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spaces-groupoids.tex
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\input{preamble}
% OK, start here.
%
\begin{document}
\title{Groupoids in Algebraic Spaces}
\maketitle
\phantomsection
\label{section-phantom}
\tableofcontents
\section{Introduction}
\label{section-introduction}
\noindent
This chapter is devoted to generalities concerning groupoids in algebraic
spaces. We recommend reading the beautiful paper \cite{K-M} by Keel and Mori.
\medskip\noindent
A lot of what we say here is a repeat of what we said in the chapter
on groupoid schemes, see
Groupoids, Section \ref{groupoids-section-introduction}.
The discussion of quotient stacks is new here.
\section{Conventions}
\label{section-conventions}
\noindent
The standing assumption is that all schemes are contained in
a big fppf site $\Sch_{fppf}$. And all rings $A$ considered
have the property that $\Spec(A)$ is (isomorphic) to an
object of this big site.
\medskip\noindent
Let $S$ be a scheme and let $X$ be an algebraic space over $S$.
In this chapter and the following we will write $X \times_S X$
for the product of $X$ with itself (in the category of algebraic
spaces over $S$), instead of $X \times X$.
\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{Notation}
\label{section-notation}
\noindent
Let $S$ be a scheme; this will be our base scheme and all algebraic spaces
will be over $S$. Let $B$ be an algebraic space over $S$; this will be our
base algebraic space, and often other algebraic spaces, and schemes will
be over $B$. If we say that $X$ is an algebraic space over $B$, then we
mean that $X$ is an algebraic space over $S$ which comes equipped with
structure morphism $X \to B$. Moreover, we try to reserve the letter $T$
to denote a ``test'' scheme over $B$. In other words $T$ is a scheme
which comes equipped with a structure morphism $T \to B$.
In this situation we denote $X(T)$ for the set of $T$-valued points of
$X$ {\it over} $B$. In a formula:
$$
X(T) = \Mor_B(T, X).
$$
Similarly, given a second algebraic space $Y$ over $B$ we set
$$
X(Y) = \Mor_B(Y, X).
$$
Suppose we are given algebraic spaces $X$, $Y$ over $B$ as above and a
morphism $f : X \to Y$ over $B$. For any scheme $T$ over $B$ we get an
induced map of sets
$$
f : X(T) \longrightarrow Y(T)
$$
which is functorial in the scheme $T$ over $B$.
As $f$ is a map of sheaves on $(\Sch/S)_{fppf}$ over the
sheaf $B$ it is clear that $f$ determines and is determined by this rule.
More generally, we use the same notation for maps between fibre
products. For example, if
$X$, $Y$, $Z$ are algebraic spaces over $B$, and if
$m : X \times_B Y \to Z \times_B Z$ is
a morphism of algebraic spaces over $B$, 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
Finally, given two maps
$f, g : X \to Y$ of algebraic spaces over $B$, if the induced maps
$f, g : X(T) \to Y(T)$ are equal for every scheme $T$ over $B$, then
$f = g$, and hence also $f, g : X(Z) \to Y(Z)$ are equal for
every third algebraic space $Z$ over $B$. Hence, for example,
to check the axioms for an group algebraic space $G$ over $B$, it
suffices to check commutativity of diagram on $T$-valued points
where $T$ is a scheme over $B$ as we do in
Definition \ref{definition-group-space}
below.
\section{Equivalence relations}
\label{section-equivalence-relations}
\noindent
Please refer to
Groupoids, Section \ref{groupoids-section-equivalence-relations}
for notation.
\begin{definition}
\label{definition-equivalence-relation}
Let $B \to S$ as in Section \ref{section-notation}.
Let $U$ be an algebraic space over $B$.
\begin{enumerate}
\item A {\it pre-relation} on $U$ over $B$ is any morphism
$j : R \to U \times_B U$ of algebraic spaces over $B$.
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 $B$ is a monomorphism
$j : R \to U \times_B U$ of algebraic spaces over $B$.
