definition}
\label{definition-free-action}
Let $S$, $G \to S$, and $X \to S$ as in
Definition \ref{definition-action-group-scheme}.
Let $a : G \times_S X \to X$ be an action of $G$ on $X/S$.
We say the action is {\it free} if for every scheme $T$ over $S$
the action $a : G(T) \times X(T) \to X(T)$ is a free action of
the group $G(T)$ on the set $X(T)$.
\end{definition}
\begin{lemma}
\label{lemma-free-action}
Situation as in Definition \ref{definition-free-action},
The action $a$ is free if and only if
$$
G \times_S X \to X \times_S X, \quad (g, x) \mapsto (a(g, x), x)
$$
is a monomorphism.
\end{lemma}
\begin{proof}
Immediate from the definitions.
\end{proof}
\section{Principal homogeneous spaces}
\label{section-principal-homogeneous}
\noindent
In
Cohomology on Sites, Definition \ref{sites-cohomology-definition-torsor}
we have defined a torsor for a sheaf of groups on a site.
Suppose $\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}$ is a
topology and $(G, m)$ is a group scheme over $S$. Since $\tau$ is stronger than
the canonical topology (see
Descent, Lemma \ref{descent-lemma-fpqc-universal-effective-epimorphisms})
we see that $\underline{G}$ (see
Sites, Definition \ref{sites-definition-representable-sheaf})
is a sheaf of groups on $(\Sch/S)_\tau$.
Hence we already know what it means to have a
torsor for $\underline{G}$ on $(\Sch/S)_\tau$. A special situation
arises if this sheaf is representable. In the following definitions
we define directly what it means for the representing scheme to be a
$G$-torsor.
\begin{definition}
\label{definition-pseudo-torsor}
Let $S$ be a scheme.
Let $(G, m)$ be a group scheme over $S$.
Let $X$ be a scheme over $S$, and let
$a : G \times_S X \to X$ be an action of $G$ on $X$.
\begin{enumerate}
\item We say $X$ is a {\it pseudo $G$-torsor} or that $X$ is
{\it formally principally homogeneous under $G$} if the induced
morphism of schemes $G \times_S X \to X \times_S X$,
$(g, x) \mapsto (a(g, x), x)$ is an isomorphism of schemes over $S$.
\item A pseudo $G$-torsor $X$ is called {\it trivial} if there exists
an $G$-equivariant isomorphism $G \to X$ over $S$ where $G$ acts on
$G$ by left multiplication.
\end{enumerate}
\end{definition}
\noindent
It is clear that if $S' \to S$ is a morphism of schemes then
the pullback $X_{S'}$ of a pseudo $G$-torsor over $S$ is a
pseudo $G_{S'}$-torsor over $S'$.
\begin{lemma}
\label{lemma-characterize-trivial-pseudo-torsors}
In the situation of
Definition \ref{definition-pseudo-torsor}.
\begin{enumerate}
\item The scheme $X$ is a pseudo $G$-torsor if and only if for every scheme
$T$ over $S$ the set $X(T)$ is either empty or the action of the group $G(T)$
on $X(T)$ is simply transitive.
\item A pseudo $G$-torsor $X$ is trivial if and only if the morphism
$X \to S$ has a section.
\end{enumerate}
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{definition}
\label{definition-principal-homogeneous-space}
Let $S$ be a scheme.
Let $(G, m)$ be a group scheme over $S$.
Let $X$ be a pseudo $G$-torsor over $S$.
\begin{enumerate}
\item We say $X$ is a {\it principal homogeneous space}
or a {\it $G$-torsor} if there exists a fpqc covering\footnote{This means
that the default type of torsor is a pseudo torsor which is trivial on an
fpqc covering. This is the definition in \cite[Expos\'e IV, 6.5]{SGA3}.
It is a little bit inconvenient for us as we most often work in the fppf
topology.}
$\{S_i \to S\}_{i \in I}$ such that each
$X_{S_i} \to S_i$ has a section (i.e., is a trivial pseudo $G_{S_i}$-torsor).
\item Let $\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}$.
We say $X$ is a {\it $G$-torsor in the $\tau$ topology}, or a
{\it $\tau$ $G$-torsor}, or simply a {\it $\tau$ torsor}
if there exists a $\tau$ covering $\{S_i \to S\}_{i \in I}$
such that each $X_{S_i} \to S_i$ has a section.
\item If $X$ is a $G$-torsor, then we say that it is
{\it quasi-isotrivial} if it is a torsor for the \'etale topology.
\item If $X$ is a $G$-torsor, then we say that it is
{\it locally trivial} if it is a torsor for the Zariski topology.
\end{enumerate}
\end{definition}
\noindent
We sometimes say ``let $X$ be a $G$-torsor over $S$'' to indicate that
$X$ is a scheme over $S$ equipped with an action of $G$ which turns it
into a principal homogeneous space over $S$.
Next we show that this agrees with the notation introduced earlier
when both apply.
\begin{lemma}
\label{lemma-torsor}
Let $S$ be a scheme.
Let $(G, m)$ be a group scheme over $S$.
Let $X$ be a scheme over $S$, and let
$a : G \times_S X \to X$ be an action of $G$ on $X$.
Let $\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}$.
Then $X$ is a $G$-torsor in the $\tau$-topology if and only if
$\underline{X}$ is a $\underline{G}$-torsor on $(\Sch/S)_\tau$.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{remark}
\label{remark-fun-with-torsors}
Let $(G, m)$ be a group scheme over the scheme $S$.
In this situation we have the following natural types of questions:
\begin{enumerate}
\item If $X \to S$ is a pseudo $G$-torsor and $X \to S$ is surjective,
then is $X$ necessarily a $G$-torsor?
\item Is every $\underline{G}$-torsor on $(\Sch/S)_{fppf}$
representable? In other words, does every $\underline{G}$-torsor
come from a fppf $G$-torsor?
\item Is every $G$-torsor an
fppf (resp.\ smooth, resp.\ \'etale, resp.\ Zariski) torsor?
\end{enumerate}
In general the answers to these questions is no. To get a positive answer
we need to impose additional conditions on $G \to S$.
For example:
If $S$ is the spectrum of a field, then the answer to (1) is yes
because then $\{X \to S\}$ is a fpqc covering trivializing $X$.
If $G \to S$ is affine, then the answer to (2) is yes
(insert future reference here).
If $G = \text{GL}_{n, S}$ then the answer to (3) is yes
and in fact any $\text{GL}_{n, S}$-torsor is locally trivial
(insert future reference here).
\end{remark}
\section{Equivariant quasi-coherent sheaves}
\label{section-equivariant}
\noindent
We think of ``functions'' as dual to ``space''. Thus for a morphism of spaces
the map on functions goes the other way. Moreover, we think of the
sections of a sheaf of modules as ``functions''. This leads us naturally
to the direction of the arrows chosen in the following definition.
\begin{definition}
\label{definition-equivariant-module}
Let $S$ be a scheme, let $(G, m)$ be a group scheme over $S$, and
let $a : G \times_S X \to X$ be an action of the group scheme $G$
on $X/S$. An {\it $G$-equivariant quasi-coherent $\mathcal{O}_X$-module},
or simply a {\it equivariant quasi-coherent $\mathcal{O}_X$-module},
is a pair $(\mathcal{F}, \alpha)$, where $\mathcal{F}$ is a quasi-coherent
$\mathcal{O}_X$-module, and $\alpha$ is a $\mathcal{O}_{G \times_S X}$-module
map
$$
\alpha : a^*\mathcal{F} \longrightarrow \text{pr}_1^*\mathcal{F}
$$
where $\text{pr}_1 : G \times_S X \to X$ is the projection
such that
\begin{enumerate}
\item the diagram
$$
\xymatrix{
(1_G \times a)^*\text{pr}_1^*\mathcal{F} \ar[r]_-{\text{pr}_{12}^*\alpha} &
\text{pr}_2^*\mathcal{F} \\
(1_G \times a)^*a^*\mathcal{F} \ar[u]^{(1_G \times a)^*\alpha} \ar@{=}[r] &
(m \times 1_X)^*a^*\mathcal{F} \ar[u]_{(m \times 1_X)^*\alpha}
}
$$
is a commutative in the category of
$\mathcal{O}_{G \times_S G \times_S X}$-modules, and
\item the pullback
$$
(e \times 1_X)^*\alpha : \mathcal{F} \longrightarrow \mathcal{F}
$$
is the identity map.
