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ws2023/groupschemes-lecture
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ws2023/gruppenschemata/lec.pdf
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\documentclass{lecture}
\usepackage{standalone}
\usepackage{tikz}
\usepackage{subcaption}
\title{Groupschemes}
\author{\Large{Christian Dahlhausen}\\[5mm]
Notes typed by\\[1mm]
Christian Merten\\
}
\date{WiSe 2023}
\begin{document}
\newgeometry{right=15mm, left=15mm}
\maketitle
\restoregeometry
\tableofcontents
\input{lec01}
\input{lec02}
\bibliographystyle{alpha}
\bibliography{refs}
\end{document}
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\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}
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\documentclass{lecture}
\begin{document}
\section{Useful statements on schemes}
Let $k$ be a field.
\begin{definition}
Let $\mathcal{P}$ be a property of schemes over fields. For
a $k$-scheme $X$ we say
\emph{$X$ is geometrically} $\mathcal{P}$ if for all field extensions
$K / k$ the base change $X_K \to \mathrm{Spec}\ K$ is $\mathcal{P}$.
\end{definition}
\begin{bsp}
The $\R$-scheme $X = \mathrm{Spec}\left( \R[x]/(x^2 + 1) \right) $
is irreducible but not geometrically irreducible.
\end{bsp}
\begin{satz}[]
For a $k$-scheme $X$ the following are equvialent:
\begin{enumerate}[(i)]
\item $X$ is geometrically reduced
\item for every reduced $k$-scheme $Y$, the fibre product $X \times_k Y$ is reduced.
\item $X$ is reduced and for every generic point $\eta \in X$ of an
irreducible component of $X$, the field extension
$\kappa(\eta) / k$ is separable.
\item There exists a perfect field $\Omega$ and an extension $\Omega / k$ such that
$X_{\Omega}$ is reduced.
\item For all finite and purely inseparable field extensions $K / k$,
the base change $X_K$ is reduced.
\end{enumerate}
\label{prop:char-geom-red}
\end{satz}
\begin{proof}
Reducedness is a local property, so without loss of generality $X = \mathrm{Spec}\ A$. Moreover
we may assume that $X$ itself is reduced. Let
$\left\{ \eta_i \right\}_{i \in I}$ be the set of generic points of irreducible components
of $X$. Then we obtain an inclusion
\[
A \hookrightarrow \prod_{i \in I} \underbrace{\kappa(\eta_i)}_{= S_i^{-1} A}
.\] We claim that for any field extension $L / k$ the ring $A \otimes_k L$ is reduced
if and only if for all $i \in I$ the ring $\kappa(\eta_i) \otimes_k L$ is reduced.
\begin{proof}[proof of the claim]
$(\Rightarrow)$: follows since forming the nilradical commutes with localisations.
$(\Leftarrow)$: We have
\[
A \otimes_k L \hookrightarrow \left( \prod_{i \in I}^{} \kappa(\eta_i) \right)
\otimes_k L
\hookrightarrow \prod_{i \in I}^{} \kappa(\eta_i) \otimes_k L
.\]
\end{proof}
The claim immediatly implies the equivalence of (iii), (iv), (v) and (1). Since
(ii) trivially implies (i). It remains to show that (iii) implies (2).
Without loss of generality we may take $Y = \mathrm{Spec}\ B$
and set $\{\lambda_j\}_{j \in J}$ to be the generic points of $Y$. Then we obtain
\[
A \otimes_k B \hookrightarrow
A\otimes_k \left( \prod_{j \in J} \kappa(\lambda_j) \right)
\hookrightarrow
\left( \prod_{i \in I} \kappa(\eta_i) \right)
\otimes_k
\left( \prod_{j \in J} \kappa(\lambda_j) \right)
\hookrightarrow
\prod_{i,j}^{} \underbrace{\kappa(\eta_i) \otimes_k \kappa(\eta_j) }_{\text{reduced}}
.\]
\end{proof}
\begin{korollar}
If $k$ is perfect, then
reduced and geometrically reduced are equivalent.
\end{korollar}
\begin{bem}[]
The statements in \ref{prop:char-geom-red} also hold when
\emph{reduced} is replaced by \emph{irreducible} or \emph{integral}.