\item A {\it pre-equivalence relation} is a pre-relation
$j : R \to U \times_B U$ such that the image of
$j : R(T) \to U(T) \times U(T)$ is an equivalence relation for
all schemes $T$ over $B$.
\item We say a morphism $R \to U \times_B U$ of algebraic spaces over $B$
is an {\it equivalence relation on $U$ over $B$}
if and only if for every $T$ over $B$ 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 $B \to S$ as in Section \ref{section-notation}.
Let $U$ be an algebraic space over $B$.
Let $j : R \to U \times_B U$ be a pre-relation.
Let $g : U' \to U$ be a morphism of algebraic spaces over $B$.
Finally, set
$$
R' = (U' \times_B U')\times_{U \times_B U} R
\xrightarrow{j'}
U' \times_B U'
$$
Then $j'$ is a pre-relation on $U'$ over $B$.
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 $B \to S$ as in Section \ref{section-notation}.
Let $U$ be an algebraic space over $B$.
Let $j : R \to U \times_B U$ be a pre-relation.
Let $g : U' \to U$ be a morphism of algebraic spaces over $B$.
The pre-relation $j' : R' \to U' \times_B U'$ 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 $B \to S$ as in Section \ref{section-notation}.
Let $j : R \to U \times_B U$ be a pre-relation of algebraic spaces over $B$.
Consider the relation on $|U|$ 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 $t(r') = y$, $s(r') = z$.
We may pick a field $K$ such that $r$ and $r'$ can be
represented by morphisms $r, r' : \Spec(K) \to R$
with $s \circ r = t \circ r'$.
Denote $x = t \circ r$, $y = s \circ r = t \circ r'$,
and $z = s \circ r'$, so $x, y, z : \Spec(K) \to U$.
By construction $(x, y) \in j(R(K))$ and
$(y, z) \in j(R(K))$. Since $j$ is a pre-equivalence relation
we see that also $(x, z) \in j(R(K))$.
This clearly implies that $x \sim z$.
\medskip\noindent
The proof that $\sim$ is reflexive and symmetric is omitted.
\end{proof}
\section{Group algebraic spaces}
\label{section-group-spaces}
\noindent
Please refer to
Groupoids, Section \ref{groupoids-section-group-schemes}
for notation.
\begin{definition}
\label{definition-group-space}
Let $B \to S$ as in Section \ref{section-notation}.
\begin{enumerate}
\item A {\it group algebraic space over $B$} is a pair $(G, m)$, where
$G$ is an algebraic space over $B$ and $m : G \times_B G \to G$ is
a morphism of algebraic spaces over $B$ with the following property:
For every scheme $T$ over $B$ the pair $(G(T), m)$ is a group.
\item A {\it morphism $\psi : (G, m) \to (G', m')$ of
group algebraic spaces over $B$}
is a morphism $\psi : G \to G'$ of algebraic spaces over $B$ such that for
every $T/B$ 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 algebraic space over the algebraic space $B$.
By the discussion in
Groupoids, Section \ref{groupoids-section-group-schemes}
we obtain morphisms of algebraic spaces over $B$
(identity) $e : B \to G$ and (inverse) $i : B \to B$ such that
for every $T$ the quadruple $(G(T), m, e, i)$ satisfies the
axioms of a group.
\medskip\noindent
Let $(G, m)$, $(G', m')$ be group algebraic spaces over $B$.
Let $f : G \to G'$ be a morphism of algebraic spaces over $B$.
It follows from the definition that $f$ is a morphism
of group algebraic spaces over $B$ if and only if the following diagram
is commutative:
$$
\xymatrix{
G \times_B G \ar[r]_-{f \times f} \ar[d]_m &
G' \times_B G' \ar[d]^m \\
G \ar[r]^f & G'
}
$$
\begin{lemma}
\label{lemma-base-change-group-space}
Let $B \to S$ as in Section \ref{section-notation}.