\end{enumerate}
For explanation compare with the relevant diagrams of
Equation (\ref{equation-action}).
\end{definition}
\noindent
Note that the commutativity of the first diagram guarantees that
$(e \times 1_X)^*\alpha$ is an idempotent operator on $\mathcal{F}$,
and hence condition (2) is just the condition that it is an isomorphism.
\begin{lemma}
\label{lemma-pullback-equivariant}
Let $S$ be a scheme. Let $G$ be a group scheme over $S$.
Let $f : X \to Y$ be a $G$-equivariant morphism between $S$-schemes
endowed with $G$-actions. Then pullback $f^*$ given by
$(\mathcal{F}, \alpha) \mapsto (f^*\mathcal{F}, (1_G \times f)^*\alpha)$
defines a functor from the category of $G$-equivariant sheaves on
$X$ to the category of quasi-coherent $G$-equivariant sheaves on $Y$.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\section{Groupoids}
\label{section-groupoids}
\noindent
Recall that a groupoid is a category in which every morphism
is an isomorphism, see
Categories, Definition \ref{categories-definition-groupoid}.
Hence a groupoid has a set of objects $\text{Ob}$,
a set of arrows $\text{Arrows}$, a {\it source} and {\it target}
map $s, t : \text{Arrows} \to \text{Ob}$, and a {\it composition law}
$c : \text{Arrows} \times_{s, \text{Ob}, t} \text{Arrows}
\to \text{Arrows}$.
These maps satisfy exactly the following axioms
\begin{enumerate}
\item (associativity) $c \circ (1, c) = c \circ (c, 1)$ as maps
$\text{Arrows} \times_{s, \text{Ob}, t}
\text{Arrows} \times_{s, \text{Ob}, t}
\text{Arrows} \to \text{Arrows}$,
\item (identity) there exists a map $e : \text{Ob} \to \text{Arrows}$
such that
\begin{enumerate}
\item $s \circ e = t \circ e = \text{id}$ as maps $\text{Ob} \to \text{Ob}$,
\item $c \circ (1, e \circ s) = c \circ (e \circ t, 1) = 1$
as maps $\text{Arrows} \to \text{Arrows}$,
\end{enumerate}
\item (inverse) there exists a map $i : \text{Arrows} \to \text{Arrows}$
such that
\begin{enumerate}
\item $s \circ i = t$, $t \circ i = s$ as maps $\text{Arrows} \to \text{Ob}$,
and
\item $c \circ (1, i) = e \circ t$ and $c \circ (i, 1) = e \circ s$
as maps $\text{Arrows} \to \text{Arrows}$.
\end{enumerate}
\end{enumerate}
If this is the case the maps $e$ and $i$ are uniquely determined and
$i$ is a bijection. Note that if $(\text{Ob}', \text{Arrows}', s', t', c')$
is a second groupoid category, then a functor
$f : (\text{Ob}, \text{Arrows}, s, t, c) \to
(\text{Ob}', \text{Arrows}', s', t', c')$
is given by a pair of set maps $f : \text{Ob} \to \text{Ob}'$ and
$f : \text{Arrows} \to \text{Arrows}'$ such that
$s' \circ f = f \circ s$, $t' \circ f = f \circ t$, and
$c' \circ (f, f) = f \circ c$. The compatibility with identity and
inverse is automatic. We will use this below.
(Warning: The compatibility with identity
has to be imposed in the case of general categories.)
\begin{definition}
\label{definition-groupoid}
Let $S$ be a scheme.
\begin{enumerate}
\item A {\it groupoid scheme over $S$}, or simply a
{\it groupoid over $S$} is a
quintuple $(U, R, s, t, c)$ where
$U$ and $R$ are schemes over $S$, and
$s, t : R \to U$ and $c : R \times_{s, U, t} R \to R$
are morphisms of schemes over $S$ with the
following property: For any scheme
$T$ over $S$ the quintuple
$$
(U(T), R(T), s, t, c)
$$
is a groupoid category in the sense described above.
\item A {\it morphism
$f : (U, R, s, t, c) \to (U', R', s', t', c')$
of groupoid schemes over $S$} is given by morphisms
of schemes $f : U \to U'$ and $f : R \to R'$ with the
following property: For any scheme
$T$ over $S$ the maps $f$ define a functor from the
groupoid category $(U(T), R(T), s, t, c)$ to the
groupoid category $(U'(T), R'(T), s', t', c')$.
\end{enumerate}
\end{definition}
\noindent
Let $(U, R, s, t, c)$ be a groupoid over $S$.
Note that, by the remarks preceding the definition and the Yoneda lemma,
there are unique morphisms of schemes
$e : U \to R$ and
$i : R \to R$ over $S$ such that for every scheme $T$ over $S$
the induced map $e : U(T) \to R(T)$ is the identity, and
$i : R(T) \to R(T)$ is the inverse of
the groupoid category. The septuple $(U, R, s, t, c, e, i)$
satisfies commutative diagrams corresponding to each of the
axioms (1), (2)(a), (2)(b), (3)(a) and (3)(b) above, and conversely
given a septuple with this property the quintuple $(U, R, s, t, c)$
is a groupoid scheme. Note that $i$ is an isomorphism,
and $e$ is a section of both $s$ and $t$.
Moreover, given a groupoid scheme over $S$ we denote
$$
j = (t, s) : R \longrightarrow U \times_S U
$$
which is compatible with our conventions in Section
\ref{section-equivalence-relations} above.
We sometimes say ``let $(U, R, s, t, c, e, i)$ be a
groupoid over $S$'' to stress the existence of identity and
inverse.
\begin{lemma}
\label{lemma-groupoid-pre-equivalence}
Given a groupoid scheme $(U, R, s, t, c)$ over $S$
the morphism $j : R \to U \times_S U$ is a pre-equivalence
relation.
\end{lemma}
\begin{proof}
Omitted.
This is a nice exercise in the definitions.
\end{proof}
\begin{lemma}
\label{lemma-equivalence-groupoid}
Given an equivalence relation $j : R \to U$ over $S$
there is a unique way to extend it to a groupoid
$(U, R, s, t, c)$ over $S$.
\end{lemma}
\begin{proof}
Omitted.
This is a nice exercise in the definitions.
\end{proof}
\begin{lemma}
\label{lemma-diagram}
Let $S$ be a scheme.
Let $(U, R, s, t, c)$ be a groupoid over $S$.
In the commutative diagram
$$
\xymatrix{
& U & \\
R \ar[d]_s \ar[ru]^t &
R \times_{s, U, t} R
\ar[l]^-{\text{pr}_0} \ar[d]^{\text{pr}_1} \ar[r]_-c &
R \ar[d]^s \ar[lu]_t \\
U & R \ar[l]_t \ar[r]^s & U
}
$$
the two lower squares are fibre product squares.
Moreover, the triangle on top (which is really a square)
is also cartesian.
\end{lemma}
\begin{proof}
Omitted.
Exercise in the definitions and the functorial point of
view in algebraic geometry.
\end{proof}
\begin{lemma}
\label{lemma-diagram-pull}
Let $S$ be a scheme.
Let $(U, R, s, t, c, e, i)$ be a groupoid over $S$.
