From 8967b2dcc6db7bb7d270e907cd7a394c97efe8a2 Mon Sep 17 00:00:00 2001 From: arnaucube Date: Mon, 29 Dec 2025 21:54:53 +0100 Subject: [PATCH] complete Noetherian rings/modules notes, add Hilbert basis theorem, some Noetherian exercises (#2) * add proof for split exact sequences * port exercise R.2.9 * Noetherian rings: add ex. 3.2 & 3.5 * Hilbert basis theorem, Noetherian module properties * add Noetherian exercises 3.3 & 3.4 --- .github/workflows/typos.toml | 5 + commutative-algebra-notes.pdf | Bin 313487 -> 333808 bytes commutative-algebra-notes.tex | 430 +++++++++++++++++++++++++++++++++- galois-theory-notes.pdf | Bin 346340 -> 346493 bytes galois-theory-notes.tex | 8 +- 5 files changed, 431 insertions(+), 12 deletions(-) diff --git a/.github/workflows/typos.toml b/.github/workflows/typos.toml index 19ff2b7..de9c696 100644 --- a/.github/workflows/typos.toml +++ b/.github/workflows/typos.toml @@ -1,4 +1,9 @@ [default.extend-words] +iddeal = "ideal" +iddeals = "ideals" +allpha = 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a/commutative-algebra-notes.tex +++ b/commutative-algebra-notes.tex @@ -80,7 +80,7 @@ \maketitle \begin{abstract} - Notes taken while studying Commutative Algebra, mostly from Atiyah \& MacDonald book \cite{am} and Reid's book \cite{reid}. + Notes taken while studying Commutative Algebra, mostly from Atiyah \& MacDonald book \cite{am} and Reid's book \cite{reid}. For the exercises, I follow the assignments listed at \cite{mit-course}. Usually while reading books and papers I take handwritten notes in a notebook, this document contains some of them re-written to $LaTeX$. @@ -178,7 +178,7 @@ $$A \supset \mM ~\text{or}~ (A, \mM) ~\text{or}~ (A, \mM, K)$$ \end{defn} -\subsection{$\mathbb{Z}$ and $K[X]$, two Principal Ideal Domains} +\subsection{Z and K[X], two Principal Ideal Domains} \begin{lemma}{} @@ -696,11 +696,124 @@ $A[\psi] \subset End M$ is the subring geneerated by $A$ and the action of $\psi \end{proof} -\begin{prop}{AM.2.10} \label{2.10} - Split exact sequence. TODO +\subsection{Sequences} + +\begin{defn}{R.2.9.a}{Exact Sequence} + Let a sequence of homomorphisms +$$L \stackrel{\alpha}{\longrightarrow} M \stackrel{\beta}{\longrightarrow} N$$ +It is \emph{exact} at $M$ if $im(\alpha)=ker(\beta)$. + +ie. $\beta \circ \alpha = 0$ and $\alpha$ maps surjectively to +$ker(\beta)$. +\end{defn} + +\begin{defn}{R.2.9.b}{Short Exact Sequence (s.e.s.)} \label{2.9} +$$0 \longrightarrow L \stackrel{\alpha}{\longrightarrow} M \stackrel{\beta}{\longrightarrow} N \longrightarrow 0$$ +is exact $\Longleftrightarrow~ L \subset M$ and $N=M / L$. + +Properties: +\begin{itemize} + \item $\alpha$ injective + \item $\beta$ surjective + \item $\alpha:~ L \Longrightarrow ker \beta$ + \item $\beta$ induces $M/\alpha(L) \longrightarrow N$ +\end{itemize} +\end{defn} + +\begin{prop}{R.2.10}{Split exact sequence} \label{2.10} + For the previous s.e.s., 3 equivalent conditions: + \begin{enumerate}[i.] + \item $\exists$ isomorphism $M \cong L \oplus N$, with + \begin{align*} + \alpha:~ &m \longmapsto (m,0)\\ + \beta:~ &(m, n) \longmapsto n + \end{align*} + + \item $\exists$ a \emph{section} of $\beta$, that is, a map $s: N \longrightarrow M$ such that $\beta \circ s = id_N$ + \item $\exists$ a \emph{retraction} of $\alpha$, that is, a map $r: M \longrightarrow L$ such that $r \circ \alpha = id_L$ + \end{enumerate} + + If all i, ii, iii are satisfied, it