Let $(G, m)$ be a group algebraic space over $B$.
Let $B' \to B$ be a morphism of algebraic spaces.
The pullback $(G_{B'}, m_{B'})$ is a group algebraic space over $B'$.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\section{Properties of group algebraic spaces}
\label{section-properties-group-spaces}
\noindent
In this section we collect some simple properties of group algebraic spaces
which hold over any base.
\begin{lemma}
\label{lemma-group-scheme-separated}
Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.
Let $G$ be a group algebraic space over $B$.
Then $G \to B$ is separated (resp.\ quasi-separated, resp.\ locally separated)
if and only if the identity morphism $e : B \to G$ is a closed immersion
(resp.\ quasi-compact, resp.\ an immersion).
\end{lemma}
\begin{proof}
We recall that by
Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-section-immersion}
we have that $e$ is a closed immersion (resp.\ quasi-compact, resp.\ an
immersion) if $G \to B$ is separated (resp.\ quasi-separated, resp.\ locally
separated).
For the converse, consider the diagram
$$
\xymatrix{
G \ar[r]_-{\Delta_{G/B}} \ar[d] &
G \times_B G \ar[d]^{(g, g') \mapsto m(i(g), g')} \\
B \ar[r]^e & G
}
$$
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
$\Delta_{G/B}$ is a base change of $e$. Hence if $e$ is a
closed immersion (resp.\ quasi-compact, resp.\ an immersion) so is
$\Delta_{G/B}$, see
Spaces, Lemma \ref{spaces-lemma-base-change-immersions}
(resp.\ Morphisms of Spaces, Lemma
\ref{spaces-morphisms-lemma-base-change-quasi-compact},
resp.\ Spaces, Lemma \ref{spaces-lemma-base-change-immersions}).
\end{proof}
\section{Examples of group algebraic spaces}
\label{section-examples-group-spaces}
\noindent
If $G \to S$ is a group scheme over the base scheme $S$, then the base
change $G_B$ to any algebraic space $B$ over $S$ is an group algebraic
space over $B$ by
Lemma \ref{lemma-base-change-group-space}.
We will frequently use this in the examples below.
\begin{example}[Multiplicative group algebraic space]
\label{example-multiplicative-group}
Let $B \to S$ as in Section \ref{section-notation}.
Consider the functor which associates to any scheme $T$ over
$B$ the group $\Gamma(T, \mathcal{O}_T^*)$
of units in the global sections of the structure sheaf.
This is representable by the group algebraic space
$$
\mathbf{G}_{m, B} = B \times_S \mathbf{G}_{m, S}
$$
over $B$. Here $\mathbf{G}_{m, S}$ is the multiplicative group scheme
over $S$, see
Groupoids, Example \ref{groupoids-example-multiplicative-group}.
\end{example}
\begin{example}[Roots of unity as a group algebraic space]
\label{example-roots-of-unity}
Let $B \to S$ as in Section \ref{section-notation}.
Let $n \in \mathbf{N}$. Consider the functor which associates
to any scheme $T$ over $B$ the subgroup of $\Gamma(T, \mathcal{O}_T^*)$
consisting of $n$th roots of unity.
This is representable by the group algebraic space
$$
\mu_{n, B} = B \times_S \mu_{n, S}
$$
over $B$. Here $\mu_{n, S}$ is the group scheme of $n$th roots of
unity over $S$, see
Groupoids, Example \ref{groupoids-example-roots-of-unity}.
\end{example}
\begin{example}[Additive group algebraic space]
\label{example-additive-group}
Let $B \to S$ as in Section \ref{section-notation}.
Consider the functor which associates
to any scheme $T$ over $B$ the group $\Gamma(T, \mathcal{O}_T)$
of global sections of the structure sheaf.
This is representable by the group algebraic space
$$
\mathbf{G}_{a, B} = B \times_S \mathbf{G}_{a, S}
$$
over $B$. Here $\mathbf{G}_{a, S}$ is the additive group scheme
over $S$, see
Groupoids, Example \ref{groupoids-example-additive-group}.