The diagram
\begin{equation}
\label{equation-pull}
\xymatrix{
R \times_{t, U, t} R
\ar@<1ex>[r]^-{\text{pr}_1} \ar@<-1ex>[r]_-{\text{pr}_0}
\ar[d]_{(\text{pr}_0, c \circ (i, 1))} &
R \ar[r]^t \ar[d]^{\text{id}_R} &
U \ar[d]^{\text{id}_U} \\
R \times_{s, U, t} R
\ar@<1ex>[r]^-c \ar@<-1ex>[r]_-{\text{pr}_0} \ar[d]_{\text{pr}_1} &
R \ar[r]^t \ar[d]^s &
U \\
R \ar@<1ex>[r]^s \ar@<-1ex>[r]_t &
U
}
\end{equation}
is commutative. The two top rows are isomorphic via the vertical maps given.
The two lower left squares are cartesian.
\end{lemma}
\begin{proof}
The commutativity of the diagram follows from the axioms of a groupoid.
Note that, in terms of groupoids, the top left vertical arrow assigns to
a pair of morphisms $(\alpha, \beta)$ with the same target, the pair
of morphisms $(\alpha, \alpha^{-1} \circ \beta)$. In any groupoid
this defines a bijection between
$\text{Arrows} \times_{t, \text{Ob}, t} \text{Arrows}$
and
$\text{Arrows} \times_{s, \text{Ob}, t} \text{Arrows}$. Hence the second
assertion of the lemma.
The last assertion follows from Lemma \ref{lemma-diagram}.
\end{proof}
\section{Quasi-coherent sheaves on groupoids}
\label{section-groupoids-quasi-coherent}
\noindent
See the introduction of Section \ref{section-equivariant} for our
choices in direction of arrows.
\begin{definition}
\label{definition-groupoid-module}
Let $S$ be a scheme, let $(U, R, s, t, c)$ be a groupoid scheme over $S$.
A {\it quasi-coherent module on $(U, R, s, t, c)$}
is a pair $(\mathcal{F}, \alpha)$, where $\mathcal{F}$ is a quasi-coherent
$\mathcal{O}_U$-module, and $\alpha$ is a $\mathcal{O}_R$-module
map
$$
\alpha : t^*\mathcal{F} \longrightarrow s^*\mathcal{F}
$$
such that
\begin{enumerate}
\item the diagram
$$
\xymatrix{
& \text{pr}_1^*t^*\mathcal{F} \ar[r]_-{\text{pr}_1^*\alpha} &
\text{pr}_1^*s^*\mathcal{F} \ar@{=}[rd] & \\
\text{pr}_0^*s^*\mathcal{F} \ar@{=}[ru] & & & c^*s^*\mathcal{F} \\
& \text{pr}_0^*t^*\mathcal{F} \ar[lu]^{\text{pr}_0^*\alpha} \ar@{=}[r] &
c^*t^*\mathcal{F} \ar[ru]_{c^*\alpha}
}
$$
is a commutative in the category of
$\mathcal{O}_{R \times_{s, U, t} R}$-modules, and
\item the pullback
$$
e^*\alpha : \mathcal{F} \longrightarrow \mathcal{F}
$$
is the identity map.
\end{enumerate}
Compare with the commutative diagrams of Lemma \ref{lemma-diagram}.
\end{definition}
\noindent
The commutativity of the first diagram forces the operator $e^*\alpha$
to be idempotent. Hence the second condition can be reformulated as saying
that $e^*\alpha$ is an isomorphism. In fact, the condition implies that
$\alpha$ is an isomorphism.
\begin{lemma}
\label{lemma-isomorphism}
Let $S$ be a scheme, let $(U, R, s, t, c)$ be a groupoid scheme over $S$.
If $(\mathcal{F}, \alpha)$ is a quasi-coherent module on $(U, R, s, t, c)$
then $\alpha$ is an isomorphism.
\end{lemma}
\begin{proof}
Pull back the commutative diagram of
Definition \ref{definition-groupoid-module}
by the morphism $(i, 1) : R \to R \times_{s, U, t} R$.
Then we see that $i^*\alpha \circ \alpha = s^*e^*\alpha$.
Pulling back by the morphism $(1, i)$ we obtain the relation
$\alpha \circ i^*\alpha = t^*e^*\alpha$. By the second assumption
these morphisms are the identity. Hence $i^*\alpha$ is an inverse of
$\alpha$.
\end{proof}
\begin{lemma}
\label{lemma-pullback}
Let $S$ be a scheme. Consider a morphism
$f : (U, R, s, t, c) \to (U', R', s', t', c')$
of groupoid schemes over $S$. Then pullback $f^*$ given by
$$
(\mathcal{F}, \alpha) \mapsto (f^*\mathcal{F}, f^*\alpha)
$$
defines a functor from the category of quasi-coherent sheaves on
$(U', R', s', t', c')$ to the category of quasi-coherent sheaves on
$(U, R, s, t, c)$.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{lemma}
\label{lemma-pushforward}
Let $S$ be a scheme. Consider a morphism
$f : (U, R, s, t, c) \to (U', R', s', t', c')$
of groupoid schemes over $S$. Assume that
\begin{enumerate}
\item $f : U \to U'$ is quasi-compact and quasi-separated,
\item the square
$$
\xymatrix{
R \ar[d]_t \ar[r]_f & R' \ar[d]^{t'} \\
U \ar[r]^f & U'
}
$$
is cartesian, and
\item $s'$ and $t'$ are flat.
\end{enumerate}
Then pushforward $f_*$ given by
$$
(\mathcal{F}, \alpha) \mapsto (f_*\mathcal{F}, f_*\alpha)
$$
defines a functor from the category of quasi-coherent sheaves on
$(U, R, s, t, c)$ to the category of quasi-coherent sheaves on
$(U', R', s', t', c')$ which is right adjoint to pullback as defined in
Lemma \ref{lemma-pullback}.
\end{lemma}
\begin{proof}
Since $U \to U'$ is quasi-compact and quasi-separated we see that
$f_*$ transforms quasi-coherent sheaves into quasi-coherent sheaves
(Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}).
Moreover, since the squares
$$
\vcenter{
\xymatrix{
R \ar[d]_t \ar[r]_f & R' \ar[d]^{t'} \\
U \ar[r]^f & U'
}
}
\quad\text{and}\quad
\vcenter{
\xymatrix{
R \ar[d]_s \ar[r]_f & R' \ar[d]^{s'} \\
U \ar[r]^f & U'
}
}
$$
are cartesian we find that $(t')^*f_*\mathcal{F} = f_*t^*\mathcal{F}$
and $(s')^*f_*\mathcal{F} = f_*s^*\mathcal{F}$ , see
Cohomology of Schemes, Lemma
\ref{coherent-lemma-flat-base-change-cohomology}.
Thus it makes sense to think of $f_*\alpha$ as a map
$(t')^*f_*\mathcal{F} \to (s')^*f_*\mathcal{F}$. A similar argument
shows that $f_*\alpha$ satisfies the cocycle condition.
The functor is adjoint to the pullback functor since pullback
and pushforward on modules on ringed spaces are adjoint.
Some details omitted.
\end{proof}
\begin{lemma}
\label{lemma-colimits}
Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.
The category of quasi-coherent modules on $(U, R, s, t, c)$ has colimits.
\end{lemma}
\begin{proof}
Let $i \mapsto (\mathcal{F}_i, \alpha_i)$ be a diagram over the index
category $\mathcal{I}$. We can form the colimit
$\mathcal{F} = \colim \mathcal{F}_i$
which is a quasi-coherent sheaf on $U$, see
Schemes, Section \ref{schemes-section-quasi-coherent}.
Since colimits commute with pullback we see that
$s^*\mathcal{F} = \colim s^*\mathcal{F}_i$ and similarly
$t^*\mathcal{F} = \colim t^*\mathcal{F}_i$. Hence we can set
$\alpha = \colim \alpha_i$. We omit the proof that $(\mathcal{F}, \alpha)$
is the colimit of the diagram in the category of quasi-coherent modules
on $(U, R, s, t, c)$.