is a split exact sequence. \end{prop} +\begin{proof} + Intuitively, when a s.e.s. \emph{splits} it means that the middle module $M$ is the direct sum of the other (outer) two modules, ie. $M = L \oplus N$. + + \begin{itemize} + \item[(i to ii, iii)] + if $M \cong L \oplus N$ such that $\alpha:~ m \longmapsto (m,0),~~ \beta:~ s(m, n) \longmapsto n$, we can define the maps + + for ii: + \begin{align*} + s:~ N &\longrightarrow L \oplus N\\ + s(n) &\longmapsto (0, n) + \end{align*} + + Then $\beta(s(n))=\beta(0,n)$, so $\beta \circ s = id_N$. -\section{Noetherian rings} + for iii: + \begin{align*} + r:~ L \oplus N &\longrightarrow L\\ + r(m,n) &\longmapsto m + \end{align*} + + Then $r(\alpha(m))=r(m,0)$, so $r \circ \alpha = id_L$. + + \item[(ii to i)] + assume $s: N \longrightarrow M$ such that $\beta \circ s = id_M$ + + Want to show $M \cong im(\alpha) \oplus im(s)$. + + $\forall m \in M$, consider $m - s(\beta(m))$, apply $\beta$ to it:\\ + $\beta(m - s(\beta(m))) = \beta(m) - (\beta \circ s)(\beta(m)) = \beta(m) - \beta(m) = 0$ + + Since $ker(\beta) = im(\alpha),~~ \exists! l \in L ~\text{such that}~ \alpha(l) = m - s(\beta(m))$. + + Thus $m = \alpha(l) + s(\beta(m))$. + + \vspace{0.3cm} + Now, suppose $x \in im(\alpha) \cap im(s)$, then $x = \alpha(l)=s(n)$, apply $\beta$ to it: $\beta(\alpha(l)) = \beta(s(n)) ~\Longrightarrow~ 0=n$. + + If $n=0$, then $s(n)=0$, so the intersection is $\{0\}$. + + \vspace{0.3cm} + Define + \begin{align*} + \phi: L \oplus N &\longrightarrow M\\ + \phi(l,n) &\longmapsto \alpha(l)+s(n) + \end{align*} + This isomorphism satisfies the required conditions. + + \item[(iii to i)] similar to the previous one. + \end{itemize} + + \vspace{0.3cm} + TL;DR:\\ + +$$ +0 \longrightarrow L + \substack{ + \stackrel{\alpha}{\longrightarrow}\\ + \stackrel{\longleftarrow}{r} + } + \substack{M \\[0.5ex] \cong L \oplus N} + \substack{ + \stackrel{\beta}{\longrightarrow}\\ + \stackrel{\longleftarrow}{s} + } + N \longrightarrow 0 +$$ +\begin{align*} + \alpha:~ &l \longmapsto (l,0)\\ + r:~ &(m,n) \longmapsto m\\ + \alpha &\circ r = id_L\\ + \beta:~ &(l,n) \longmapsto n\\ + s:~ &n \longmapsto (0,n)\\ + \beta &\circ s = id_N +\end{align*} + +\end{proof} + +\section{Noetherian rings (and modules)} \begin{defn}{}{Ascending Chain Condition} A partially orddered set $\Sigma$ has the \emph{ascending chain condition} (a.c.c.) if every chain @@ -717,8 +830,8 @@ $\Longrightarrow~ \Sigma$ has the a.c.c. iff every non-empty subset $S \subset \ \item the set $\Sigma$ of ideals of $A$ has the a.c.c.; in other words, every increasing chain of ideals $$I_1 \subset I_2 \subset \ldots \subset I_k \subset \ldots$$ eventually stops, that is $I_k = I_{k+1}=\ldots$ for some $k$. - \item every nonempty set $S$ of iddeals has a maximal element - \item every iddeal $I \subset A$ is finitely generated + \item every nonempty set $S$ of ideals has a maximal element + \item every ideal $I \subset A$ is finitely generated \end{enumerate} If these conditions hold, then $A$ is \emph{Noetherian}. \end{defn} @@ -741,7 +854,7 @@ As in with rings, it is equivalent to say that -\begin{prop}{R.3.4.P} +\begin{prop}{R.3.4.P}\label{R.3.4.P} Let $0 \longrightarrow L \xrightarrow{\ \alpha \ } M \xrightarrow{\ \beta \ } N \longrightarrow 0$ be a s.e.s. (split exact sequence, \ref{2.10}). Then, $M$ is Noetherian $\Longleftrightarrow~ L$ and $N$ are Noetherian. @@ -765,7 +878,7 @@ As