\end{example}
\begin{example}[General linear group algebraic space]
\label{example-general-linear-group}
Let $B \to S$ as in Section \ref{section-notation}.
Let $n \geq 1$.
Consider the functor which associates
to any scheme $T$ over $B$ 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 group algebraic space
$$
\text{GL}_{n, B} = B \times_S \text{GL}_{n, S}
$$
over $B$. Here $\mathbf{G}_{m, S}$ is the general linear group scheme
over $S$, see
Groupoids, Example \ref{groupoids-example-general-linear-group}.
\end{example}
\begin{example}
\label{example-determinant}
Let $B \to S$ as in Section \ref{section-notation}.
Let $n \geq 1$.
The determinant defines a morphisms of group algebraic spaces
$$
\det : \text{GL}_{n, B} \longrightarrow \mathbf{G}_{m, B}
$$
over $B$. It is the base change of the determinant morphism
over $S$ from
Groupoids, Example \ref{groupoids-example-determinant}.
\end{example}
\begin{example}[Constant group algebraic space]
\label{example-constant-group}
Let $B \to S$ as in Section \ref{section-notation}.
Let $G$ be an abstract group. Consider the functor
which associates to any scheme $T$ over $B$ 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
group algebraic space
$$
G_B = B \times_S G_S
$$
over $B$. Here $G_S$ is the constant group scheme introduced in
Groupoids, Example \ref{groupoids-example-constant-group}.
\end{example}
\section{Actions of group algebraic spaces}
\label{section-action-group-space}
\noindent
Please refer to
Groupoids, Section \ref{groupoids-section-action-group-scheme}
for notation.
\begin{definition}
\label{definition-action-group-space}
Let $B \to S$ as in Section \ref{section-notation}.
Let $(G, m)$ be a group algebraic space over $B$.
Let $X$ be an algebraic space over $B$.
\begin{enumerate}
\item An {\it action of $G$ on the algebraic space $X/B$} is
a morphism $a : G \times_B X \to X$ over $B$ such that
for every scheme $T$ over $B$ 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 algebraic spaces over $B$ 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 algebraic spaces over $B$ such
that for every $T$ over $B$ 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_B G \times_B X \ar[r]_-{1_G \times a} \ar[d]_{m \times 1_X} &
G \times_B X \ar[d]^a \\
G \times_B X \ar[r]^a & X
}
\quad
\xymatrix{
G \times_B 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
$$
\xymatrix{
G \times_B X \ar[r]_-{\text{id} \times f} \ar[d]_a &
G \times_B Y \ar[d]^a \\
X \ar[r]^f & Y
}
$$
commutes.
\begin{definition}
\label{definition-free-action}
Let $B \to S$, $G \to B$, and $X \to B$ as in
Definition \ref{definition-action-group-space}.
Let $a : G \times_B X \to X$ be an action of $G$ on $X/B$.
We say the action is {\it free} if for every scheme $T$ over $B$
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
$$
G \times_B X \to X \times_B X, \quad (g, x) \mapsto (a(g, x), x)
$$
is a monomorphism of algebraic spaces.
\end{lemma}
\begin{proof}
Immediate from the definitions.
\end{proof}
\section{Principal homogeneous spaces}
\label{section-principal-homogeneous}
\noindent
This section is the analogue of
Groupoids, Section \ref{groupoids-section-principal-homogeneous}.
We suggest reading that section first.
\begin{definition}
\label{definition-pseudo-torsor}
Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.
Let $(G, m)$ be a group algebraic space over $B$.
Let $X$ be an algebraic space over $B$, and let
$a : G \times_B X \to X$ 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 $G \times_B X \to X \times_B X$,
$(g, x) \mapsto (a(g, x), x)$ is an isomorphism.
\item A pseudo $G$-torsor $X$ is called {\it trivial} if there exists
an $G$-equivariant isomorphism $G \to X$ over $B$ where $G$ acts on
$G$ by left multiplication.