\end{proof}
\begin{lemma}
\label{lemma-abelian}
Let $S$ be a scheme.
Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.
If $s$, $t$ are flat, then the category of quasi-coherent modules on
$(U, R, s, t, c)$ is abelian.
\end{lemma}
\begin{proof}
Let $\varphi : (\mathcal{F}, \alpha) \to (\mathcal{G}, \beta)$ be a
homomorphism of quasi-coherent modules on $(U, R, s, t, c)$. Since
$s$ is flat we see that
$$
0 \to s^*\Ker(\varphi)
\to s^*\mathcal{F} \to s^*\mathcal{G} \to s^*\Coker(\varphi) \to 0
$$
is exact and similarly for pullback by $t$. Hence $\alpha$ and $\beta$
induce isomorphisms
$\kappa : t^*\Ker(\varphi) \to s^*\Ker(\varphi)$ and
$\lambda : t^*\Coker(\varphi) \to s^*\Coker(\varphi)$
which satisfy the cocycle condition. Then it is straightforward to
verify that $(\Ker(\varphi), \kappa)$ and
$(\Coker(\varphi), \lambda)$ are a kernel and cokernel in the
category of quasi-coherent modules on $(U, R, s, t, c)$. Moreover,
the condition $\Coim(\varphi) = \Im(\varphi)$ follows
because it holds over $U$.
\end{proof}
\section{Colimits of quasi-coherent modules}
\label{section-colimits}
\noindent
In this section we prove some technical results saying that under
suitable assumptions every quasi-coherent module on a groupoid is
a filtered colimit of ``small'' quasi-coherent modules.
\begin{lemma}
\label{lemma-construct-quasi-coherent}
Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.
Assume $s, t$ are flat, quasi-compact, and quasi-separated.
For any quasi-coherent module $\mathcal{G}$ on $U$, there exists
a canonical isomorphism
$\alpha : t^*t_*s^*\mathcal{G} \to s^*t_*s^*\mathcal{G}$
which turns $(t_*s^*\mathcal{G}, \alpha)$ into a quasi-coherent module
on $(U, R, s, t, c)$. This construction defines a functor
$$
\QCoh(\mathcal{O}_U) \longrightarrow \QCoh(U, R, s, t, c)
$$
which is a right adjoint to the forgetful functor
$(\mathcal{F}, \beta) \mapsto \mathcal{F}$.
\end{lemma}
\begin{proof}
The pushforward of a quasi-coherent module along a quasi-compact and
quasi-separated morphism is quasi-coherent, see Schemes, Lemma
\ref{schemes-lemma-push-forward-quasi-coherent}. Hence $t_*s^*\mathcal{G}$
is quasi-coherent. With notation as in Lemma \ref{lemma-diagram} we have
$$
t^*t_*s^*\mathcal{G} =
\text{pr}_{0, *}c^* s^*\mathcal{G} =
\text{pr}_{0, *}\text{pr}_1^*s^*\mathcal{G} =
s^*t_*s^*\mathcal{G}
$$
The middle equality because $s \circ c = s \circ \text{pr}_1$ as
morphisms $R \times_{s, U, t} R \to U$, and the first and the last
equality because we know that base change and pushforward commute in
these steps by Cohomology of Schemes, Lemma
\ref{coherent-lemma-flat-base-change-cohomology}.
\medskip\noindent
To verify the cocycle condition of Definition \ref{definition-groupoid-module}
for $\alpha$ and the adjointness property we describe the construction
$\mathcal{G} \mapsto (\mathcal{G}, \alpha)$ in another way.
Consider the groupoid scheme
$(R, R \times_{s, U, s} R, \text{pr}_0, \text{pr}_1, \text{pr}_{02})$
associated to the equivalence relation $R \times_{s, U, s} R$
on $R$, see Lemma \ref{lemma-equivalence-groupoid}.
There is a morphism
$$
f :
(R, R \times_{s, U, s} R, \text{pr}_1, \text{pr}_0, \text{pr}_{02})
\longrightarrow
(U, R, s, t, c)
$$
of groupoid schemes given by $t : R \to U$ and $R \times_{t, U, t} R \to R$
given by $(r_0, r_1) \mapsto r_0 \circ r_1^{-1}$ (we omit the verification
of the commutativity of the required diagrams). Since
$t, s : R \to U$ are quasi-compact, quasi-separated, and flat,
and since we have a cartesian square
$$
\xymatrix{
R \times_{s, U, s} R \ar[d]_{\text{pr}_0}
\ar[rr]_-{(r_0, r_1) \mapsto r_0 \circ r_1^{-1}} & & R \ar[d]^t \\
R \ar[rr]^t & & U
}
$$
by Lemma \ref{lemma-diagram-pull} it follows that
Lemma \ref{lemma-pushforward} applies to $f$. Note that
$$
\QCoh(R, R \times_{s, U, s} R, \text{pr}_1, \text{pr}_0, \text{pr}_{02})
= \QCoh(\mathcal{O}_U)
$$
by the theory of descent of quasi-coherent sheaves as $\{t : R \to U\}$
is an fpqc covering, see
Descent, Proposition \ref{descent-proposition-fpqc-descent-quasi-coherent}.
Observe that pullback along $f$ agrees with the forgetful functor and
that pushforward agrees with the construction that assigns to
$\mathcal{G}$ the pair $(\mathcal{G}, \alpha)$. We omit the precise
verifications. Thus the lemma follows from Lemma \ref{lemma-pushforward}.
\end{proof}
\begin{lemma}
\label{lemma-push-pull}
Let $f : Y \to X$ be a morphism of schemes. Let $\mathcal{F}$
be a quasi-coherent $\mathcal{O}_X$-module, let $\mathcal{G}$
be a quasi-coherent $\mathcal{O}_Y$-module, and let
$\varphi : \mathcal{G} \to f^*\mathcal{F}$ be a module map. Assume
\begin{enumerate}
\item $\varphi$ is injective,
\item $f$ is quasi-compact, quasi-separated, flat, and surjective,
\item $X$, $Y$ are locally Noetherian, and
\item $\mathcal{G}$ is a coherent $\mathcal{O}_Y$-module.
\end{enumerate}
Then $\mathcal{F} \cap f_*\mathcal{G}$ defined as the pullback
$$
\xymatrix{
\mathcal{F} \ar[r] & f_*f^*\mathcal{F} \\
\mathcal{F} \cap f_*\mathcal{G} \ar[u] \ar[r] &
f_*\mathcal{G} \ar[u]
}
$$
is a coherent $\mathcal{O}_X$-module.
\end{lemma}
\begin{proof}
We will freely use the characterization of coherent modules of
Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-Noetherian}
as well as the fact that coherent modules form a Serre subcategory
of $\QCoh(\mathcal{O}_X)$, see
Cohomology of Schemes,
Lemma \ref{coherent-lemma-coherent-Noetherian-quasi-coherent-sub-quotient}.
If $f$ has a section $\sigma$, then we see that
$\mathcal{F} \cap f_*\mathcal{G}$ is contained in the image of
$\sigma^*\mathcal{G} \to \sigma^*f^*\mathcal{F} = \mathcal{F}$,
hence coherent. In general, to show that $\mathcal{F} \cap f_*\mathcal{G}$
is coherent, it suffices the show that
$f^*(\mathcal{F} \cap f_*\mathcal{G})$ is coherent (see
Descent, Lemma \ref{descent-lemma-finite-type-descends}).
Since $f$ is flat this is equal to $f^*\mathcal{F} \cap f^*f_*\mathcal{G}$.
Since $f$ is flat, quasi-compact, and quasi-separated we see
$f^*f_*\mathcal{G} = p_*q^*\mathcal{G}$ where $p, q : Y \times_X Y \to Y$
are the projections, see
Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}.