\end{bem}
\begin{satz}
Let $f\colon X \to Y$ be a morphism of schemes that is locally of finite presentation.
Then $f$ is open if and only if
for every point $x \in X$ and every point $y' \in Y$ with
$y = f(x) \in \overline{\{y'\} }$ there exists
$x' \in X$ with $x \in \overline{\{x'\} }$ such that $f(x') = y'$.
\label{prop:open-stab-gener}
\end{satz}
\begin{proof}
Assume $X = \mathrm{Spec}\ B$ and $Y = \mathrm{Spec}\ A$.
$(\Rightarrow)$: Then set
\[
Z \coloneqq \mathrm{Spec}\ \mathcal{O}_{X,x}
\cap \bigcap_{t \in B \setminus \mathfrak{p}_x} D(t)
.\] Since $f$ is open, $y' \in f(D(t))$ for all $t \in B \setminus \mathfrak{p}_x$.
Set $f_t \coloneqq f|_{D(t)}$. Then $f_t ^{-1}(y') \neq \emptyset$. For sake
of contradiction suppose that $y' \not\in f(Z)$. Then set
$g\colon \mathrm{Spec}\ \mathcal{O}_{X,x} \to X \xrightarrow{f} Y$.
Therefore
\[
\emptyset = g^{-1}(y') = \mathrm{Spec}\ \left( \mathcal{O}_{X,x} \otimes_A \kappa(y') \right)
.\] Thus
\[
0 = \mathcal{O}_{X,x} \otimes_A \kappa(y')
= \operatorname{colim}_{t \in B \setminus \mathfrak{p}_x}
\underbrace{B_t \otimes_A \kappa(y')}_{\neq 0}
\] which is a contradiction.
$(\Leftarrow)$:
Show $f(X) \subseteq Y$ is open. By Chevalley's theorem (\cite{gw}, 10.70),
the image $f(X)$ is constructible. In the noetherian case
use that open is equivalent to constructible and stable under generalizations
(\cite{gw}, 10.17). In the general case write $A$ as a colimit of noetherian rings and
conclude by careful general nonsense.
\end{proof}
\begin{lemma}
Let $f\colon X \to Y$ be flat, $x \in X$, $y = f(x)$, $y' \in Y$ a
generalization of $y$. Then there exists a generalization $x'$ of $x$ such that
$f(x') = y'$.
\label{lemma:flat-stable-gener}
\end{lemma}
\begin{proof}
Set $A = \mathcal{O}_{Y,y}$, $B = \mathcal{O}_{X,x}$ and
$\varphi\colon A \to B$. Since $y \in \text{im}(f)$
we have $\mathfrak{m}_yB \neq B$ and
$B$ is faithfully flat $A$-module (since $\varphi$ is local and flat). Thus
\[
0 \neq B \otimes_A \kappa(y')
,\] i.e. $f^{-1}(y') \cap \mathrm{Spec}\ B \neq \emptyset$.
\end{proof}
\begin{korollar}
Let $f\colon X \to Y$ be flat and locally of finite presentation. Then $f$ is universally
open.
\end{korollar}
\begin{proof}
From \ref{prop:open-stab-gener} and \ref{lemma:flat-stable-gener} follows
that flat and locally of finite presentation implies open. Since the former
two properties are stable under base change, the result follows.
\end{proof}
\begin{korollar}
Let $f\colon X \to S$ be locally of finite presentation. If
$|S|$ is discrete, then every morphism $X \to S$ is universally open.
\end{korollar}
\begin{definition}[]
Let $f\colon X \to Y$. We say
\begin{enumerate}[(i)]
\item $f$ is \emph{flat in $x \in X$} if
$f_x^{\#}\colon \mathcal{O}_{Y,f(x)} \to \mathcal{O}_{X,x}$ is flat.
\item $f$ is \emph{flat} if
$f$ is flat in every point.
\end{enumerate}
\end{definition}
\begin{bsp}[]
\begin{enumerate}[(1)]
\item $X \to \mathrm{Spec}\ k$ is flat.
\item $\mathbb{A}_{Y}^{n} \to Y$ and
$\mathbb{P}_{Y}^{n} \to Y$ are flat.
\item Let $f\colon Z \hookrightarrow Y$ be a closed immersion. Then
$f$ is flat and locally of finite presentation if and only if $f$ is an open immersion.
