From 8263aa6d10cc1eaa0b6bd3d6d6e45b2f60b3d6b9 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Maximilian=20Ke=C3=9Fler?= <git@maximilian-kessler.de>
Date: Wed, 16 Feb 2022 02:52:05 +0100
Subject: [PATCH] get rid of legacy macros

---
 algebra.sty                                     | 10 ++++------
 inputs/nullstellensatz_and_zariski_topology.tex |  4 ++--
 2 files changed, 6 insertions(+), 8 deletions(-)

diff --git a/algebra.sty b/algebra.sty
index 1a8c4ec..2b3f4c7 100644
--- a/algebra.sty
+++ b/algebra.sty
@@ -37,14 +37,9 @@
 \newcommand{\Vp}{\ensuremath V_{\mathbb{P}}}%\Spec}}
 \newcommand{\Pn}{\bP^n}%\Spec}}
 
-\newcommand{\npr}{\footnote{Not relevant for the exam.}}
-\newcommand{\limrel}{\footnote{Limited relevance for the exam.}} % may appear in 3x questions
-%\DeclareMathOperator{\ev}{ev}
 \DeclareMathOperator{\Ker}{Ker}
 \DeclareMathOperator{\nil}{\mathfrak{nil}}
-%\DeclareMathOperator{\rad}{\mathfrak{rad}}
-\RequirePackage{stackengine}
-\stackMath
+
 \usetikzlibrary{arrows.meta,
 	quotes, babel}
 
@@ -53,3 +48,6 @@
 \def\existsone{\exists!}
 \def\defon#1{\upharpoonright_{#1}}
 \DeclareMathOperator{\Id}{Id}
+
+% \RequirePackage{stackengine}
+% \stackMath
diff --git a/inputs/nullstellensatz_and_zariski_topology.tex b/inputs/nullstellensatz_and_zariski_topology.tex
index db3c572..6d94183 100644
--- a/inputs/nullstellensatz_and_zariski_topology.tex
+++ b/inputs/nullstellensatz_and_zariski_topology.tex
@@ -1549,7 +1549,7 @@ Let $R = \mathfrak{k}[X_1,\ldots,X_n]$ and $I \subseteq R$ an ideal.
 \end{remark}
 
 \subsubsection{Application to the property of being a UFD}
-\begin{proposition}\limrel
+\begin{proposition}
     Let $R$ be a Noetherian domain. Then $R$ is a UFD iff every $\fp \in \Spec R$ with $\hght(\fp)= 1$\footnote{In other words, every  $\subseteq$-minimal element of the set of non-zero prime ideals of $R$ } is a principal ideal.
 \end{proposition}
 \begin{proof}
@@ -1565,7 +1565,7 @@ Let $R = \mathfrak{k}[X_1,\ldots,X_n]$ and $I \subseteq R$ an ideal.
     Thus $\fp = pR$ for some prime element $p$. We have $p | f$ since $f \in \fp$. As $f$ is irreducible, $p$ and $f$ are multiplicatively equivalent. Thus $f$ is a prime element.
 \end{proof}
 
-\subsection{The Jacobson radical}\limrel
+\subsection{The Jacobson radical}
 \begin{proposition}
     For a ring $A, \bigcap_{\mathfrak{m} \in \MaxSpec A} \mathfrak{m} = \{a \in A | \forall x \in  A ~ 1 - ax \in A^{\times }\} \text{\reflectbox{$\coloneqq$}} \rad(A)$, the \vocab{Jacobson radical}  of $A$.
 \end{proposition}