Since $p$ has a section we win.
\end{proof}
\noindent
Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid in schemes over $S$.
Assume that $U$ is locally Noetherian. In the lemma below we say that a
quasi-coherent sheaf $(\mathcal{F}, \alpha)$ on $(U, R, s, t, c)$ is
{\it coherent} if $\mathcal{F}$ is a coherent $\mathcal{O}_U$-module.
\begin{lemma}
\label{lemma-colimit-coherent}
Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.
Assume that
\begin{enumerate}
\item $U$, $R$ are Noetherian,
\item $s, t$ are flat, quasi-compact, and quasi-separated.
\end{enumerate}
Then every quasi-coherent module $(\mathcal{F}, \alpha)$ on $(U, R, s, t, c)$
is a filtered colimit of coherent modules.
\end{lemma}
\begin{proof}
We will use the characterization of Cohomology of Schemes, Lemma
\ref{coherent-lemma-coherent-Noetherian} of coherent modules on locally
Noetherian scheme without further mention. Write
$\mathcal{F} = \colim \mathcal{H}_i$ with $\mathcal{H}_i$ coherent, see
Properties, Lemma \ref{properties-lemma-directed-colimit-finite-presentation}.
Given a quasi-coherent sheaf $\mathcal{H}$ on $U$ we denote $t_*s^*\mathcal{H}$
the quasi-coherent sheaf on $(U, R, s, t, c)$ of
Lemma \ref{lemma-construct-quasi-coherent}. There is an adjunction map
$\mathcal{F} \to t_*s^*\mathcal{F}$ in $\QCoh(U, R, s, t, c)$.
Consider the pullback diagram
$$
\xymatrix{
\mathcal{F} \ar[r] & t_*s^*\mathcal{F} \\
\mathcal{F}_i \ar[r] \ar[u] & t_*s^*\mathcal{H}_i \ar[u]
}
$$
in other words $\mathcal{F}_i = \mathcal{F} \cap t_*s^*\mathcal{H}_i$.
Then $\mathcal{F}_i$ is coherent by Lemma \ref{lemma-push-pull}.
On the other hand, the diagram above is a pullback diagram in
$\QCoh(U, R, s, t, c)$ also as restriction to $U$ is an
exact functor by (the proof of) Lemma \ref{lemma-abelian}. Finally,
because $t$ is quasi-compact and quasi-separated we see that
$t_*$ commutes with colimits (see
Cohomology of Schemes, Lemma \ref{coherent-lemma-colimit-cohomology}).
Hence $t_*s^*\mathcal{F} = \colim t_*\mathcal{H}_i$ and hence
$\mathcal{F} = \colim \mathcal{F}_i$ as desired.
\end{proof}
\noindent
Here is a curious lemma that is useful when working with groupoids
on fields. In fact, this is the standard argument to prove that any
representation of an algebraic group is a colimit of finite dimensional
representations.
\begin{lemma}
\label{lemma-colimit-finite-type}
Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.
Assume that
\begin{enumerate}
\item $U$, $R$ are affine,
\item there exist $e_i \in \mathcal{O}_R(R)$ such that
every element $g \in \mathcal{O}_R(R)$ can be uniquely written as
$\sum s^*(f_i)e_i$ for some $f_i \in \mathcal{O}_U(U)$.
\end{enumerate}
Then every quasi-coherent module $(\mathcal{F}, \alpha)$ on $(U, R, s, t, c)$
is a filtered colimit of finite type quasi-coherent modules.
\end{lemma}
\begin{proof}
The assumption means that $\mathcal{O}_R(R)$ is a free
$\mathcal{O}_U(U)$-module via $s$ with basis $e_i$. Hence
for any quasi-coherent $\mathcal{O}_U$-module $\mathcal{G}$
we see that $s^*\mathcal{G}(R) = \bigoplus_i \mathcal{G}(U)e_i$.
We will write $s(-)$ to indicate pullback of sections by $s$ and
similarly for other morphisms.
Let $(\mathcal{F}, \alpha)$ be a quasi-coherent module on
$(U, R, s, t, c)$. Let $\sigma \in \mathcal{F}(U)$. By the above
we can write
$$
\alpha(t(\sigma)) = \sum s(\sigma_i) e_i
$$
for some unique $\sigma_i \in \mathcal{F}(U)$ (all but finitely many
are zero of course). We can also write
$$
c(e_i) = \sum \text{pr}_1(f_{ij}) \text{pr}_0(e_j)
$$
as functions on $R \times_{s, U, t}R$. Then the commutativity of the diagram
in Definition \ref{definition-groupoid-module} means that
$$
\sum \text{pr}_1(\alpha(t(\sigma_i))) \text{pr}_0(e_i)
=
\sum \text{pr}_1(s(\sigma_i)f_{ij}) \text{pr}_0(e_j)
$$
(calculation omitted). Picking off the coefficients of $\text{pr}_0(e_l)$
we see that $\alpha(t(\sigma_l)) = \sum s(\sigma_i)f_{il}$. Hence
the submodule $\mathcal{G} \subset \mathcal{F}$ generated by the
elements $\sigma_i$ defines a finite type quasi-coherent module
preserved by $\alpha$. Hence it is a subobject of $\mathcal{F}$ in
$\QCoh(U, R, s, t, c)$. This submodule contains $\sigma$
(as one sees by pulling back the first relation by $e$). Hence we win.
\end{proof}
\noindent
We suggest the reader skip the rest of this section. Let $S$ be a scheme.
Let $(U, R, s, t, c)$ be a groupoid in schemes over $S$. Let $\kappa$ be a
cardinal. In the following we will say that a quasi-coherent sheaf
$(\mathcal{F}, \alpha)$ on $(U, R, s, t, c)$ is $\kappa$-generated if
$\mathcal{F}$ is a $\kappa$-generated $\mathcal{O}_U$-module, see
Properties, Definition \ref{properties-definition-kappa-generated}.
\begin{lemma}
\label{lemma-set-of-iso-classes}
Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.
Let $\kappa$ be a cardinal.
There exists a set $T$ and a family $(\mathcal{F}_t, \alpha_t)_{t \in T}$ of
$\kappa$-generated quasi-coherent modules on $(U, R, s, t, c)$
such that every $\kappa$-generated quasi-coherent module on
$(U, R, s, t, c)$ is isomorphic to one of the $(\mathcal{F}_t, \alpha_t)$.
\end{lemma}
\begin{proof}
For each quasi-coherent module $\mathcal{F}$ on $U$ there is a
(possibly empty) set of maps $\alpha : t^*\mathcal{F} \to s^*\mathcal{F}$
such that $(\mathcal{F}, \alpha)$ is a quasi-coherent modules on
$(U, R, s, t, c)$. By
Properties, Lemma \ref{properties-lemma-set-of-iso-classes}
there exists a set of isomorphism classes of $\kappa$-generated
quasi-coherent $\mathcal{O}_U$-modules.
\end{proof}
\begin{lemma}
\label{lemma-colimit-kappa}
Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.
Assume that $s, t$ are flat. There exists a
cardinal $\kappa$ such that every quasi-coherent module
$(\mathcal{F}, \alpha)$ on $(U, R, s, t, c)$
is the directed colimit of its $\kappa$-generated
quasi-coherent submodules.
\end{lemma}
\begin{proof}
In the statement of the lemma and in this proof
a {\it submodule} of a quasi-coherent module $(\mathcal{F}, \alpha)$
is a quasi-coherent submodule $\mathcal{G} \subset \mathcal{F}$
such that $\alpha(t^*\mathcal{G}) = s^*\mathcal{G}$ as subsheaves of
$s^*\mathcal{F}$. This makes sense because since $s, t$ are flat the
pullbacks $s^*$ and $t^*$ are exact, i.e., preserve subsheaves.
The proof will be a repeat of the proof of
Properties, Lemma \ref{properties-lemma-colimit-kappa}.
