diff --git a/notes.cls b/notes.cls index d6f3165..75540bc 100644 --- a/notes.cls +++ b/notes.cls @@ -41,6 +41,41 @@ }, } +\usetikzlibrary{quotes, angles, math} +\pgfplotsset{ + compat=1.15, + axis lines = middle, + ticks = none, + %default 2d plot/.style={% + % ticks=none, + % axis lines = middle, + % grid=both, + % minor tick num=4, + % grid style={line width=.1pt, draw=gray!10}, + % major grid style={line width=.2pt,draw=gray!50}, + % axis lines=middle, + % enlargelimits={abs=0.2} +} + +\newcounter{curve} + +\NewDocumentCommand{\algebraiccurve}{ O{} O{$#5 = 0$} O{-4:4} O{-4:4} m }{ + \addplot[id=curve\arabic{curve}, raw gnuplot, smooth, #1] function{% + f(x,y) = #5; + set xrange [#3]; + set yrange [#4]; + set view 0,0; + set isosample 1000,1000; + set size square; + set cont base; + set cntrparam levels incre 0,0.1,0; + unset surface; + splot f(x,y) + }; + \addlegendentry{#2} + \stepcounter{curve} +}% + % PAGE GEOMETRY \geometry{ top=1.2in,bottom=1.4in,left=1.3in,right=1.3in, diff --git a/ws2022/rav/lecture b/ws2022/rav/lecture index 868e4ca..40409ed 160000 --- a/ws2022/rav/lecture +++ b/ws2022/rav/lecture @@ -1 +1 @@ -Subproject commit 868e4cab60566f5d8dfaeb565c65f4041751c40a +Subproject commit 40409ed872ac4db0567ba9fac20e2dd955744e27 diff --git a/ws2024/gruppenschemata/lec.pdf b/ws2024/gruppenschemata/lec.pdf new file mode 100644 index 0000000..66b3059 Binary files /dev/null and b/ws2024/gruppenschemata/lec.pdf differ diff --git a/ws2024/gruppenschemata/lec.tex b/ws2024/gruppenschemata/lec.tex new file mode 100644 index 0000000..8ec6e75 --- /dev/null +++ b/ws2024/gruppenschemata/lec.tex @@ -0,0 +1,24 @@ +\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} diff --git a/ws2024/gruppenschemata/lec01.pdf b/ws2024/gruppenschemata/lec01.pdf new file mode 100644 index 0000000..1e9afa5 Binary files /dev/null and b/ws2024/gruppenschemata/lec01.pdf differ diff --git a/ws2024/gruppenschemata/lec01.tex b/ws2024/gruppenschemata/lec01.tex new file mode 100644 index 0000000..a8cade8 --- /dev/null +++ b/ws2024/gruppenschemata/lec01.tex @@ -0,0 +1,226 @@ +\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} diff --git a/ws2024/gruppenschemata/lecture.cls b/ws2024/gruppenschemata/lecture.cls new file mode 100644 index 0000000..e604222 --- /dev/null +++ b/ws2024/gruppenschemata/lecture.cls @@ -0,0 +1,280 @@ +\ProvidesClass{lecture} +\LoadClass[a4paper]{book} + +\RequirePackage{faktor} +\RequirePackage{xparse} +\RequirePackage{stmaryrd} +\RequirePackage[utf8]{inputenc} +\RequirePackage[T1]{fontenc} +\RequirePackage{textcomp} +\RequirePackage{babel} +\RequirePackage{amsmath, amssymb, amsthm} +\RequirePackage{mdframed} +\RequirePackage{tikz-cd} +\RequirePackage{geometry} +\RequirePackage{import} +\RequirePackage{pdfpages} +\RequirePackage{transparent} +\RequirePackage{xcolor} +\RequirePackage{array} +\RequirePackage[shortlabels]{enumitem} +\RequirePackage{tikz} +\RequirePackage{pgfplots} +\RequirePackage[pagestyles, nobottomtitles]{titlesec} +\RequirePackage{listings} +\RequirePackage{mathtools} +\RequirePackage{forloop} +\RequirePackage{totcount} +\RequirePackage[hidelinks, unicode]{hyperref} %[unicode, hidelinks]{hyperref} +\RequirePackage{bookmark} +\RequirePackage{wasysym} +\RequirePackage{environ} +\RequirePackage{stackrel} +\RequirePackage{subcaption} + +\usetikzlibrary{quotes, angles, math} +\pgfplotsset{ + compat=1.15, + axis lines = middle, + ticks = none, + %default 2d plot/.style={% + % ticks=none, + % axis lines = middle, + % grid=both, + % minor tick num=4, + % grid style={line width=.1pt, draw=gray!10}, + % major grid style={line width=.2pt,draw=gray!50}, + % axis lines=middle, + % enlargelimits={abs=0.2} +} + +\newcounter{curve} + +\NewDocumentCommand{\algebraiccurve}{ O{} O{$#5 = 0$} O{-4:4} O{-4:4} m }{ + \addplot[id=curve\arabic{curve}, raw gnuplot, smooth, #1] function{% + f(x,y) = #5; + set xrange [#3]; + set yrange [#4]; + set view 0,0; + set isosample 1000,1000; + set size square; + set cont base; + set cntrparam levels incre 0,0.1,0; + unset surface; + splot f(x,y) + }; + \addlegendentry{#2} + \stepcounter{curve} +}% +%\newcommand{\algebraiccurve}[3][][hi]{% +% %\addlegendentry{#2} +% \stepcounter{curve} +%}% + +\geometry{ + bottom=35mm +} + +%\DeclareOption*{\PassOptionsToClass{\CurrentOption}{article}} +\DeclareOption{uebung}{ + \makeatletter + \lhead{\@title} + \rhead{\@author} + \makeatother +} +\ProcessOptions\relax + +% PARAGRAPH no indent but skip +%\setlength{\parskip}{3mm} +%\setlength{\parindent}{0mm} + +\newtheorem{satz}{Proposition}[chapter] +\newtheorem{theorem}[satz]{Theorem} +\newtheorem{lemma}[satz]{Lemma} +\newtheorem{korollar}[satz]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[satz]{Definition} + +\newtheorem{bsp}[satz]{Example} +\newtheorem{bem}[satz]{Remark} +\newtheorem{aufgabe}[satz]{Exercise} + +\counterwithin{figure}{chapter} + +% enable aufgaben counting +%\regtotcounter{aufgabe} + +\newcommand{\N}{\mathbb{N}} +\newcommand{\R}{\mathbb{R}} +\newcommand{\Z}{\mathbb{Z}} +\newcommand{\Q}{\mathbb{Q}} +\newcommand{\C}{\mathbb{C}} + +% I(V(I)) + +\newcommand{\ivkr}{\mathcal{I}_k(\mathcal{V}_{(k^{(r)})^n}(I))} + +% HEADERS + +%\newpagestyle{main}[\small]{ +% \setheadrule{.55pt}% +% \sethead[\thepage]% even-left +% []% even-center +% [\thechapter~\chaptertitle]% even-right +% {\thesection~\sectiontitle}% odd-left +% {}% odd-center +% {\thepage}% odd-right +%} +%\pagestyle{main} + +\newcommand{\incfig}[1]{% + \def\svgwidth{\columnwidth} + \import{./figures/}{#1.pdf_tex} +} +\pdfsuppresswarningpagegroup=1 + +% horizontal rule +\newcommand\hr{ + \noindent\rule[0.5ex]{\linewidth}{0.5pt} +} + +% code listings, define style +\lstdefinestyle{mystyle}{ + commentstyle=\color{gray}, + keywordstyle=\color{blue}, + numberstyle=\tiny\color{gray}, + stringstyle=\color{black}, + basicstyle=\ttfamily\footnotesize, + breakatwhitespace=false, + breaklines=true, + captionpos=b, + keepspaces=true, + numbers=left, + numbersep=5pt, + showspaces=false, + showstringspaces=false, + showtabs=false, + tabsize=2 +} + +% activate my colour style +\lstset{style=mystyle} + +% better stackrel +\let\oldstackrel\stackrel +\renewcommand{\stackrel}[3][]{% + \oldstackrel[\mathclap{#1}]{\mathclap{#2}}{#3} +}% + +% integral d sign +\makeatletter \renewcommand\d[2][]{\ensuremath{% + \,\mathrm{d}^{#1}#2\@ifnextchar^{}{\@ifnextchar\d{}{\,}}}} +\makeatother + +% remove page before chapters +\let\cleardoublepage=\clearpage + +%josua +\newcommand{\norm}[1]{\left\Vert#1\right\Vert} + +% contradiction +\newcommand{\contr}{\text{\Large\lightning}} + +% people seem to prefer varepsilon over epsilon +\renewcommand{\epsilon}{\varepsilon} + +\ExplSyntaxOn + +% S-tackrelcompatible