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