\documentclass{lecture}

\title{Groupschemes}

\begin{document}

\chapter{Introduction}

Literature: Görtz-Wedhorn: Algebraic Geometry I and II

The goal of this lecture is a brief introduction to the theory of group schemes.

\begin{definition}[Group object]
    Let $\mathcal{C}$ be a category with finite products. A \emph{group object in $\mathcal{C}$} is the
    data $(G, m, e, i)$ where
    \begin{itemize}
        \item $G$ is an object of $\mathcal{C}$
        \item $m\colon G \times G \to G$ is the multiplication map
        \item $e\colon 1 \to G$ is the unit
        \item $i\colon G \to G$ is the inversion map
    \end{itemize}
    such that the following diagrams commute
    \[
    \begin{tikzcd}
        G \times G \times G \arrow{r}{m \times \text{id}} \arrow{d}{\text{id} \times m} &
        G \times G \arrow{d}{m} \\
        G \times G \arrow{r}{m} & G
    \end{tikzcd}, \quad
    \begin{tikzcd}
        G \times G \arrow{r}{m} & G \arrow{dl} \\
        G \times 1 \arrow{u}{\text{id} \times e}
    \end{tikzcd}
    \text{ and }
    \begin{tikzcd}
        G \arrow{r}{\text{id} \times i} \arrow{d} & G \times G \arrow{d}{m} \\
        1 \arrow{r}{e} & G
    \end{tikzcd}
    .\] $G$ is called \emph{commutative}, if additionally the diagram
    \[
    \begin{tikzcd}
        G \times G \arrow{d}{m} \arrow{r}{\text{swap}} & G \times G \arrow{dl}{m} \\
        G &
    \end{tikzcd}
    \] commutes.

    A \emph{morphism of group objects} $(G, m, e, i) \to (G', m', e', i')$
    is a morphism $f\colon G \to G'$ in $\mathcal{C}$ such that the diagrams
    \[
    \begin{tikzcd}
        G \times G \arrow{r}{f \times f} \arrow{d}{m} & G' \times G' \arrow{d}{m'} \\
        G \arrow{r}{f} & G'
    \end{tikzcd}
    , \quad
    \begin{tikzcd}
        G \arrow{r}{f} & G' \\
        1 \arrow{u}{e} \arrow[swap]{ur}{e'}
    \end{tikzcd}
    \text{ and }
    \begin{tikzcd}
        G \arrow{d}{i} \arrow{r}{f} & G' \arrow{d}{i'} \\
        G \arrow{r}{f} & G'
    \end{tikzcd}
    .\]
    This defines the category $\operatorname{Grp}(\mathcal{C})$ of group objects in $\mathcal{C}$.
\end{definition}

\begin{bsp}[]
    \begin{itemize}
        \item $\operatorname{Grp}(\text{Set}) \simeq \mathrm{Grp}$
        \item $\operatorname{Grp}(\text{Grp}) \simeq \mathrm{Ab}$
        \item $\operatorname{Grp}(\text{Ab}) \simeq ?$
        \item $\operatorname{Grp}(Top) \simeq \text{topological Groups}$
        \item $\operatorname{Grp}(Mfd_{\infty}) \simeq \text{Lie Groups}$
    \end{itemize}
\end{bsp}

\begin{definition}[group scheme]
    Let $S$ be a scheme.
    An \emph{S-group scheme} or an \emph{S-group} is a group object in the category of schemes over $S$.
\end{definition}

\begin{bem}
    Let $S$ be a scheme. The structure of a group scheme over $S$ on a $S$-scheme $G$ is equivalent to a
    factorisation of the functor of points
    \[
    \begin{tikzcd}
        \mathrm{Sch}_S \arrow{r} \arrow[dashed]{d} & \mathrm{Set} \\
        \mathrm{Grp} \arrow{ur}
    \end{tikzcd}
    \] via the forgetful functor from groups to sets.
\end{bem}

\begin{bsp}
    Let $S$ be a scheme.
    \begin{enumerate}[(i)]
        \item Let $\Gamma$ be a group. Then $G = \Gamma_S$ where
            $G(T) \coloneqq \{ \text{ locally constant maps $T \to \Gamma$ }\} $
        \item (additive group) $\mathbb{G}_{a, S}$ where $\mathbb{G}_{a,S}(T) = \mathcal{O}_T(T)$. We have
            $\mathbb{G}_{a, S} \simeq \mathbb{A}^{1}_S$.
        \item (multiplicative group) $\mathbb{G}_{m, S}$ where
            $\mathbb{G}_{m, S}(T) \coloneqq \mathcal{O}_{T}(T)^{\times}$.
        \item (roots of unity) $\mu_{n, S}$ ($n \ge 1$) where
            $\mu_{n,S}(T) = \{ x \in \mathcal{O}_T(T)^{\times }  \mid x^{n} = 1\}$.
        \item $S = \mathrm{Spec}(R)$. $\mathrm{GL}_{n,R} = \mathrm{Spec}(A)$ where
            $A = R[T_{ij}  \mid 1 \le i, j \le n][\mathrm{det}^{-1}]$ where
            $\mathrm{det} = \sum_{\sigma \in S_n} \mathrm{sgn}(\sigma) T_{1 \sigma(1)} \cdots T_{n \sigma(n)}$.
            We obtain $\mathrm{GL}_{n,S}$ by base changing $\mathrm{GL}_{n, \Z}$.
    \end{enumerate}
\end{bsp}