\end{enumerate}
\end{bsp}
\begin{satz}
The following holds
\begin{enumerate}[(i)]
\item $\mathrm{Spec}\ B \to \mathrm{Spec}\ A$ is flat if and only if $A \to B$ is flat.
\item Flatness is stable under base change and composition.
\item Flatness is local on the source and the target.
\item Open immersions are flat.
\item A morphism $f\colon X \to Y$ is flat if and only if
for every $y \in Y$ the canonical morphism
\[
X \times_Y \mathrm{Spec}(\mathcal{O}_{X,y})
\to \mathrm{Spec}(\mathcal{O}_{Y,y})
\] is flat.
\end{enumerate}
\end{satz}
\begin{definition}
A morphism $f\colon X \to Y$ is called \emph{faithfully flat} if
$f$ is flat and surjective.
\end{definition}
\begin{bsp}[]
$\mathrm{Spec}\ \overline{k} \to \mathrm{Spec}\ k$ is faithfully flat.
\end{bsp}
\begin{lemma}
Let $\mathcal{C}$ be a category with equalizers, $F\colon \mathcal{C} \to \mathcal{D}$ a
conservative (i.e. reflects isomorphisms) functor that commutes with equalizers. Then
$F$ is faithful.
\label{lemma:cons-eq-faithful}
\end{lemma}
\begin{proof}
Left as an exercise to the reader.
\end{proof}
\begin{satz}
Is $f\colon X \to Y$ faithfully flat, then
$f^{*}\colon \mathrm{QCoh}(Y) \to \mathrm{QCoh}(X)$ faithful.
\label{prop:faithfully-flat-faithful-pullback}
\end{satz}
\begin{proof}
Can be deduced from \ref{lemma:cons-eq-faithful}. The details are left to the reader.
\end{proof}
\begin{bem}[Faithfully flat descent]
The statement from \ref{prop:faithfully-flat-faithful-pullback} can be
- from a carefully selected viewpoint - viewn as the statement
that the functor $X \mapsto \mathrm{QCoh}(X)$ satisfies the sheaf condition
for faithfully flat and quasicompact morphisms, i.e. that the diagram
\[
\begin{tikzcd}
\mathrm{QCoh}(Y)
\arrow{r}{f^{*}}
& \mathrm{QCoh}(X)
\arrow[yshift=2pt]{r}{\text{pr}_1^{*}}
\arrow[swap, yshift=-2pt]{r}{\text{pr}_2^{*}}
&\mathrm{QCoh}(X \times_Y X)
\arrow[yshift=4pt]{r}
\arrow[yshift=0pt]{r}
\arrow[yshift=-4pt]{r}
&
\underbrace{\mathrm{QCoh}(X \times_Y X \times_Y X)}_{\text{corresponds to the cocycle condition}}
\end{tikzcd}
\] is a limit diagram.
\end{bem}
\begin{satz}[\cite{gw}, 14.53]
Let $f\colon X \to Y$ be a $S$-morphism and
$g\colon S' \to S$ faithfully flat and quasicompact.
Denote by $f' = f \times_S S'$. If $f'$ is
\begin{enumerate}[(i)]
\item (locally) of finite type or (locally) of finite presentation,
\item isomorphism / monomorphism,
\item open / closed / quasicompact immersion,
\item proper / affine / finite,
\end{enumerate}
then $f$ has the same property.
\end{satz}
\end{document}
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%\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}}
ws2023/gruppenschemata/refs.bib
-15
View File
@@ -1,15 +0,0 @@
@book {gw,
AUTHOR = {G\"{o}rtz, Ulrich and Wedhorn, Torsten},
TITLE = {Algebraic geometry {I}},
SERIES = {Advanced Lectures in Mathematics},
NOTE = {Schemes with examples and exercises},
PUBLISHER = {Vieweg + Teubner, Wiesbaden},
YEAR = {2010},
PAGES = {viii+615},
ISBN = {978-3-8348-0676-5},
MRCLASS = {14-01},
MRNUMBER = {2675155},
MRREVIEWER = {C\'{\i}cero\ Carvalho},
DOI = {10.1007/978-3-8348-9722-0},
URL = {https://doi.org/10.1007/978-3-8348-9722-0},
}