groupschemes: first lecture

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,

ws2022/rav/lecture

@@ -1 +1 @@
-Subproject commit 868e4cab60566f5d8dfaeb565c65f4041751c40a
+Subproject commit 40409ed872ac4db0567ba9fac20e2dd955744e27

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}

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}

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}}