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Then $G \to \Spec k$ is separated. +\end{lemma} + +\begin{proof} + Let $\pi \colon G \to S$ the structure morphism. Then + $\pi$ is separated if and only if $e\colon S \to G$ is a closed immersion. For + any $x \in \mathrm{im}(e) \in G$, choose an affine open neighbourhood + $x \in U = \Spec A \subseteq G$. + Then $\pi|_{U} \circ e = \mathrm{id}_S$, hence the induced map + $A \xrightarrow{\Gamma(e)} k$ has a section $\Gamma(\pi|_U)$ and is therefore + surjective. Thus $e$ is a closed immersion. +\end{proof} + +\begin{satz} + Let $G$ be a group scheme locally of finite type over $k$. Then + $G$ is smooth over $k$ if and only if $G$ is geometrically reduced. +\end{satz} + +\begin{proof} + The first direction is immediate, since smoothness is invariant under base change and + smooth over a field implies reduced. + Conversely, for any field extension $\ell / k$ by a prior result + $G$ is smooth over $k$ if and only if $G$ is smooth over $\ell$. Thus + we may assume $k = \bar k$. By \ref{idk} and \ref{idk}, we obtain + $G_{\mathrm{sm}} \neq \emptyset$. By the transitive action + of $G(k)$ on $G$, every closed point is smooth. Since + \[ + G_{(0)} = \{ g \in G \mid \mathrm{dim} \overline{\{g\}} = 0 \} + \] is very dense in $G$ and $G_{\mathrm{sm}} \subseteq G$ is open, the result follows. +\end{proof} + +\begin{lemma} + Let $k$ be perfect and $G$ a group scheme locally of finite type over $k$. Then + the induced reduced subscheme $G_{\mathrm{red}}$ is a subgroup scheme of $G$. +\end{lemma} + +\begin{proof} + Since $(-)_{\mathrm{red}}$ is a functor, we obtain + $i\colon G_{\mathrm{red}} \to G_{\mathrm{red}}$ and + $e\colon S \to G_{\mathrm{red}}$. By \ref{idk}, + reduced is equivalent to geometrically reduced since $k$ is perfect. Thus + $G_{\mathrm{red}} \times_k G_{\mathrm{red}}$ is reduced and we obtain + \[ + \begin{tikzcd} + G x_k G \arrow{r}{m} & G \\ + G_{\mathrm{red}} \times_k G_{\mathrm{red}} \arrow{u} + \arrow[dashed]{r} & G_{\mathrm{red}} \arrow{u} + \end{tikzcd} + .\] +\end{proof} + +\begin{korollar} + If $k$ is perfect and $G$ a group scheme locally of finite type over $k$. Then + $G_{\mathrm{red}}$ is smooth over $k$. +\end{korollar} + +\begin{lemma} + Let $G$ be locally of finite type over $k$. Then $G$ is geometrically irreducible + if (and only if) $G$ is connected. +\end{lemma} + +\begin{proof} + Since $G(k) \neq \emptyset$, we have a morphism + $\Spec k \to G$ and $\Spec k$ is geometrically connected. Thus $G$ is geometrically connected. + We may therefore assume $k = \bar k$. Since the statement is purely topological, we may + further assume that $G$ is reduced and thus smooth over $k$. Hence + $G$ is regular by \ref{idk}, in particular for every $g \in G$ the local ring + $\mathcal{O}_{G,g}$ is regular and hence an integral domain. Since $G$ is locally noetherian + and connected, the claim follows. +\end{proof} + +\begin{definition} + An \emph{abelian variety} over $k$ is a connected, geometrically reduced + and proper $k$-group scheme. +\end{definition} + +\begin{bem} + Abelian varieties are smooth and geometrically integral. +\end{bem} + +\begin{bsp} + Elliptic curves are abelian varieties of dimension $1$. +\end{bsp} + +The goal is now to show that abelian varieties are commutative group schemes. + +\begin{lemma} + Let $X$ be a proper, geometrically connected and geometrically reduced $k$-scheme and + $Y$ an affine $k$-scheme. Then every morphism $X \xrightarrow{f} Y$ factors over a + $k$-valued point of $Y$. + \label{lemma:constant-of-proper-conn-irred-affine} +\end{lemma} + +\begin{proof} + By the Liouville theorem for schemes, the global + sections of $\mathcal{O}_{X_{\bar k}}$ is $\bar k$. Since + $k \to \bar k$ is flat, we obtain + \[ + \Gamma(X, \mathcal{O}_X) \otimes_k \bar k + \xlongrightarrow{\simeq} \Gamma(X_{\bar k}, \mathcal{O}_{X_{\bar k}}) + .