We urge the reader to read that proof first.
\medskip\noindent
Choose an affine open covering $U = \bigcup_{i \in I} U_i$.
For each pair $i, j$ choose affine open coverings
$$
U_i \cap U_j = \bigcup\nolimits_{k \in I_{ij}} U_{ijk}
\quad\text{and}\quad
s^{-1}(U_i) \cap t^{-1}(U_j) = \bigcup\nolimits_{k \in J_{ij}} W_{ijk}.
$$
Write $U_i = \Spec(A_i)$, $U_{ijk} = \Spec(A_{ijk})$,
$W_{ijk} = \Spec(B_{ijk})$.
Let $\kappa$ be any infinite cardinal $\geq$ than the cardinality
of any of the sets $I$, $I_{ij}$, $J_{ij}$.
\medskip\noindent
Let $(\mathcal{F}, \alpha)$ be a quasi-coherent module on $(U, R, s, t, c)$.
Set $M_i = \mathcal{F}(U_i)$, $M_{ijk} = \mathcal{F}(U_{ijk})$.
Note that
$$
M_i \otimes_{A_i} A_{ijk} = M_{ijk} = M_j \otimes_{A_j} A_{ijk}
$$
and that $\alpha$ gives isomorphisms
$$
\alpha|_{W_{ijk}} :
M_i \otimes_{A_i, t} B_{ijk}
\longrightarrow
M_j \otimes_{A_j, s} B_{ijk}
$$
see
Schemes, Lemma \ref{schemes-lemma-widetilde-pullback}.
Using the axiom of choice we choose a map
$$
(i, j, k, m) \mapsto S(i, j, k, m)
$$
which associates to every $i, j \in I$, $k \in I_{ij}$ or $k \in J_{ij}$
and $m \in M_i$ a finite subset $S(i, j, k, m) \subset M_j$
such that we have
$$
m \otimes 1 = \sum\nolimits_{m' \in S(i, j, k, m)} m' \otimes a_{m'}
\quad\text{or}\quad
\alpha(m \otimes 1) = \sum\nolimits_{m' \in S(i, j, k, m)} m' \otimes b_{m'}
$$
in $M_{ijk}$ for some $a_{m'} \in A_{ijk}$ or $b_{m'} \in B_{ijk}$.
Moreover, let's agree that $S(i, i, k, m) = \{m\}$ for all
$i, j = i, k, m$ when $k \in I_{ij}$. Fix such a collection $S(i, j, k, m)$
\medskip\noindent
Given a family $\mathcal{S} = (S_i)_{i \in I}$ of subsets
$S_i \subset M_i$ of cardinality at most $\kappa$ we set
$\mathcal{S}' = (S'_i)$ where
$$
S'_j = \bigcup\nolimits_{(i, j, k, m)\text{ such that }m \in S_i}
S(i, j, k, m)
$$
Note that $S_i \subset S'_i$. Note that $S'_i$ has cardinality at most
$\kappa$ because it is a union over a set of cardinality at most $\kappa$
of finite sets. Set $\mathcal{S}^{(0)} = \mathcal{S}$,
$\mathcal{S}^{(1)} = \mathcal{S}'$ and by induction
$\mathcal{S}^{(n + 1)} = (\mathcal{S}^{(n)})'$. Then set
$\mathcal{S}^{(\infty)} = \bigcup_{n \geq 0} \mathcal{S}^{(n)}$.
Writing $\mathcal{S}^{(\infty)} = (S^{(\infty)}_i)$ we see that
for any element $m \in S^{(\infty)}_i$ the image of $m$ in
$M_{ijk}$ can be written as a finite sum $\sum m' \otimes a_{m'}$
with $m' \in S_j^{(\infty)}$. In this way we see that setting
$$
N_i = A_i\text{-submodule of }M_i\text{ generated by }S^{(\infty)}_i
$$
we have
$$
N_i \otimes_{A_i} A_{ijk} = N_j \otimes_{A_j} A_{ijk}
\quad\text{and}\quad
\alpha(N_i \otimes_{A_i, t} B_{ijk}) = N_j \otimes_{A_j, s} B_{ijk}
$$
as submodules of $M_{ijk}$ or $M_j \otimes_{A_j, s} B_{ijk}$.
Thus there exists a quasi-coherent submodule
$\mathcal{G} \subset \mathcal{F}$ with $\mathcal{G}(U_i) = N_i$
such that $\alpha(t^*\mathcal{G}) = s^*\mathcal{G}$ as submodules
of $s^*\mathcal{F}$. In other words,
$(\mathcal{G}, \alpha|_{t^*\mathcal{G}})$ is a submodule of
$(\mathcal{F}, \alpha)$.
Moreover, by construction $\mathcal{G}$ is $\kappa$-generated.
\medskip\noindent
Let $\{(\mathcal{G}_t, \alpha_t)\}_{t \in T}$ be the set of
$\kappa$-generated quasi-coherent submodules of $(\mathcal{F}, \alpha)$.
If $t, t' \in T$ then $\mathcal{G}_t + \mathcal{G}_{t'}$ is also a
$\kappa$-generated quasi-coherent submodule as it is the image of the map
$\mathcal{G}_t \oplus \mathcal{G}_{t'} \to \mathcal{F}$.
Hence the system (ordered by inclusion) is directed.
The arguments above show that every section of $\mathcal{F}$ over $U_i$
is in one of the $\mathcal{G}_t$ (because we can start with $\mathcal{S}$
such that the given section is an element of $S_i$). Hence
$\colim_t \mathcal{G}_t \to \mathcal{F}$ is both injective and surjective
as desired.
\end{proof}
\section{Groupoids and group schemes}
\label{section-groupoids-group-schemes}
\noindent
There are many ways to construct a groupoid out of an action $a$
of a group $G$ on a set $V$. We choose the one where we think
of an element $g \in G$ as an arrow with source $v$ and target $a(g, v)$.
This leads to the following construction for group actions of
schemes.
\begin{lemma}
\label{lemma-groupoid-from-action}
Let $S$ be a scheme.
Let $Y$ be a scheme over $S$.
Let $(G, m)$ be a group scheme over $Y$ with
identity $e_G$ and inverse $i_G$.
Let $X/Y$ be a scheme over $Y$ and let $a : G \times_Y X \to X$
be an action of $G$ on $X/Y$.
Then we get a groupoid scheme $(U, R, s, t, c, e, i)$ over $S$
in the following manner:
\begin{enumerate}
\item We set $U = X$, and $R = G \times_Y X$.
\item We set $s : R \to U$ equal to $(g, x) \mapsto x$.
\item We set $t : R \to U$ equal to $(g, x) \mapsto a(g, x)$.
\item We set $c : R \times_{s, U, t} R \to R$ equal to
$((g, x), (g', x')) \mapsto (m(g, g'), x')$.
\item We set $e : U \to R$ equal to $x \mapsto (e_G(x), x)$.
\item We set $i : R \to R$ equal to $(g, x) \mapsto (i_G(g), a(g, x))$.
\end{enumerate}
\end{lemma}
\begin{proof}
Omitted. Hint: It is enough to show that this works on the set
level. For this use the description above the lemma describing
$g$ as an arrow from $v$ to $a(g, v)$.
\end{proof}
\begin{lemma}
\label{lemma-action-groupoid-modules}
Let $S$ be a scheme.
Let $Y$ be a scheme over $S$.
Let $(G, m)$ be a group scheme over $Y$.
Let $X$ be a scheme over $Y$ and let $a : G \times_Y X \to X$
be an action of $G$ on $X$ over $Y$. Let $(U, R, s, t, c)$ be
the groupoid scheme constructed in Lemma \ref{lemma-groupoid-from-action}.
The rule
$(\mathcal{F}, \alpha) \mapsto (\mathcal{F}, \alpha)$ defines
an equivalence of categories between $G$-equivariant
$\mathcal{O}_X$-modules and the category of quasi-coherent
modules on $(U, R, s, t, c)$.