in with rings, it is equivalent to say that $$L \cap M_1 = L \cap M_2 ~\text{and}~ \beta(M_1) = \beta(M_2) ~\Longrightarrow~ M_1 = M_2$$ \end{lemma} \begin{proof} - if $m\in M_2$, then $\beta(m) \in \beta(M_1) = \beta(M_2)$, so that there is an $n \in M_1$ such that $\beta(m)=\beta(m)$. + if $m\in M_2$, then $\beta(m) \in \beta(M_1) = \beta(M_2)$, so that there is an $n \in M_1$ such that $\beta(m)=\beta(n)$. Then $\beta(m-n)=0$, so that $$m - n \in M_2 \cap ker(\beta)=M_1 \cap ker(\beta)$$ @@ -773,6 +886,114 @@ As in with rings, it is equivalent to say that Hence $m \in M_1$, thus $M_1 = M_2$. \end{proof} + + +\begin{cor}{R.3.5}{Properties of Noetherian modules.}\label{R.3.5} + \begin{enumerate}[i.] + \item if $\forall i \in [r],~~M_i$ are Noetherian modules, then + $\bigoplus_{i=1}^r M_i$ is Noetherian. + \item if $A$ a Noetherian ring, then an $A$-module $M$ is Noetherian iff it is finite over $A$. + \item if $A$ a Noetherian ring, $M$ a finite module, then any submodule $N \subset M$ is again finite. + \item if $A$ a Noetherian ring, and $\psi: A \longrightarrow B$ a ring homomorphism such that $B$ is a finite $A$-module, then $B$ is a Noetherian ring. + \end{enumerate} +\end{cor} +\begin{proof} + \begin{enumerate}[i.] + \item a direct sum $M_1 \oplus M_2$ is a particular case of an exact sequence. + + Then, Proposition \ref{R.3.4.P} proves this statement when $r=2$. The case $r>2$ follows by induction. + + \item if $M$ finite, then $\exists~$ surjective homomorphism + $$A^r \longrightarrow M \longrightarrow 0$$ + for some $r$, so that $M$ is a quotient + $$M \cong A^r / N$$ + for some submodule $N \subset A^r$. + + $A^r$ is a Noetherian module by i., so $M$ is Noetherian due Proposition \ref{R.3.4.P}. + + Conversely, $M$ Noetherian implies $M$ finite. + + item as in previous implications:\\ + $M$ finite and $A$ Noetherian $\Longrightarrow$ $M$ is Noetherian,\\ + $\Longrightarrow$ since $N \subseteq M$, then $N$ is Noetherian too\\ + $\Longrightarrow$ which implies that $N$ is a finite $A$-module. + + \item $B$ is Noetherian as an $A$-module; but ideals of $B$ are submodules of $B$ as an $A$-submodule, so that $B$ is a Noetherian ring. + \end{enumerate} +\end{proof} + + +\vspace{0.5cm} +\begin{thm}{R.3.6}{Hilbert basis theorem} \label{hilbert-basis} + if $A$ a Noetherian ring, then so is the polynomial ring $A[x]$. +\end{thm} +\begin{proof} + Prove that any ideal $I \subset A[x]$ is fingen. + + Define auxiliary sets $J_n \subset A$ by + $$J_n = \{ a \in A ~|~ \exists f \in I ~\text{s.th.}~ f = a x^n + b_{n-1}x^{n-1} + \ldots b_0 \}$$ + ie. $J_n$ is the set of leading coefficients of $I$ of degree $n$. + + $J_n$ is an ideal, since $I$ is an ideal. + + $J_n \subset J_{n+1}$, since for $f \in I$ also $x f \in I$. + + Therefore $J_1 \subset J_2 \subset \ldots \subset J_k \subset \ldots$ is an increasing chain of ideals. + + Using the assumption that $A$ is Noetherian, deduce that $J_n = J_{n+1}$ for some $n$. + + For each $m \leq n, ~~ J_m \subset A$ is fingen, ie. + $$J_m = (a_{m,1}, \ldots a_{m, r_m})$$ + + By definition of $J_m$, for each $a_{m,j}$ with $1 \leq j \leq r_m$,\\ + $\exissts$ a polynomial $f_{m, j} \in I$ of degree $m$ having the leading coefficient $a_{m, j}$. + + $$\Longrightarrow~~ \{ f_{m,j} \}_{m(BkD89ejR2IeT`_ zZQ+R1To;qef=NZ3RE1WCaG{FHyto)CGJZdK)$S@TN=51DvfkkGvva|E2)H56e^`L? 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