From 75b0f40bc932e0789623ffa5f6bbc2443a7d03cf Mon Sep 17 00:00:00 2001 From: Derived Cat Date: Fri, 29 Dec 2023 04:35:59 -0500 Subject: [PATCH] finitely generated object --- algebraic_construction.tex | 122 ++++++++++++++++++++++++++++++++----- 1 file changed, 107 insertions(+), 15 deletions(-) diff --git a/algebraic_construction.tex b/algebraic_construction.tex index a4efe40..34e94c4 100644 --- a/algebraic_construction.tex +++ b/algebraic_construction.tex @@ -97,7 +97,7 @@ \chapter*{Notation Conventions} \item $\mathsf{Ring}$: the category of rings. \item $\mathsf{CRing}$: the category of commutative rings. \item $\mathsf{Fld}$: the category of fields. - \item $R\text{-}\mathsf{Mod}$: the category of $R$-modules, where $R\in \mathrm{Ob}\left(\mathsf{Ring}\right)$. + \item $R\text{-}\mathsf{Mod}$: the category of left $R$-modules, where $R\in \mathrm{Ob}\left(\mathsf{Ring}\right)$. \item $K\text{-}\mathsf{Vect}$: the category of $K$-vector spaces, where $K\in \mathrm{Ob}\left(\mathsf{Fld}\right)$. \item $R\text{-}\mathsf{Alg}$: the category of associative $R$-algebras, where $R\in \mathrm{Ob}\left(\mathsf{CRing}\right)$. \item $R\text{-}\mathsf{CAlg}$: the category of commutative $R$-algebras, where $R\in \mathrm{Ob}\left(\mathsf{CRing}\right)$. @@ -257,6 +257,30 @@ \section{Function} \item $A\subseteq f^{-1}(f(A))\ $, $B\supseteq f(f^{-1}(B))$. \end{itemize} } +\prop{Equivalent Characterization of Injections}{ + Let $f:X\to Y$ be a map. The following are equivalent: + \begin{enumerate}[(i)] + \item $f$ is injective. + \item $f$ has a left inverse: there exists a map $h:Y\to X$ such that $h\circ f=\mathrm{id}_X$. + \item $f$ is left-cancellable: if $f\circ g_1=f\circ g_2$, then $g_1=g_2$. + \end{enumerate} +} +\prop{Equivalent Characterization of Surjections}{ + Let $f:X\to Y$ be a map. The following are equivalent: + \begin{enumerate}[(i)] + \item $f$ is surjective. + \item $f$ has a right inverse: there exists a map $h:Y\to X$ such that $f\circ h=\mathrm{id}_Y$. + \item $f$ is right-cancellable: if $g_1\circ f=g_2\circ f$, then $g_1=g_2$. + \end{enumerate} +} + +\prop{}{ + Suppose $f:X\to Y$ and $g:Y\to Z$ are maps. We have + \begin{enumerate}[(i)] + \item If $g\circ f$ is injective, then $f$ is injective. + \item If $g\circ f$ is surjective, then $g$ is surjective. + \end{enumerate} +} \section{Grothendieck Universe} \dfn{Grothendieck Universe}{ @@ -2446,7 +2470,7 @@ \section{Basic Concepts} \chapter{Commutative Ring} \section{Basic Concepts} -A commutative ring $R$ is a commutative $R$-algebra. And we have a categorical isomorphism +A commutative ring $R$ is a commutative $R$-algebra and accordingly a commutative $\mathbb{Z}$-algebra. Furthermore we have a categorical isomorphism \[ \mathsf{CRing}\cong \mathbb{Z}\text{-}\mathsf{CAlg}. \] @@ -2751,6 +2775,11 @@ \section{Polynomial Ring} } \section{Construction} +\subsection{Free Object} +\dfn{Free Commutative Ring}{ + Since $\mathsf{CRing}\cong \mathbb{Z}\text{-}\mathsf{CAlg}$, the \textbf{free commutative ring} on a set $X$ is isomorphic to the polynomial ring $\mathbb{Z}[X]$, which coincides with the free commutative $\mathbb{Z}$-algebra on $X$. +} +\subsection{Localization} \dfn{Multuplicative Subset}{ Let $R$ be a commutative ring. A subset $S\subseteq R$ is called \textbf{multiplicative} if $S$ is monoid under the multiplication of $R$, i.e. \begin{enumerate}[(i)] @@ -3160,7 +3189,7 @@ \section{Basic Concepts} \end{tikzcd} \] } -In particular, homomorphism $R\to S$ makes $S$ an $R$-module. +In particular, ring homomorphism $R\to S$ makes $S$ an $R$-module. \dfn{Noetherian Module}{ Let $R$ be a commutative ring, and let $M$ be an $R$-module. We say $M$ is \textbf{Noetherian} if one of the following equivalent conditions holds: @@ -3195,9 +3224,24 @@ \subsection{Free