\end{lemma}
\begin{proof}
The assertion makes sense because $t = a$ and $s = \text{pr}_1$
as morphisms $R = G \times_Y X \to X$, see
Definitions \ref{definition-equivariant-module} and
\ref{definition-groupoid-module}.
Using the translation in Lemma \ref{lemma-groupoid-from-action}
the commutativity requirements
of the two definitions match up exactly.
\end{proof}
\section{The stabilizer group scheme}
\label{section-stabilizer}
\noindent
Given a groupoid scheme we get a group scheme as follows.
\begin{lemma}
\label{lemma-groupoid-stabilizer}
Let $S$ be a scheme.
Let $(U, R, s, t, c)$ be a groupoid over $S$.
The scheme $G$ defined by the cartesian square
$$
\xymatrix{
G \ar[r] \ar[d] & R \ar[d]^{j = (t, s)} \\
U \ar[r]^-{\Delta} & U \times_S U
}
$$
is a group scheme over $U$ with composition law
$m$ induced by the composition law $c$.
\end{lemma}
\begin{proof}
This is true because in a groupoid category the
set of self maps of any object forms a group.
\end{proof}
\noindent
Since $\Delta$ is an immersion we see that $G = j^{-1}(\Delta_{U/S})$
is a locally closed subscheme of $R$. Thinking of it in this way,
the structure morphism $j^{-1}(\Delta_{U/S}) \to U$ is induced by
either $s$ or $t$ (it is the same), and $m$ is induced by $c$.
\begin{definition}
\label{definition-stabilizer-groupoid}
Let $S$ be a scheme.
Let $(U, R, s, t, c)$ be a groupoid over $S$.
The group scheme $j^{-1}(\Delta_{U/S})\to U$
is called the {\it stabilizer of the groupoid scheme
$(U, R, s, t, c)$}.
\end{definition}
\noindent
In the literature the stabilizer group scheme is often denoted $S$
(because the word stabilizer starts with an ``s'' presumably);
we cannot do this since we have already used $S$ for the base scheme.
\begin{lemma}
\label{lemma-groupoid-action-stabilizer}
Let $S$ be a scheme.
Let $(U, R, s, t, c)$ be a groupoid over $S$, and let $G/U$ be its stabilizer.
Denote $R_t/U$ the scheme $R$ seen as a scheme over $U$ via the
morphism $t : R \to U$.
There is a canonical left action
$$
a : G \times_U R_t \longrightarrow R_t
$$
induced by the composition law $c$.
\end{lemma}
\begin{proof}
In terms of points over $T/S$ we define $a(g, r) = c(g, r)$.
\end{proof}
\begin{lemma}
\label{lemma-groupoid-action-stabilizer-pseudo-torsor}
Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme
over $S$. Let $G$ be the stabilizer group scheme of $R$.
Let
$$
G_0 = G \times_{U, \text{pr}_0} (U \times_S U) = G \times_S U
$$
as a group scheme over $U \times_S U$. The action of $G$ on $R$ of
Lemma \ref{lemma-groupoid-action-stabilizer}
induces an action of $G_0$ on $R$ over $U \times_S U$
which turns $R$ into a pseudo $G_0$-torsor over $U \times_S U$.
\end{lemma}
\begin{proof}
This is true because in a groupoid category $\mathcal{C}$ the set
$\Mor_\mathcal{C}(x, y)$ is a principal homogeneous set
under the group $\Mor_\mathcal{C}(y, y)$.
\end{proof}
\begin{lemma}
\label{lemma-fibres-j}
Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.
Let $p \in U \times_S U$ be a point. Denote
$R_p$ the scheme theoretic fibre of $j = (t, s) : R \to U \times_S U$.
If $R_p \not = \emptyset$, then the action
$$
G_{0, \kappa(p)} \times_{\kappa(p)} R_p \longrightarrow R_p
$$
(see
Lemma \ref{lemma-groupoid-action-stabilizer-pseudo-torsor})
which turns $R_p$ into a $G_{\kappa(p)}$-torsor over $\kappa(p)$.
\end{lemma}
\begin{proof}
The action is a pseudo-torsor by the lemma cited in the statement.
And if $R_p$ is not the empty scheme, then $\{R_p \to p\}$
is an fpqc covering which trivializes the pseudo-torsor.
\end{proof}
\section{Restricting groupoids}
\label{section-restrict-groupoid}
\noindent
Consider a (usual) groupoid
$\mathcal{C} = (\text{Ob}, \text{Arrows}, s, t, c)$.
Suppose we have a map of sets $g : \text{Ob}' \to \text{Ob}$.
Then we can construct a groupoid
$\mathcal{C}' = (\text{Ob}', \text{Arrows}', s', t', c')$
by thinking of a morphism between elements $x', y'$ of $\text{Ob}'$
as a morphism in $\mathcal{C}$ between $g(x'), g(y')$.
In other words we set
$$
\text{Arrows}' =
\text{Ob}'
\times_{g, \text{Ob}, t}
\text{Arrows}
\times_{s, \text{Ob}, g}
\text{Ob}'.
$$
with obvious choices for $s'$, $t'$, and $c'$. There is a canonical
functor $\mathcal{C}' \to \mathcal{C}$ which is fully faithful,
but not necessarily essentially surjective. This groupoid $\mathcal{C}'$
endowed with the functor $\mathcal{C}' \to \mathcal{C}$
is called the {\it restriction} of the groupoid
$\mathcal{C}$ to $\text{Ob}'$.
\begin{lemma}
\label{lemma-restrict-groupoid}
Let $S$ be a scheme.
Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.
Let $g : U' \to U$ be a morphism of schemes.
Consider the following diagram
$$
\xymatrix{
R' \ar[d] \ar[r] \ar@/_3pc/[dd]_{t'} \ar@/^1pc/[rr]^{s'}&
R \times_{s, U} U' \ar[r] \ar[d] &
U' \ar[d]^g \\
U' \times_{U, t} R \ar[d] \ar[r] &
R \ar[r]^s \ar[d]_t &
U \\
U' \ar[r]^g &
U
}
$$
where all the squares are fibre product squares. Then there is a
canonical composition law $c' : R' \times_{s', U', t'} R' \to R'$
such that $(U', R', s', t', c')$ is a groupoid scheme over
$S$ and such that $U' \to U$, $R' \to R$ defines a morphism
$(U', R', s', t', c') \to (U, R, s, t, c)$ of groupoid schemes over $S$.
Moreover, for any scheme $T$ over $S$ the functor of groupoids
$$
(U'(T), R'(T), s', t', c') \to (U(T), R(T), s, t, c)
$$
is the restriction (see above) of $(U(T), R(T), s, t, c)$ via the map
$U'(T) \to U(T)$.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{definition}
\label{definition-restrict-groupoid}
Let $S$ be a scheme.
Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.
Let $g : U' \to U$ be a morphism of schemes.
The morphism of groupoids
$(U', R', s', t', c') \to (U, R, s, t, c)$
constructed in Lemma \ref{lemma-restrict-groupoid} is called
the {\it restriction of $(U, R, s, t, c)$ to $U'$}.
We sometime use the notation $R' = R|_{U'}$ in this case.
\end{definition}
\begin{lemma}
\label{lemma-restrict-groupoid-relation}
The notions of restricting groupoids and
(pre-)equivalence relations defined in Definitions
\ref{definition-restrict-groupoid} and \ref{definition-restrict-relation}
agree via the constructions of
Lemmas \ref{lemma-groupoid-pre-equivalence} and
\ref{lemma-equivalence-groupoid}.
\end{lemma}
\begin{proof}
What we are saying here is that $R'$ of
Lemma \ref{lemma-restrict-groupoid} is also
equal to
$$
R' = (U' \times_S U')\times_{U \times_S U} R
\longrightarrow
U' \times_S U'
$$
In fact this might have been a clearer way to state that lemma.
