diff --git a/2021_Algebra_I.tex b/2021_Algebra_I.tex index 2f5bc54..0f057bd 100644 --- a/2021_Algebra_I.tex +++ b/2021_Algebra_I.tex @@ -31,7 +31,7 @@ Fields which are not assumed to be algebraically closed have been renamed (usual \pagebreak -\subsection{Finiteness conditions} +\section{Finiteness conditions} \subsection{Finitely generated and Noetherian modules} @@ -313,7 +313,7 @@ is injective. $n$ and the $a_i$ can be chosen such that $A$ is finite over the i This contradicts the minimality of $n$, as $B$ can be generated by $< n$ elements $b_i$. \end{proof} -\subsection{The Nullstellensatz and the Zariski topology} +\section{The Nullstellensatz and the Zariski topology} \subsection{The Nullstellensatz} %LECTURE 1 Let $\mathfrak{k}$ be a field, $R \coloneqq \mathfrak{k}[X_1,\ldots,X_n], I \subseteq R$ an ideal. @@ -1913,7 +1913,7 @@ $\rad(A) = f A$ where $f = \prod_{i=1}^{n} p_i$. % Lecture 11 -\subsection{Projective spaces} +\section{Projective spaces} Let $\mathfrak{l}$ be any field. \begin{definition} For a $\mathfrak{l}$-vector space $V$, let $\mathbb{P}(V)$ be the set of one-dimensional subspaces of $V$. @@ -2278,7 +2278,7 @@ If $P_e $ with $e < d$ was $\neq 0$, it could not be a multiple of $P$ contradic % Lecture 13 -\subsection{Varieties} +\section{Varieties} \subsection{Sheaves} @@ -2927,7 +2927,7 @@ Original (Noether normalization) Artin-Tate Uncountable fields \begin{landscape} -\subsection{Übersicht} +\section{Übersicht} {\rowcolors{2}{gray!10}{white} \begin{longtable}{lll} \end{longtable} diff --git a/algebra.sty b/algebra.sty index 8be0c37..79187c1 100644 --- a/algebra.sty +++ b/algebra.sty @@ -14,8 +14,6 @@ \RequirePackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry} -\setlength{\droptitle}{-1.0cm} - \RequirePackage[normalem]{ulem} \RequirePackage{pdflscape} \RequirePackage{longtable}