\end{enumerate}
\end{definition}
\noindent
It is clear that if $B' \to B$ is a morphism of algebraic spaces then
the pullback $X_{B'}$ of a pseudo $G$-torsor over $B$ is a
pseudo $G_{B'}$-torsor over $B'$.
\begin{lemma}
\label{lemma-characterize-trivial-pseudo-torsors}
In the situation of
Definition \ref{definition-pseudo-torsor}.
\begin{enumerate}
\item The algebraic space $X$ is a pseudo $G$-torsor if and only if for
every scheme $T$ over $B$ 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
$X \to B$ 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 $B$ be an algebraic space over $S$.
Let $(G, m)$ be a group algebraic space over $B$.
Let $X$ be a pseudo $G$-torsor over $B$.
\begin{enumerate}
\item We say $X$ is a
{\it principal homogeneous space}, or more precisely a
{\it principal homogeneous $G$-space over $B$}
if there exists a fpqc covering\footnote{The default type of torsor in
Groupoids, Definition \ref{groupoids-definition-principal-homogeneous-space}
is a pseudo torsor which is trivial on an fpqc covering.
Since $G$, as an algebraic space, can be seen a sheaf of groups
there already is a notion of a $G$-torsor which corresponds
to fppf-torsor, see
Lemma \ref{lemma-torsor}.
Hence we use ``principal homogeneous space'' for a pseudo torsor which
is fpqc locally trivial, and we try to avoid using the word torsor in
this situation.}
$\{B_i \to B\}_{i \in I}$ such that each
$X_{B_i} \to B_i$ has a section (i.e., is a trivial pseudo $G_{B_i}$-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 $\{B_i \to B\}_{i \in I}$
such that each $X_{B_i} \to B_i$ 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 $G$-principal homogeneous space over $B$''
to indicate that $X$ is an algebraic space over $B$ equipped with an
action of $G$ which turns it into a principal homogeneous space over $B$.
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 algebraic space over $S$.
Let $X$ be an algebraic space over $S$, and let
$a : G \times_S X \to X$ be an action of $G$ on $X$.
Then
$X$ is a $G$-torsor in the $fppf$-topology in the sense of
Definition \ref{definition-principal-homogeneous-space}
if and only if
$X$ is a $G$-torsor on $(\Sch/S)_{fppf}$
in the sense of
Cohomology on Sites, Definition \ref{sites-cohomology-definition-torsor}.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\section{Equivariant quasi-coherent sheaves}
\label{section-equivariant}
\noindent
Please compare with
Groupoids, Section \ref{groupoids-section-equivariant}.
\begin{definition}
\label{definition-equivariant-module}
Let $B \to S$ as in Section \ref{section-notation}.
Let $(G, m)$ be a group algebraic space over $B$, and
let $a : G \times_B X \to X$ be an action of $G$
on the algebraic space $X$ over $B$.
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 $\mathcal{O}_{G \times_B X}$-module
map
$$
\alpha : a^*\mathcal{F} \longrightarrow \text{pr}_1^*\mathcal{F}
$$
where $\text{pr}_1 : G \times_B X \to X$ is the projection
such that
\begin{enumerate}
\item the diagram
$$
\xymatrix{
(1_G \times a)^*\text{pr}_2^*\mathcal{F} \ar[r]_-{\text{pr}_{12}^*\alpha} &
\text{pr}_2^*\mathcal{F} \\
(1_G \times a)^*a^*\mathcal{F} \ar[u]^{(1_G \times a)^*\alpha} \ar@{=}[r] &
(m \times 1_X)^*a^*\mathcal{F} \ar[u]_{(m \times 1_X)^*\alpha}
}
$$
is a commutative in the category of
$\mathcal{O}_{G \times_B G \times_B X}$-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 $B \to S$ as in Section \ref{section-notation}.
Let $G$ be a group algebraic space over $B$.