\] Since $k \to \bar k$ is even faithfully flat, we obtain + $\Gamma(X, \mathcal{O}_X) \simeq k$. + + Choose an embedding $Y \hookrightarrow \mathbb{A}_k^{(I)}$. Then a + morphism $f\colon X \to Y$ is equivalent to a morphism + $X \xrightarrow{f} Y \hookrightarrow \mathbb{A}_k^{(I)}$, which is equivalent + to the datum of a family of $e_i \in \Gamma(X, \mathcal{O}_X)$ which + corresponds to a morphism + $\Spec k \xrightarrow{e} \mathbb{A}_k^{(I)}$. Thus by construction we obtain + a factorisation + \[ + \begin{tikzcd} + X \arrow{r}{f} \arrow[dashed]{d} & Y \arrow{r} & \mathbb{A}^{(I)} \\ + \Spec k \arrow{rru} + \end{tikzcd} + \] where the dashed arrow is induced from the isomorphism $\Gamma(X, \mathcal{O}_X) \simeq k$. +\end{proof} + +\begin{lemma}[Rigidity] + Let $X$ be a geometrically reduced, geometrically connected and proper $k$-scheme + with $X(k) \neq \emptyset$. Let further $Y$ be an integral scheme over $k$, $Z$ + be a separated $k$-scheme and $f\colon X \times_k Y \to Z$ a morphism such that + there exists $y \in Y(k)$ such that + $f|_{X_{y}}$ factors via a $k$-point $z \in Z(k)$. Then + $f$ factors via $\mathrm{pr}_2$. + \label{lemma:rigidity} +\end{lemma} + +\begin{proof} + Consider the composition + \[ + g\colon X \times_k Y \xrightarrow{pr_2} Y \simeq \Spec k \times_k Y + \xrightarrow{(x_0, \mathrm{id})} X \times_k Y \xrightarrow{f} Z + \] where $x_0$ is an arbitrarily chosen $k$-rational point of $X$. + It remains to show that $f = g$. Choose an open affine + neighbourhood $z \in U \subseteq Z$. Then + $X_y = \mathrm{pr}_2^{-1}(y) \subseteq f^{-1}(U)$. Since + $X$ is proper, $\mathrm{pr}_2$ is a closed map. Thus there + exists a $y \in V \subseteq Y$ open + with $\mathrm{pr}_2^{-1}(V) \subseteq f^{-1}(U)$. For + any $y' \in V$, we obtain + \[ + \begin{tikzcd} + X \times_k Y \arrow{r}{f} & Z \\ + X_{y'} \arrow[dashed, swap]{d}{\alpha(y')} + \arrow[hookrightarrow]{u} \arrow[dashed]{r} & U \arrow[hookrightarrow]{u} \\ + U \times_k \kappa(y') \arrow{ur} + \end{tikzcd} + .\] By \ref{lemma:constant-of-proper-conn-irred-affine}, the morphism + $\alpha(y')$ factors over a $\kappa(y')$-valued point. Thus + $f$ and $g$ agree on the dense open subset $X \times_k V$. By reduced-to-separated, + the result follows. +\end{proof} + +\begin{korollar} + Let $A$ and $B$ be abelian varieties over $k$ + and $f$ a morphism of $k$-schemes $A \to B$. If under the induced + map $f(k)\colon A(k) \to B(k)$ the identity $e_A$ is mapped to $e_B$. + \label{cor:av-group-homs} +\end{korollar} + +\begin{proof} + Consider the composition + \[ + g\colon A \times_k A \xrightarrow{(f \circ m_A) \times (i_B \circ m_A \circ (f \times f))} + B \times_k B + \xrightarrow{m_B} + B + .\] It remains to show that the image of $g$ is precisely $\{e_B\} $. By + assumption $f(e_A) = e_B$ and thus + \[ + g(\{e_A\} \times_k A) = \{ e_B\} = g(A \times_k \{e_A\}) + .\] By repeated application of \ref{lemma:rigidity}, $g$ factors + via $\mathrm{pr}_1$ and $\mathrm{pr}_2$. Thus $g$ is constant and $e_B$ is in the image. +\end{proof} + +\begin{korollar} + Every abelian variety is commutative. +\end{korollar} + +\begin{proof} + Apply \ref{cor:av-group-homs} on $i\colon A \to A$. +\end{proof} + +\end{document}