\end{proof}
\begin{lemma}
\label{lemma-restrict-stabilizer}
Let $S$ be a scheme.
Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.
Let $g : U' \to U$ be a morphism of schemes.
Let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ via $g$.
Let $G$ be the stabilizer of $(U, R, s, t, c)$ and let
$G'$ be the stabilizer of $(U', R', s', t', c')$.
Then $G'$ is the base change of $G$ by $g$, i.e.,
there is a canonical identification $G' = U' \times_{g, U} G$.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\section{Invariant subschemes}
\label{section-invariant}
\noindent
In this section we discuss briefly the notion of an invariant subscheme.
\begin{definition}
\label{definition-invariant-open}
Let $(U, R, s, t, c)$ be a groupoid scheme over the base scheme $S$.
\begin{enumerate}
\item A subset $W \subset U$ is {\it set-theoretically $R$-invariant}
if $t(s^{-1}(W)) \subset W$.
\item An open $W \subset U$ is {\it $R$-invariant} if
$t(s^{-1}(W)) \subset W$.
\item A closed subscheme $Z \subset U$ is called {\it $R$-invariant}
if $t^{-1}(Z) = s^{-1}(Z)$. Here we use the scheme theoretic inverse image, see
Schemes, Definition \ref{schemes-definition-inverse-image-closed-subscheme}.
\item A monomorphism of schemes $T \to U$ is {\it $R$-invariant} if
$T \times_{U, t} R = R \times_{s, U} T$ as schemes over $R$.
\end{enumerate}
\end{definition}
\noindent
For subsets and open subschemes $W \subset U$ the $R$-invariance
is also equivalent to requiring that $s^{-1}(W) = t^{-1}(W)$
as subsets of $R$. If $W \subset U$ is an $R$-equivariant open subscheme
then the restriction of $R$ to $W$ is just $R_W = s^{-1}(W) = t^{-1}(W)$.
Similarly, if $Z \subset U$ is an $R$-invariant closed subscheme, then
the restriction of $R$ to $Z$ is just $R_Z = s^{-1}(Z) = t^{-1}(Z)$.
\begin{lemma}
\label{lemma-constructing-invariant-opens}
Let $S$ be a scheme.
Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.
\begin{enumerate}
\item For any subset $W \subset U$ the subset $t(s^{-1}(W))$
is set-theoretically $R$-invariant.
\item If $s$ and $t$ are open, then for every open $W \subset U$
the open $t(s^{-1}(W))$ is an $R$-invariant open subscheme.
\item If $s$ and $t$ are open and quasi-compact, then $U$ has an open
covering consisting of $R$-invariant quasi-compact open subschemes.
\end{enumerate}
\end{lemma}
\begin{proof}
Part (1) follows from
Lemmas \ref{lemma-pre-equivalence-equivalence-relation-points} and
\ref{lemma-groupoid-pre-equivalence}, namely, $t(s^{-1}(W))$
is the set of points of $U$ equivalent to a point of $W$.
Next, assume $s$ and $t$ open and $W \subset U$ open.
Since $s$ is open the set $W' = t(s^{-1}(W))$ is an open subset of $U$.
Finally, assume that $s$, $t$ are both open and quasi-compact.
Then, if $W \subset U$ is a quasi-compact open, then also
$W' = t(s^{-1}(W))$ is a quasi-compact open, and invariant by the
discussion above. Letting $W$ range over all affine opens of $U$
we see (3).
\end{proof}
\begin{lemma}
\label{lemma-first-observation}
Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.
Assume $s$ and $t$ quasi-compact and flat and $U$ quasi-separated.
Let $W \subset U$ be quasi-compact open. Then $t(s^{-1}(W))$
is an intersection of a nonempty family of quasi-compact open subsets of $U$.
\end{lemma}
\begin{proof}
Note that $s^{-1}(W)$ is quasi-compact open in $R$.
As a continuous map $t$ maps the quasi-compact subset
$s^{-1}(W)$ to a quasi-compact subset $t(s^{-1}(W))$.
As $t$ is flat and $s^{-1}(W)$ is closed under generalization,
so is $t(s^{-1}(W))$, see
(Morphisms, Lemma \ref{morphisms-lemma-generalizations-lift-flat} and
Topology, Lemma \ref{topology-lemma-lift-specializations-images}).
Pick a quasi-compact open $W' \subset U$ containing $t(s^{-1}(W))$. By
Properties, Lemma \ref{properties-lemma-quasi-compact-quasi-separated-spectral}
we see that $W'$ is a spectral space (here we use that $U$ is quasi-separated).
Then the lemma follows from
Topology, Lemma \ref{topology-lemma-make-spectral-space}
applied to $t(s^{-1}(W)) \subset W'$.
\end{proof}
\begin{lemma}
\label{lemma-second-observation}
Assumptions and notation as in Lemma \ref{lemma-first-observation}.
There exists an $R$-invariant open $V \subset U$ and a quasi-compact
open $W'$ such that $W \subset V \subset W' \subset U$.
\end{lemma}
\begin{proof}
Set $E = t(s^{-1}(W))$. Recall that $E$ is set-theoretically $R$-invariant
(Lemma \ref{lemma-constructing-invariant-opens}).
By Lemma \ref{lemma-first-observation} there exists a quasi-compact
open $W'$ containing $E$. Let $Z = U \setminus W'$ and consider
$T = t(s^{-1}(Z))$. Observe that $Z \subset T$ and that
$E \cap T = \emptyset$ because $s^{-1}(E) = t^{-1}(E)$ is disjoint
from $s^{-1}(Z)$. Since $T$ is the image of the closed subset
$s^{-1}(Z) \subset R$ under the quasi-compact morphism $t : R \to U$
we see that any point $\xi$ in the closure $\overline{T}$
is the specialization of a point of $T$, see
Morphisms, Lemma \ref{morphisms-lemma-reach-points-scheme-theoretic-image} (and
Morphisms, Lemma \ref{morphisms-lemma-quasi-compact-scheme-theoretic-image}
to see that the scheme theoretic image is the closure of the image).
Say $\xi' \leadsto \xi$ with $\xi' \in T$. Suppose that $r \in R$ and
$s(r) = \xi$. Since $s$ is flat we can find a specialization $r' \leadsto r$
in $R$ such that $s(r') = \xi'$
(Morphisms, Lemma \ref{morphisms-lemma-generalizations-lift-flat}).
Then $t(r') \leadsto t(r)$. We conclude that $t(r') \in T$ as $T$
is set-theoretically invariant by
Lemma \ref{lemma-constructing-invariant-opens}.
Thus $\overline{T}$ is a set-theoretically $R$-invariant closed subset
and $V = U \setminus \overline{T}$ is the open we are
looking for. It is contained in $W'$ which finishes the proof.
\end{proof}
\section{Quotient sheaves}
\label{section-quotient-sheaves}
\noindent
Let $\tau \in \{Zariski, \etale, fppf, smooth, syntomic\}$.
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 $\Sch_\tau$
(see Topologies, Section \ref{topologies-section-procedure}).
Then we can consider the functors
$$
h_U, h_R :
(\Sch/S)_\tau^{opp}
\longrightarrow
\textit{Sets}.
$$
These are sheaves, see
Descent, Lemma \ref{descent-lemma-fpqc-universal-effective-epimorphisms}.
The morphism $j$ induces a map $j : h_R \to h_U \times h_U$.
For each object $T \in \Ob((\Sch/S)_\tau)$
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
\begin{equation}
\label{equation-quotient-presheaf}
(\Sch/S)_\tau^{opp}
\longrightarrow
\textit{Sets}, \quad
T \longmapsto U(T)/\sim_T
\end{equation}
\begin{definition}
\label{definition-quotient-sheaf}
Let $\tau$, $S$, and the pre-relation $j : R \to U \times_S U$ be as above.