Let $f : X \to Y$ be a $G$-equivariant morphism between
algebraic spaces over $B$ 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 in algebraic spaces}
\label{section-groupoids}
\noindent
Please refer to
Groupoids, Section \ref{groupoids-section-groupoids}
for notation.
\begin{definition}
\label{definition-groupoid}
Let $B \to S$ as in Section \ref{section-notation}.
\begin{enumerate}
\item A {\it groupoid in algebraic spaces over $B$} is a
quintuple $(U, R, s, t, c)$ where
$U$ and $R$ are algebraic spaces over $B$, and
$s, t : R \to U$ and $c : R \times_{s, U, t} R \to R$
are morphisms of algebraic spaces over $B$ with the
following property: For any scheme $T$ over $B$ the quintuple
$$
(U(T), R(T), s, t, c)
$$
is a groupoid category.
\item A {\it morphism
$f : (U, R, s, t, c) \to (U', R', s', t', c')$
of groupoids in algebraic spaces over $B$} is given by morphisms
of algebraic spaces $f : U \to U'$ and $f : R \to R'$ over $B$
with the following property: For any scheme
$T$ over $B$ 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 in algebraic spaces over $B$.
Note that there are unique morphisms of algebraic spaces
$e : U \to R$ and $i : R \to R$ over $B$ such that for every scheme $T$
over $B$ 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) of
Groupoids, Section \ref{groupoids-section-groupoids}.
Conversely given a septuple with this property the quintuple $(U, R, s, t, c)$
is a groupoid in algebraic spaces over $B$. Note that $i$ is an isomorphism,
and $e$ is a section of both $s$ and $t$.
Moreover, given a groupoid in algebraic spaces over $B$ we denote
$$
j = (t, s) : R \longrightarrow U \times_B U
$$
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 in algebraic spaces over $B$'' to stress the existence of identity and
inverse.
\begin{lemma}
\label{lemma-groupoid-pre-equivalence}
Let $B \to S$ as in Section \ref{section-notation}.
Given a groupoid in algebraic spaces $(U, R, s, t, c)$ over $B$
the morphism $j : R \to U \times_B U$ 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}
Let $B \to S$ as in Section \ref{section-notation}.
Given an equivalence relation $j : R \to U$ over $B$
there is a unique way to extend it to a groupoid in algebraic spaces
$(U, R, s, t, c)$ over $B$.
\end{lemma}
\begin{proof}
Omitted.
This is a nice exercise in the definitions.
\end{proof}
\begin{lemma}
\label{lemma-diagram}
Let $B \to S$ as in Section \ref{section-notation}.
Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.
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 $B \to S$ be as in Section \ref{section-notation}.
Let $(U, R, s, t, c, e, i)$ be a groupoid in algebraic spaces over $B$.
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 \times 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
Please compare with
Groupoids, Section \ref{groupoids-section-groupoids-quasi-coherent}.
\begin{definition}
\label{definition-groupoid-module}
Let $B \to S$ as in Section \ref{section-notation}.
Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.
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 $B \to S$ as in Section \ref{section-notation}.
Consider a morphism
$f : (U, R, s, t, c) \to (U', R', s', t', c')$
of groupoid in algebraic spaces over $B$. 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-colimits}
Let $B \to S$ be as in Section \ref{section-notation}.
Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.
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
Properties of Spaces, Lemma
\ref{spaces-properties-lemma-properties-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 $B \to S$ as in Section \ref{section-notation}.
Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.
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
$$
0 \to s^*\text{Ker}(\varphi)
\to s^*\mathcal{F} \to s^*\mathcal{G} \to s^*\text{Coker}(\varphi) \to 0
$$
is exact and similarly for pullback by $t$. Hence $\alpha$ and $\beta$
induce isomorphisms
$\kappa : t^*\text{Ker}(\varphi) \to s^*\text{Ker}(\varphi)$ and
$\lambda : t^*\text{Coker}(\varphi) \to s^*\text{Coker}(\varphi)$