update groupschemes lecture

.gitmodules

@@ -13,3 +13,9 @@
 [submodule "ws2022/rav/lecture"]
 	path = ws2022/rav/lecture
 	url = git@git.mathi.uni-heidelberg.de:cmerten/ravlecture
+[submodule "ws2023/groupschemes"]
+	path = ws2023/groupschemes 
+	url = https://git.flavigny.de/christian/groupschemes-lecture
+[submodule "ws2023/groupschemes-lecture"]
+	path = ws2023/groupschemes-lecture
+	url = https://git.flavigny.de/christian/groupschemes-lecture

ws2023/gruppenschemata/lec.tex

@@ -1,24 +0,0 @@
-\documentclass{lecture}
-
-\usepackage{standalone}
-\usepackage{tikz}
-\usepackage{subcaption}
-
-\title{Groupschemes}
-\author{Christian Dahlhausen\\[5mm]
-Transcript of\\[1mm]
-Christian Merten (\href{mailto:cmerten@mathi.uni-heidelberg.de}{cmerten@mathi.uni-heidelberg.de})\\
-}
-\date{WiSe 2023}
-
-\begin{document}
-
-\newgeometry{right=15mm, left=15mm}
-\maketitle
-\restoregeometry
-
-\tableofcontents
-
-\input{lec01}
-
-\end{document}

ws2023/gruppenschemata/lec01.tex

@@ -1,226 +0,0 @@
-\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}

ws2023/gruppenschemata/lecture.cls

@@ -1,280 +0,0 @@
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-\DeclareOption{uebung}{
-    \makeatletter
-    \lhead{\@title}
-    \rhead{\@author}
-    \makeatother
-}
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-% PARAGRAPH no indent but skip
-%\setlength{\parskip}{3mm}
-%\setlength{\parindent}{0mm}
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-\newtheorem{satz}{Proposition}[chapter]
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-    \def\svgwidth{\columnwidth}
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-% S-tackrelcompatible ALIGN environment
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-% to put additional padding on both sides of relation symbols
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-{
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-% starred version that does no equation numbering
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-      \int_incr:N \l_idx_int
-  }
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-  { (\c{simeq}&|&\c{simeq}|\c{leq}&|&\c{leq}|\c{geq}&|&\c{geq}|\c{iff}&|&\c{iff}|\c{impliedby}&|&\c{impliedby}|\c{implies}&|&\c{implies}|\c{approx}&|&\c{approx}|\c{equiv}&|&\c{equiv}|=&|&=|\c{le}&|&\c{le}|\c{ge}&|&\c{ge}|&\c{stackrel}(\[.*?\])?{.*?}{.*?}|\c{stackrel}(\[.*?\])?{.*?}{.*?}&|&\c{neq}|\c{neq}&|>&|&>|<&|&<) }
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