replace groupschemes files by submodule

.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/groupschemes-lecture

@@ -0,0 +1 @@
+Subproject commit 215e46c90439d658fca78964fd6aab556b0e0976

ws2023/gruppenschemata/lec.tex

@@ -1,28 +0,0 @@
-\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}

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/lec02.tex

@@ -1,252 +0,0 @@
-\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}

ws2023/gruppenschemata/lecture.cls

@@ -1,280 +0,0 @@
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-\LoadClass[a4paper]{book}
-
-\RequirePackage{faktor}
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-\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}
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-\RequirePackage{wasysym}
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-
-\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}}

ws2023/gruppenschemata/refs.bib

@@ -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},
-}