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 \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.