From f0f54c8f41bd6d95d25e63b949cb48f6a868cd98 Mon Sep 17 00:00:00 2001 From: arnaucube Date: Mon, 29 Dec 2025 18:18:01 +0100 Subject: [PATCH] add Noetherian exercises 3.3 & 3.4 --- commutative-algebra-notes.pdf | Bin 329307 -> 333808 bytes commutative-algebra-notes.tex | 70 +++++++++++++++++++++++++++++++++- 2 files changed, 69 insertions(+), 1 deletion(-) diff --git a/commutative-algebra-notes.pdf b/commutative-algebra-notes.pdf index 1b03d28e4df3134ab112082c175365fa91e29b12..bae5bb72540603c0281a7148902a602db3a2a89d 100644 GIT binary patch delta 64174 zcmYhhQ*@wR7p)oFwq3Dp+pbuZitRTxE4H1AZQB*www?a|(WCpEn{~6-n78{`bI$c> zjGGz^R7z!WNd^{14mir$g`pKVF3tp0bZ~Yyj>Nh+T0nc#ew`BqxHil$je#Od$wdnh zaQ$4VZyK647p`MRj-DD6H6Ken;CQ`uKTv7DC{tuLI$6Vx;wOq5+1|XC&Y)a|mGkrJ zzk2=hcse-!kxtED8u&Xo>nCL(ZU&~zFh_ET^fcBOkLOmECR;*+g8loQ{kBtsI;%*I z#`0Bt6R=XAx#ijr)nM3LB*R@AvechBiL0SXRnEMCL7=eK+?@G( zd_Oo#HW6rwi%GAfT3$ZJ%6@lIf3C-OEyN{&ub{+(y&w&WsK7?V(1|EmDhfZp9>-=J 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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 $LaTeX$. @@ -1343,6 +1343,74 @@ $0 \longrightarrow L \stackrel{\alpha}{\longrightarrow} M \stackrel{\beta}{\long while having $M_1 \neq M_2$. \end{proof} + + +\begin{ex}{R.3.3} +Let $A$ a ring, $I_1, \ldots, I_k$ ideals such that each $A/I_i$ is a Noetherian ring. +Prove that $\bigoplus A/I_i$ is a Noetherian $A$-module, and deduce that if $\bigcap I_i = 0$ then $A$ is also Noetherian. +\end{ex} +\begin{proof} +\begin{enumerate}[i.] + \item by Corollary \ref{R.3.5} (i), if $M_i$ Noetherian modules, then $\bigoplus M_i$ is Noetherian. + $\Longrightarrow$ thus $\bigoplus A/I_i$ is Noetherian. + \item Take the canoncial homomorphism + $$\phi: A \longrightarrow \bigoplus_{i=1}^n A/ I_i$$ + by $\phi(a) = (a+I_1, a+I_2, \ldots, a+I_n)$. + + $\phi$ is injective: $ker(\phi)= \{ a \in A | a \in I_i \forall i \}$. + + Since we're given $\cap I_i = 0$, then $ker(\phi)=\cap I_i$, and $\phi$ is injective. + + Thus, $\phi$ is the isomorphism $A \cong im(\phi)$, where $im(\phi)$ is an $A$-submodule of $\bigoplus A/I_i$. + + We know that any submodule of a Noetherian module is Noetherian, thus, since + \begin{itemize} + \item $A/I_i$ is Noetherian by hypothesis of the exercise + \item $A \cong im(\phi)$ + \item $im(\phi)$ is an $A$-submodule of $\bigoplus A/I_i$ + \end{itemize} + then, $A$ is Noetherian. +\end{enumerate} +\end{proof} + +\begin{ex}{R.3.4} + Prove that if A is a Noetherian ring and M a finite A-module, then there + exists an exact sequence $A^q \stackrel{\alpha}{\longrightarrow} A^p \stackrel{\beta}{\longrightarrow} M \longrightarrow 0$. + That is, M has a presentation as an A-module in terms of finitely many generators and relations. +\end{ex} +\begin{proof} + since $M$ fingen $~\Longrightarrow~$ generators $\{m_1, \ldots, m_2 \} \subseteq M$ span $M$. + + Let $\beta$ be a surjective $A$-linear map, which forms a free $A$-module of rank $p$ onto $M$: + \begin{align*} + \beta: A^p &\longrightarrow M\\ + (a_1, \ldots, a_p) &\longmapsto \sum_{i=1}^p a_i m_i + \end{align*} + + Let $K=ker(\beta)$. By the 1st Isomorphism Theorem, + $$M \cong A^p / K$$ + + Since $A$ is a Noetherian ring, then every free $A$-module of finite rank (eg. $A^p$) is a Noetherian module. + + Every submodule of a Noetherian module is fingen. + + $\Longrightarrow~$ since $K \subseteq A^p, ~\Longrightarrow~ K ~~(=ker(\beta))$ is fingen. + + Since $K$ fingen, let $\{k_1, \ldots, l_q\}$ be generators of $K$. + + Define $\psi: A^q \longrightarrow K$. + + Compose it with the inclusion map $i: K \longrightarrow A^p$, + $$\alpha = i \circ \psi:~ A^q \longrightarrow A^p$$ + + So we have the whole sequence $A^q \stackrel{\alpha}{\longrightarrow} A^p \stackrel{\beta}{\longrightarrow} M \longrightarrow 0$, where + \begin{itemize} + \item $\beta$ is surjective + \item $im(\alpha)=ker(\beta)$ + \end{itemize} + so that it is a exact sequence, thus, $M$ has a finite presentation. +\end{proof} + \bibliographystyle{unsrt} \bibliography{commutative-algebra-notes.bib}