Refine inertia group definition and page layout

Moved and expanded the definition of inertia group in commutative_ring.tex for better logical flow and added explanatory context. Clarified notation for submodules defined by principal ideals in module.tex. Updated page geometry and footer to display page numbers at the bottom right in preamble.tex.

Author: hooyuser

commutative_ring.tex

@@ -2276,18 +2276,6 @@ \subsection{Ramification Theory}
     \]
 \end{prf}
 
-\begin{definition}{Inertia Group}{}
-    Let $\mathcalo$ be a Dedekind domain with field of fractions $K=\mathrm{Frac}(\mathcalo)$. Let $L/K$ be a finite Galois extension and $\mathcal{O}$ be the integral closure of $\mathcalo$ in $L$. Suppose $\mathfrak{P}$ is a maximal ideal over $\mathfrak{p}\in \spec\left(\mathcalo\right)$. The \textbf{inertia group} of $\mathfrak{P}$ is defined as the kernel of the surjective group homomorphism
-    \begin{align*}
-        \xi:D_{\mathfrak{P}}&\longrightarrow \mathrm{Aut}_{\left(\kappa(\mathfrak{p})/\mathsf{Field}\right)}\left(\kappa(\mathfrak{P})/\kappa(\mathfrak{p})\right)\\
-        \sigma&\longmapsto \widetilde{\sigma}.
-    \end{align*}
-    denote by $I_{\mathfrak{P}}:=\ker \xi$. And we have the following exact sequence
-    \[
-    1\longrightarrow I_{\mathfrak{P}}\longrightarrow D_{\mathfrak{P}}\longrightarrow \mathrm{Aut}\left(\kappa(\mathfrak{P})/\kappa(\mathfrak{p})\right)\longrightarrow 1.
-    \]
-\end{definition}
-
 \begin{proposition}{}{}
     Let $\mathcalo$ be a normal integral domain with fraction field $K$. Let $L/K$ be a (possibly infinite) Galois extension. Let $G = \text{Gal}(L/K)$ and let
     $\mathcal{O}$ be the integral closure of $\mathcalo$ in $L$.
@@ -2309,6 +2297,23 @@ \subsection{Ramification Theory}
 \url{https://stacks.math.columbia.edu/tag/0BRK}
 \end{prf}
 
+This proposition enables us to define the inertia group.
+
+\begin{definition}{Inertia Group}{}
+    Let $\mathcalo$ be a Dedekind domain with field of fractions $K=\mathrm{Frac}(\mathcalo)$. Let $L/K$ be a finite Galois extension and $\mathcal{O}$ be the integral closure of $\mathcalo$ in $L$. Suppose $\mathfrak{P}$ is a maximal ideal over $\mathfrak{p}\in \spec\left(\mathcalo\right)$. The \textbf{inertia group} of $\mathfrak{P}$ is defined as the kernel of the surjective group homomorphism
+    \begin{align*}
+        \xi:D_{\mathfrak{P}}&\longrightarrow \mathrm{Aut}_{\left(\kappa(\mathfrak{p})/\mathsf{Field}\right)}\left(\kappa(\mathfrak{P})/\kappa(\mathfrak{p})\right)\\
+        \sigma&\longmapsto \widetilde{\sigma}.
+    \end{align*}
+    denote by $I_{\mathfrak{P}}:=\ker \xi$. And we have the following exact sequence
+    \[
+    1\longrightarrow I_{\mathfrak{P}}\longrightarrow D_{\mathfrak{P}}\longrightarrow \mathrm{Aut}\left(\kappa(\mathfrak{P})/\kappa(\mathfrak{p})\right)\longrightarrow 1.
+    \]
+\end{definition}
+
+Therefore, the decomposition group $D_{\mathfrak{P}}$ controls the splitting behavior of $\mathfrak{p}$ in $L$, with size $|D_{\mathfrak{P}}|=ef$. If we further assume that $\kappa(\mathfrak{P})/\kappa(\mathfrak{p})$ is a finite separable extension, then $\mathrm{Aut}\left(\kappa(\mathfrak{P})/\kappa(\mathfrak{p})\right)=\mathrm{Gal}\left(\kappa(\mathfrak{P})/\kappa(\mathfrak{p})\right)$ and $|\mathrm{Gal}\left(\kappa(\mathfrak{P})/\kappa(\mathfrak{p})\right)|=f$. Thus by the exact sequence, the inertia group $I_{\mathfrak{P}}$ has size $|I_{\mathfrak{P}}|=e$. In this case, the inertia group $I_{\mathfrak{P}}$ measures the ramification of $\mathfrak{p}$ in $L$, while $\mathrm{Gal}\left(\kappa(\mathfrak{P})/\kappa(\mathfrak{p})\right)$ measures the inertia.
+
+
 \begin{definition}{Frobenius Element}{}
     Let $\mathcalo$ be a Dedekind domain with field of fractions $K=\mathrm{Frac}(\mathcalo)$. Let $L/K$ be a finite Galois extension and $\mathcal{O}$ be the integral closure of $\mathcalo$ in $L$. Suppose $\mathfrak{P}$ is a maximal ideal over $\mathfrak{p}\in \spec\left(\mathcalo\right)$ and $\kappa(\mathfrak{p})$ is a finite field isomorphic to $\mathbb{F}_{q}$. Then $\kappa(\mathfrak{P})/\kappa(\mathfrak{p})$ is a finite Galois extension and we have the following exact sequence
     \[

module.tex

@@ -1339,6 +1339,7 @@ \section{Torsion-Free Modules}
     \[
         M[\mathfrak{a}]:=\left\{m\in M\midv \forall a\in \mathfrak{a}, am=0\right\}.
     \]
+    If $\mathfrak{a}=a R$ is a principal right ideal generated by $a\in R$, we simply write $M[a]$ instead of $M[\mathfrak{a}]$.
 \end{definition}
 \begin{prf}
     We can check that $M[\mathfrak{a}]$ is a submodule of $M$. For any $m_1, m_2\in M[\mathfrak{a}]$, we have

preamble.tex

@@ -6,7 +6,7 @@
 % --- 1. SETUP & FUNDAMENTALS (Load these first to define page/colors) ---
 % Load xcolor FIRST to avoid "Option Clash" with tikz/tcolorbox
 \usepackage[table]{xcolor} 
-\usepackage[tmargin=2cm,rmargin=1in,lmargin=1in,margin=0.85in,bmargin=1.5cm,footskip=.2in]{geometry}
+\usepackage[tmargin=2cm,rmargin=1in,lmargin=1in,margin=0.85in,bmargin=1.5cm,footskip=0.8cm]{geometry}
 \usepackage{etoolbox}
 \usepackage{anyfontsize}
 
@@ -81,14 +81,15 @@
 % Section name on the right of the header
 \fancyhead[R]{\rightmark}
 
-% Page number on the bottom right in the format "- 34 -"
-\fancyhead[C]{\raisebox{-0pt}{-\hspace{4pt}\thepage\hspace{4pt}-}}
+% Page number on the bottom right 
+\fancyfoot[R]{\thepage}
 
 % Line at the top of the header
 \renewcommand{\headrulewidth}{0.0pt}
 
 \fancypagestyle{plain}{%
   \fancyhf{} % clear all header and footer fields
+  \fancyfoot[R]{\thepage}
   \renewcommand{\headrulewidth}{0pt} % remove the header rule
   \renewcommand{\footrulewidth}{0pt} % remove the footer rule
 }