update

Author: hooyuser

algebraic_construction.tex

@@ -10,8 +10,9 @@
 
 \input{preamble}
 
-% \includeonly{set_theory, commutative_ring, associative_algebra,field, module, vector_space, valuation_theory}
-%\includeonly{ring, commutative_ring, module}
+% \includeonly{set_theory, ring, commutative_ring, associative_algebra,field, module, vector_space, valuation_theory}
+%\includeonly{category_theory}
+%\includeonly{ring, commutative_ring, module, associative_algebra, field}
 %\includeonly{module}
 %\includeonly{topological_group}
 %\includeonly{set_theory,
@@ -30,6 +31,8 @@
 
 \usepackage{leftindex}
 
+\newcommand{\divides}{\mid}
+
 
 \newcommand{\mathcalo}{\mathchoice
   {\scalebox{0.7}{$\mathcal{O}$}}% Display style

associative_algebra.tex

@@ -3,17 +3,20 @@ \chapter{Associative Algebra}
 
 \section{Basic Properties}
 \begin{definition}{Associative Algebra over Commutative Ring}{}
-    Let $R$ be a commutative ring. An \textbf{associative $R$-algebra} is a ring $A$ together with a ring homomorphism $\sigma:R\to Z(A)$, which makes $A$ an $R$-module by defining 
-    $$
-    r\cdot a=\sigma(r)a
-    $$
-    for all $r\in R$ and $a\in A$. \\
-    We can check that 
-\[
-  r\cdot (ab)=\sigma(r)ab=(r\cdot a)b=\sigma(r)ab=a\left(\sigma(r)b\right) =a(r\cdot b),
-\]
-which justifies the naming ``associative". 
+    Let $R$ be a commutative ring. An \textbf{associative $R$-algebra} is a ring $A$ together with a ring homomorphism $\varphi:R\to Z(A)$, which makes $A$ an $R$-module by defining the scalar multiplication as
+    \begin{align*}
+        R\times A &\longrightarrow A\\
+        (r,a) &\longmapsto r\cdot a:=\varphi(r)a.
+    \end{align*}
+    $\varphi:R\to Z(A)$ is called the \textbf{structure homomorphism} of $A$.
 \end{definition}
+\begin{remark}
+    We can check that 
+    \[
+      r\cdot (ab)=\sigma(r)ab=(r\cdot a)b=\sigma(r)ab=a\left(\sigma(r)b\right) =a(r\cdot b),
+    \]
+    which justifies the naming ``associative". 
+\end{remark}
 
 We usually call associative $R$-algebra as $R$-algebra for short.
 
@@ -88,7 +91,7 @@ \subsection{Graded Object}
 
 
 
-\section{Tensor Algebra}
+\subsection{Tensor Algebra}
 \begin{definition}{Tensor Algebra $T^{\bullet}(M)$}{}
     Given a $R$-module $M$, the \textbf{$k$-th tensor power of $M$} is defined as
     \begin{align*}
@@ -128,7 +131,7 @@ \section{Tensor Algebra}
 \end{proposition}
 
 
-\section{Exterior Algebra and Symmetric Algebra}
+\subsection{Exterior Algebra and Symmetric Algebra}
 \begin{definition}{Exterior Algebra $\Largewedge^{\bullet} (M)$}{}
     Given an $R$-module $M$, 
     \begin{align*}
@@ -184,7 +187,38 @@ \section{Exterior Algebra and Symmetric Algebra}
         Moreover, we have $\Largewedge^{m}(M)=0$ for all $m>n$.
     \end{enumerate}
 \end{proposition}
+\section{Integral Element}
+
+\begin{definition}{Integral Element}{}
+    Let $R$ be a commutative ring and $A$ be an $R$-algebra with structure homomorphism $\varphi:R\to Z(A)$. An element $x\in A$ is called \textbf{integral} over $R$ if there exists a monic polynomial $f\in R[T]$ such that $\leftindex^{\varphi}\!f(x)=0$.
+\end{definition}
+
+
+\begin{definition}{Generated Subalgebra}{generated_subalgebra}
+    Let $R$ be a commutative ring and $A$ be an $R$-algebra. By the universal property of $R\langle T\rangle$, there exists a unique $R$-algebra homomorphism $\psi:R\langle T\rangle\to A$ such that 
+    $\psi(T)=x$.
+    \begin{center}
+        \begin{tikzcd}[ampersand replacement=\&]
+            R\langle T\rangle\arrow[r, dashed, "\exists !\,\psi"]  \& A\\[0.3cm]
+            \{T\}\arrow[u, "\iota"] \arrow[ru, "\mathrm{const}_x"'] \&  
+        \end{tikzcd}
+    \end{center}
+    The \textbf{$R$-subalgebra of $A$ generated by $x$} is defined as 
+    \[
+    R[x]:=\psi\left(R\langle T\rangle\right)=\left\{\sum_{k=0}^n r_k x^k \in A\;\middle|\; r_k\in R\right\}.
+    \]
+\end{definition}
+
+\begin{proposition}{Equivalent Definition of Integral Element}{equivalent_definition_of_integral_element}
+    Let $R$ be a commutative ring and $A$ be an $R$-algebra. Let $R[x]$ be the \hyperref[th:generated_subalgebra]{$R$-subalgebra of $A$ generated by $x$}. Then $A$ is an $R[x]$-module. And the following statements are equivalent:
+    \begin{enumerate}[(i)]
+        \item $x$ is integral over $R$.
+        \item $R[x]$ is a finitely generated $R$-module.
+        \item There exists a faithful $R[x]$-submodule of $A$ that is finitely generated as an $R$-module and contains $x$.
+    \end{enumerate}
+\end{proposition}
 
+\
 
 \section{Determinant and Trace}
 

category_theory.tex

@@ -4197,7 +4197,7 @@ \section{Kan Extension}
 
 
 \section{Monoidal Category}
-\begin{definition}{Monoidal Category}{}
+\begin{definition}{Monoidal Category}{monoidal_category}
     A monoidal category is a category $\mathsf{V}$ equipped with
     \begin{enumerate}[(i)]
         \item Tensor product: a functor $\otimes:\mathsf{V}\times\mathsf{V}\to\mathsf{V}$.
@@ -4221,7 +4221,7 @@ \section{Monoidal Category}
         \item Unit object: an object $1\in \mathrm{Ob}(\mathsf{V})$ 
         \item An isomorphism in $\mathsf{V}$: $\iota:1\otimes 1\to 1$
     \end{enumerate}
-    such that the following two condition holds
+    such that the following two conditions holds
     \begin{enumerate}[(i)]
         \item The pentagon axiom: the following diagram commutes
     \[
@@ -4276,7 +4276,7 @@ \section{Monoidal Category}
 \end{definition}
 
 
-\begin{definition}{Symmetric Monoidal Category}{}
+\begin{definition}{Symmetric Monoidal Category}{symmetric_monoidal_category}
     A \textbf{symmetric monoidal category} is a braided monoidal category $\mathsf{V}$ satisfying the following condition:
     \[
         B_{Y,X}\circ B_{X,Y}=\mathrm{id}_{X\otimes Y}.