ALIGN environment +% some might also call it the S-uper ALIGN environment +% uses regular expressions to calculate the widest stackrel +% to put additional padding on both sides of relation symbols +\NewEnviron{salign} +{ + \begin{align} + \lec_insert_padding:V \BODY + \end{align} +} +% starred version that does no equation numbering +\NewEnviron{salign*} +{ + \begin{align*} + \lec_insert_padding:V \BODY + \end{align*} +} + +% some helper variables +\tl_new:N \l__lec_text_tl +\seq_new:N \l_lec_stackrels_seq +\int_new:N \l_stackrel_count_int +\int_new:N \l_idx_int +\box_new:N \l_tmp_box +\dim_new:N \l_tmp_dim_a +\dim_new:N \l_tmp_dim_b +\dim_new:N \l_tmp_dim_c +\dim_new:N \l_tmp_dim_needed + +% function to insert padding according to widest stackrel +\cs_new_protected:Nn \lec_insert_padding:n + { + \tl_set:Nn \l__lec_text_tl { #1 } + % get all stackrels in this align environment + \regex_extract_all:nnN { \c{stackrel}(\[.*?\])?{(.*?)}{(.*?)} } { #1 } \l_lec_stackrels_seq + % get number of stackrels + \int_set:Nn \l_stackrel_count_int { \seq_count:N \l_lec_stackrels_seq } + \int_set:Nn \l_idx_int { 1 } + \dim_set:Nn \l_tmp_dim_needed { 0pt } + % iterate over stackrels + \int_while_do:nn { \l_idx_int <= \l_stackrel_count_int } + { + % calculate width of text + \hbox_set:Nn \l_tmp_box {$\seq_item:Nn \l_lec_stackrels_seq { \l_idx_int + 1 }$} + \dim_set:Nn \l_tmp_dim_a {\box_wd:N \l_tmp_box} + \hbox_set:Nn \l_tmp_box {$\seq_item:Nn \l_lec_stackrels_seq { \l_idx_int + 2 }$} + \dim_set:Nn \l_tmp_dim_c {\box_wd:N \l_tmp_box} + \dim_set:Nn \l_tmp_dim_a {\dim_max:nn{ \l_tmp_dim_c} {\l_tmp_dim_a}} + % calculate width of relation symbol + \hbox_set:Nn \l_tmp_box {$\seq_item:Nn \l_lec_stackrels_seq { \l_idx_int + 3 }$} + \dim_set:Nn \l_tmp_dim_b {\box_wd:N \l_tmp_box} + % check if 0.5*(a-b) > minimum padding, if yes updated minimum padding + \dim_compare:nNnTF + { 1pt * \dim_ratio:nn { \l_tmp_dim_a - \l_tmp_dim_b } { 2pt } } > { \l_tmp_dim_needed } + { \dim_set:Nn \l_tmp_dim_needed { 1pt * \dim_ratio:nn { \l_tmp_dim_a - \l_tmp_dim_b } { 2pt } } } + { } + % increment list index by three, as every stackrel produces three list entries + \int_incr:N \l_idx_int + \int_incr:N \l_idx_int + \int_incr:N \l_idx_int + \int_incr:N \l_idx_int + } + % replace all relations with align characters (&) and add the needed padding + \regex_replace_all:nnN + { (\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}&|>&|&>|<&|&<) } + { \c{kern} \u{l_tmp_dim_needed} \1 \c{kern} \u{l_tmp_dim_needed} } + \l__lec_text_tl + \l__lec_text_tl + } +\cs_generate_variant:Nn \lec_insert_padding:n { V } + +\NewEnviron{leftright} +{ + \lec_replace_parens:V \BODY +} + +% function to replace parens with left right +\cs_new_protected:Nn \lec_replace_parens:n + { + \tl_set:Nn \l__lec_text_tl { #1 } + % replace all parantheses with \left( \right) + \regex_replace_all:nnN { \( } { \c{left}( } \l__lec_text_tl + \regex_replace_all:nnN { \) } { \c{right}) } \l__lec_text_tl + \regex_replace_all:nnN { \[ } { \c{left}[ } \l__lec_text_tl + \regex_replace_all:nnN { \] } { \c{right}] } \l__lec_text_tl + \l__lec_text_tl + } +\cs_generate_variant:Nn \lec_replace_parens:n { V } + +\ExplSyntaxOff + +% add one equation tag to the current line to otherwise unnumbered environment +\newcommand{\tageq}{\stepcounter{equation}\tag{\theequation}}