Object} one of the following equivariant conditions holds: \begin{enumerate}[(i)] \item there exist $x_1, \ldots, x_n \in M$ such that every element of $M$ is an $R$-linear combination of the $x_i$. - \item there exists an epimorphism $R^{\oplus n} \rightarrow M$ for some $n \in \mathbb{N}$. + \item there exists an epimorphism $R^{\oplus n} \rightarrow M$ for some $n \in \mathbb{Z}_+$. + \item there exists an exact sequence + \[ + R^{\oplus n} \rightarrow M \rightarrow 0 + \] + for some $n\in \mathbb{N}$. + \item $S\cong R^{\oplus n}/M$ for some $n\in \mathbb{Z}_+$ and some submodule $M$ of $R^{\oplus n}$. \end{enumerate} } +\dfn{Finitely Presented Module}{ + We say $M$ is a \textbf{finitely presented $R$-module} if + there exists an exact sequence + \[ + R^{\oplus m} \rightarrow R^{\oplus n} \rightarrow M \rightarrow 0 + \] + for some $m, n\in \mathbb{Z}_+$. +} + \subsection{Localization} \dfn{Localization of a Module}{ Let $R$ be a commutative ring, $S$ be a multiplicative set in $R$, and $M$ be an $R$-module. The \textbf{localization of the module} $M$ by $S$, denoted $S^{-1}M$, is an $S^{-1}R$-module that is constructed exactly as the localization of $R$, except that the numerators of the fractions belong to $M$. That is, as a set, it consists of equivalence classes, denoted $\frac{m}{s}$, of pairs $(m, s)$, where $m\in M$ and $s\in S$, and two pairs $(m, s)$ and $(n, t)$ are equivalent if there is an element $u$ in $S$ such that @@ -3399,18 +3443,39 @@ \section{Basic Properties} $$ r\cdot a=\sigma(r)a $$ - for all $r\in R$ and $a\in A$. - We usually call associative $R$-algebra as $R$-algebra for short. -} -We can check that + for all $r\in R$ and $a\in A$. \\ + We can check that \[ - r\cdot (ab)=\sigma(r)ab=(r\cdot a)b=\sigma(r)ab=a\left(\sigma(r)b\right) =a(r\cdot b). + r\cdot (ab)=\sigma(r)ab=(r\cdot a)b=\sigma(r)ab=a\left(\sigma(r)b\right) =a(r\cdot b), \] +which justifies the naming ``associative". +} +We usually call associative $R$-algebra as $R$-algebra for short. + +\prop{Commutative Ring homomorphism $R\to S$ induces functor $S\text{-}\mathsf{Alg}\to R\text{-}\mathsf{Alg}$}{ + Let $R$ and $S$ be commutative rings with a ring homomorphism $f: R\to S$. Then every $S$-algebra $A$ is an $R$-algebra by defining $ra = f(r)a$, or equivalently through $R\to S\to Z(A)$. This defines a functor $F: S\text{-}\mathsf{Alg}\to R\text{-}\mathsf{Alg}$, which is identify map on objects and morphisms. + \[ + \begin{tikzcd}[ampersand replacement=\&] + S\text{-}\mathsf{Alg}\&[-25pt]\&[+10pt]\&[-30pt] R\text{-}\mathsf{Alg}\&[-30pt]\&[-30pt] \\ [-15pt] + A \arrow[dd, "g"{name=L, left}] + \&[-25pt] \& [+10pt] + \& [-30pt] A\arrow[dd, "g"{name=R}] \&[-30pt]\\ [-10pt] + \& \phantom{.}\arrow[r, "F", squigarrow]\&\phantom{.} \& \\[-10pt] + B \& \& \& B\& + \end{tikzcd} + \] +} +In particular, commutative ring homomorphism $R\to S$ makes $S$ an $R$-algebra. + +\section{Construction} +\subsection{Quotient Object} \dfn{Quotient Algebra}{ Let $A$ be an $R$-algebra and $\mathfrak{a}$ be a two-sided ideal of $A$. Since $\mathfrak{a}$ is an $R$-submodule of $A$, the quotient ring $A/\mathfrak{a}$ can also be endowed with an $R$-module structure, which makes $A/\mathfrak{a}$ an $R$-algebra. We call $A/\mathfrak{a}$ the \textbf{quotient algebra} of $A$ by $\mathfrak{a}$. } +\subsection{Graded Object} + \dfn{$I$-Graded Algebra over an Graded Commutative Ring}{ Let $(I,+)$ be a monoid and $R$ be a $I$-graded commutative ring with grading $(R_i)_{i\in I}$. An \textbf{$I$-graded algebra over graded ring $R$} is an $R$-algebra $A$ together with a family of subalgebras $\left(A_i\right)_{i\in I}$ such that \begin{enumerate}[(i)] @@ -3477,16 +3542,43 @@ \section{Exterior Algebra and Symmetric Algebra} \section{Commutative Algebra} \dfn{Commutative Algebra}{ - Let $R$ be a commutative ring. A \textbf{commutative $R$-algebra} is an $R$-algebra where the multiplication is commutative. + Let $R$ be a commutative ring. A \textbf{commutative $R$-algebra} is an $R$-algebra where the multiplication is commutative. Or equivalently, a commutative $R$-algebra is a commutative ring $A$ together with a ring homomorphism $R\to A$. Hence there is a category isomorphism $R\text{-}\mathsf{CAlg}\cong \left(R/\mathsf{CRing}\right)$. } \dfn{Free Commutative Algebra}{ - Let $R$ be a commutative ring. - \begin{tikzcd}[ampersand replacement=\&] - \& X \arrow[ld, "\iota"'] \arrow[rd, "f"] \& \\ - F \arrow[rr, "\hat{f}"] \& \& A - \end{tikzcd} + Let $X$ be a set and $R$ be a commutative ring. The \textbf{free commutative $R$-algebra} on $X$, denoted by $\mathrm{Free}_{R\text{-}\mathsf{CAlg}}(X)$, together with a map $\iota:X\to \mathrm{Free}_{R\text{-}\mathsf{CAlg}}(X)$, is defined by the following universal property: for any commutative $R$-algebra $A$ and any map $f:X\to A$, there exists a unique homomorphism $\widetilde{f}:\mathrm{Free}_{R\text{-}\mathsf{CAlg}}(X)\to A$ such that the following diagram commutes + \begin{center} + \begin{tikzcd}[ampersand replacement=\&] + \mathrm{Free}_{R\text{-}\mathsf{CAlg}}(X)\arrow[r, dashed, "\exists !\,\widetilde{f}"] \& A\\[0.3cm] + X\arrow[u, "\iota"] \arrow[ru, "f"'] \& + \end{tikzcd} + \end{center} + The free $R$-module $\mathrm{Free}_{R\text{-}\mathsf{CAlg}}(X)$ can be contructed as the polynomial algebra $R[X]$. +} +\dfn{Finite-type Commutative Algebra}{ + Let $R\to A$ be a commutative ring homomorphism. We say $A$ is a \textbf{finite-type $R$-algebra}, or that $R\to A$ is \textbf{of finite type}, if one of the following equivalent conditions holds: + \begin{enumerate}[(i)] + \item there exists a finite set of elements $a_1,\cdots,a_n$ of A such that every element of $A$ can be expressed as a polynomial in $a1,\cdots,an$, with coefficients in $K$. + \item there exists a finite set $X$ such that $A\cong R[X]/I$ as $R$-algebra where $I$ is an ideal of $R[X]$. + \end{enumerate} + +} + +\prop{}{ + Let $A$ be a $R$-algebra. If $A$ is finitely generated as an $R$-module, then $A$ is a finite-type $R$-algebra. +} +\pf{ + This holds because if each element of $A$ can be expressed as an $R$-linear combination of finitely many elements of $A$, then each element of $A$ can also be expressed as a polynomial in finitely many elements of $A$ with coefficients in $R$. + + An alternative proof can be given by utilizing the universal property of the free contruction. Suppose $A$ is finitely generated as an $R$-module. Then there exists some a finite set $X=\{x_1,\cdots,x_n\}$ and a surjective $R$-linear map $\varphi:R^{\oplus X}\to A$. Define $f=\varphi\circ \iota$, where $\iota:X\to R^{\oplus X}$ is the inclusion map. + \begin{center} + \begin{tikzcd}[ampersand replacement=\&] + R^{\oplus X}\arrow[r, dashed, "\exists !\,\widetilde{j}"] \&R[X]\arrow[r, dashed, "\exists !\,\widetilde{f}"] \& A\\[0.3cm] + \& X\arrow[ul, "\iota"] \arrow[u, "j"] \arrow[ru, "f:=\varphi\circ \iota"'] \& + \end{tikzcd} + \end{center} + The universal property of free $R$-module induces a unique $R$-linear map $\widetilde{j}:R^{\oplus}\to R[X]$ such that $j=\widetilde{j}\circ \iota$. And the universal property of free commutative $R$-algebra induces a unique $R$-algebra homomorphism $\widetilde{f}:R[X]\to A$ such that $f=\widetilde{f}\circ j$. Note $f=\varphi\circ \iota=\left(\widetilde{f}\circ \widetilde{j}\right)\circ \iota$. By the uniqueness of the universal property of $R^{\oplus}$, we have $\widetilde{f}\circ \widetilde{j}=\varphi$. Since $\widetilde{f}\circ \widetilde{j}$ is surjective, $\widetilde{f}$ must be surjective, which implies $A$ is a finite-type $R$-algebra. } \chapter{Field Theory}