diff --git a/.gitmodules b/.gitmodules index 3f1c529..b0fea84 100644 --- a/.gitmodules +++ b/.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 diff --git a/ws2023/groupschemes-lecture b/ws2023/groupschemes-lecture new file mode 160000 index 0000000..215e46c --- /dev/null +++ b/ws2023/groupschemes-lecture @@ -0,0 +1 @@ +Subproject commit 215e46c90439d658fca78964fd6aab556b0e0976 diff --git a/ws2023/gruppenschemata/lec.pdf b/ws2023/gruppenschemata/lec.pdf deleted file mode 100644 index 9021728..0000000 Binary files a/ws2023/gruppenschemata/lec.pdf and /dev/null differ diff --git a/ws2023/gruppenschemata/lec.tex b/ws2023/gruppenschemata/lec.tex deleted file mode 100644 index 353ace7..0000000 --- a/ws2023/gruppenschemata/lec.tex +++ /dev/null @@ -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} diff --git a/ws2023/gruppenschemata/lec01.pdf b/ws2023/gruppenschemata/lec01.pdf deleted file mode 100644 index 1e9afa5..0000000 Binary files a/ws2023/gruppenschemata/lec01.pdf and /dev/null differ diff --git a/ws2023/gruppenschemata/lec01.tex b/ws2023/gruppenschemata/lec01.tex deleted file mode 100644 index a8cade8..0000000 --- a/ws2023/gruppenschemata/lec01.tex +++ /dev/null @@ -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} diff --git a/ws2023/gruppenschemata/lec02.pdf b/ws2023/gruppenschemata/lec02.pdf deleted file mode 100644 index fe4e6bc..0000000 Binary files a/ws2023/gruppenschemata/lec02.pdf and /dev/null differ diff --git a/ws2023/gruppenschemata/lec02.tex b/ws2023/gruppenschemata/lec02.tex deleted file mode 100644 index fc685a2..0000000 --- a/ws2023/gruppenschemata/lec02.tex +++ /dev/null @@ -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} diff --git a/ws2023/gruppenschemata/lecture.cls b/ws2023/gruppenschemata/lecture.cls deleted file mode 100644 index e604222..0000000 --- a/ws2023/gruppenschemata/lecture.cls +++ /dev/null @@ -1,280 +0,0 @@ -\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} - 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-\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}} diff --git a/ws2023/gruppenschemata/refs.bib b/ws2023/gruppenschemata/refs.bib deleted file mode 100644 index 1c02485..0000000 --- a/ws2023/gruppenschemata/refs.bib +++ /dev/null @@ -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}, -}