\begin{lemma}
    Let $G$ be a $S$-group. Then
    $G \to S$ is separated if and only if $S \xrightarrow{e} G$ is a closed immersion.
\end{lemma}

\begin{definition}[]
    Let $R$ be a ring. A \emph{(commutative) Hopf-Algebra} over $R$ is a group object
    in $\mathrm{Alg}_R^{\text{op}}$, where $\mathrm{Alg}_R = \mathrm{CRing}_R$.
\end{definition}

\begin{bem}
    For a $R$-Hopf-Algebra $A$, we denote the canonical maps by
    \begin{itemize}
        \item $\mu \colon A \to A \otimes_R A $ (Comultiplication)
        \item $\epsilon \colon A \to R$ (Counit)
        \item $\iota\colon A \to A$ (Antipode)
    \end{itemize}
    A Hopf-Algebra is called \emph{cocommutative}, if the associated group object in $\mathrm{Alg}_R^{\text{op}}$
    kommutativ ist.
\end{bem}

\begin{bem}
    For a ring $R$, by construction we have an equivalence of categories between
    the category of affine $R$-group schemes and the opposite category of $R$-Hopf-Algebras.
\end{bem}

\begin{bsp}[]
    The additive group $\mathbb{G}_{a,R} = \mathrm{Spec}(R[t])$
    has
    \begin{itemize}
        \item comultiplication $\mu\colon R[t] \to R[t] \otimes_R R[t], t \mapsto t \otimes 1 - 1 \otimes t$.
        \item counit $\epsilon\colon R[t] \to R, t \mapsto 0$
        \item antipode $\iota\colon R[t] \to R[t], t \mapsto -t$
    \end{itemize}
\end{bsp}

\begin{proof}
    For any $R$-algebra $A$ we have $\mathbb{G}_{a,R}(A) = A$ and the diagram
    \[
    \begin{tikzcd}
        \mathbb{G}_{a, R}(A) \times \mathbb{G}_{a, R}(A) \arrow{r}{m} \arrow{d}{\simeq}
        & \mathbb{G}_{a,R}(A) \arrow{d}{\simeq} \\
        \mathrm{Hom}_R(R[s_1, s_2], A) \arrow{r}{\mu^{*}} & \mathrm{Hom}_{R}(R[t], A)
    \end{tikzcd}
    .\] 
\end{proof}

\begin{definition}
    Let $G$ be a $S$-group. A \emph{subgroupscheme} of $G$ is a subscheme $H \subseteq G$ such that
    \begin{enumerate}[1)]
        \item for all $T \in \mathrm{Sch}_S$, we have $H(T) \subseteq G(T)$ a subgroup,
        \item We have commutative diagrams
            \[
            \begin{tikzcd}
                H \times_S H \arrow{r} \arrow[dashed]{d} & G \times_S G \arrow{r}{m} & G \\
                H \arrow{urr}
            \end{tikzcd}
            \text{ and }
            \begin{tikzcd}
                S \arrow{r}{e} \arrow[dashed]{d} & G \\
                H \arrow{ur}
            \end{tikzcd}
            \]
    \end{enumerate}
    A subgroup scheme $H \subseteq G $ is \emph{normal} if $H(T)$ is a normal subgroup of $G(T)$ for all
    $T \in \mathrm{Sch}_S$.

    For a morphism $f\colon G \to G'$ of $S$-groups and a subgroup $H' \subseteq G'$, let
    $f^{-1}(H')$ be $G \times_G' H$. For $H' = {1} \xrightarrow{e} G'$, we obtain the
    \emph{kernel of $f$} and the cartesian square
    \[
        \begin{tikzcd}
            \operatorname{Ker}(f) \arrow{r} \arrow{d} & G \arrow{d}{f} \\
            S \arrow{r}{e} & G'
        \end{tikzcd}
    .\] 
\end{definition}

\begin{bem}[]
    The kernel of a homomorphism $f$ of $S$-groups is for any $S$-scheme $T$ given by
    \[
    \text{Ker}(f)(T) = \text{ker}\left(f(T)\right) 
    .\] In particular, the $\text{Ker}(f)$ is normal.
\end{bem}

\begin{definition}
    Let $G$ be a $S$-group, $T$ a $S$-scheme and $g \in G(T) = \mathrm{Hom}_S(T, G)$. Define
    the \emph{lefttranslation by $g$} as
    \[
    \begin{tikzcd}
        G_T \arrow[dashed]{d}{t_g} & \arrow[swap]{l}{=} T \times_T G_T \arrow{d}{g \times \text{id}} \\
        G_T & \arrow{l}{m} G_T \times_T G_T
    \end{tikzcd}
    .\] 
    \label{def:left-translation}
\end{definition}

\begin{bem}
    In the situation of \ref{def:left-translation}, for every $T' \xrightarrow{f} T$, the map
    \[
    t_g(T')\colon G_T(T') = G(T') \longrightarrow G(T') = G_T(T')
    \] is the lefttranslation by the element $f^{*}(g) \in G(T')$.
\end{bem}

\begin{bem}
    Consider
    \[
    \begin{tikzcd}
        G \times_S G \arrow{d}{m} \arrow{r}{(g, h) \mapsto (gh, h)} & G \times_S G \arrow{dl}{\mathrm{pr}_1} \\
        G
    \end{tikzcd}
    .\] Let $\mathcal{P}$ be a property of morphisms stable under base change and composition with
    isomorphisms. Then whenever $G \to S$ satisfies $\mathcal{P}$, then $m$ satisfies $\mathcal{P}$.
\end{bem}

\end{document}