From 7f29627348d698b72a008fac1940de0e333e6e46 Mon Sep 17 00:00:00 2001 From: arnaucube Date: Sat, 27 Dec 2025 18:25:14 +0100 Subject: [PATCH] port exercise R.2.9 --- .github/workflows/typos.toml | 5 +++ commutative-algebra-notes.pdf | Bin 320222 -> 317234 bytes commutative-algebra-notes.tex | 77 +++++++++++++++++++++++++++++----- 3 files changed, 72 insertions(+), 10 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 = "alpha" + +# strings that are not a typo: thm = "thm" # equations stuff diff --git a/commutative-algebra-notes.pdf b/commutative-algebra-notes.pdf index 5201cde176aa670dd5315ce441c17dfe253b1e75..4935616cfa645083ad709d79017f36c649e2ff3b 100644 GIT binary patch delta 125874 zcmV)XK&`*t#S^lw6A>j)L`E$!E;R}z(A34)3O6v35fcJ6Gm~I!DSzEqOLN;c5WeSE 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End M$ is the subring geneerated by $A$ and the action of $\psi \end{proof} +\subsection{Sequences} + \begin{defn}{R.2.9.a}{Exact Sequence} Let a sequence of homomorphisms $$L \stackrel{\alpha}{\longrightarrow} M \stackrel{\beta}{\longrightarrow} N$$ @@ -708,6 +710,14 @@ $ker(\beta)$. \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} @@ -720,16 +730,16 @@ is exact $\Longleftrightarrow~ L \subset M$ and $N=M / L$. \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 $r \circ \allpha = id_L$ + \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 (outter) two modules, ie. $M = L \oplus N$. + 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 \Longrightarrow ii, iii)] + \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: @@ -748,7 +758,7 @@ is exact $\Longleftrightarrow~ L \subset M$ and $N=M / L$. Then $r(\alpha(m))=r(m,0)$, so $r \circ \alpha = id_L$. - \item[(ii \Longrightarrow i)] + \item[(ii to i)] assume $s: N \longrightarrow M$ such that $\beta \circ s = id_M$ Want to show $M \cong im(\alpha) \oplus im(s)$. @@ -773,7 +783,7 @@ is exact $\Longleftrightarrow~ L \subset M$ and $N=M / L$. \end{align*} This isomorphism satisfies the required conditions. - \item[(iii \Longrightarrow i)] similar to the previous one. + \item[(iii to i)] similar to the previous one. \end{itemize} \vspace{0.3cm} @@ -820,8 +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} @@ -868,7 +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 $$m - n \in M_2 \cap ker(\beta)=M_1 \cap ker(\beta)$$ @@ -916,7 +926,7 @@ The exercises that start with \textbf{R} are the ones from the book \cite{reid}, Let $$\psi^{-1}(P) = \{ a \in A | \psi(a) \in P \} = A \cap P$$ - The claim is that $\psi^{-1}(P)$ is prime iddeal of $A$. + The claim is that $\psi^{-1}(P)$ is prime ideal of $A$. \begin{enumerate}[i.] \item show that $\psi^{-1}(P)$ is an ideal of $A$:\\ @@ -1104,6 +1114,53 @@ The exercises that start with \textbf{R} are the ones from the book \cite{reid}, \subsection{Exercises Chapter 2} +\begin{ex}{R.2.9} +$0 \longrightarrow L \stackrel{\alpha}{\longrightarrow} M \stackrel{\beta}{\longrightarrow} N \longrightarrow 0$ is a s.e.s. of $A$-modules. Prove that if $N, L$ are finite over $A$, then $M$ is finite over $A$. +\end{ex} +\begin{proof} + Denote the generators of $L$ and $N$ respectively as + \begin{align*} + \{l_1, \ldots, l_k \} &\subseteq L\\ + \{n_1, \ldots, n_p \} &\subseteq N + \end{align*} + + By s.e.s. definition, + \begin{itemize} + \item[-] $\alpha$ is injective (one-to-one), so + $$\forall l_i \in L,~ \exists~ x_i \in M ~\text{s.th.}~ \alpha(l_i)=x_i$$ + + \item[-] $\beta$ is surjective (onto), so + $$\forall n_j \in N,~ \exists~ y_j \in M ~\text{s.th.}~ \beta(y_j)=n_j$$ +\end{itemize} + + We will show that $\{x_1, \ldots, x_k, y_1, \ldots, y_p \}$ generate $M$, and thus $M$ is finite: + + Let $m \in M$, then $\beta(m) \in N$, and + $$\beta(m) = \sum_{j=1}^p a_j n_j ~~~\text{with}~ a_j \in A$$ + + Take $m' \in M$, with $m' = \sum a_j y_j$, then + $$\beta(m') = \sum a_j \beta(y_j) = \sum a_j n_j = \beta(m)$$ + + Then, since $\beta(m)=\beta(m')~~ \Longrightarrow~~ \beta(m-m')=0$, thus + $$(m-m') \in ker(\beta)$$ + + By \emph{exactness} property, since $\alpha: L \longrightarrow ker(\beta)$, we have $ker(\beta)=im(\alpha)$. + + Therefore, $\exists~ l \in L$ such that $\alpha(l)= m-m'$. + + Since $\{l_i\}_k$ generate $L$, + $$l = \sum^k b_i l_i$$ + thus + $$m-m' = \alpha(l) = \alpha(\underbrace{\sum b_i l_i}_{l}) = \sum b_i \underbrace{\alpha(l_i)}_{x_i} = \sum b_i x_i$$ + + Rearrange, + $$m = m' + \sum b_i x_i = \sum_{j=1}^p a_j y_j + \sum_{i=1}^k b_i x_i ~~~~~ \forall m \in M$$ + + So, $L$ provides $k$ generators for the kernel part of $M$, $N$ provides $p$ "lifts" for the quotient part of $M$; thus $M$ is generated by $k+p$ elements.\\ + Thus $M$ is finitely generated over $A$. + +\end{proof} + \bibliographystyle{unsrt} \bibliography{commutative-algebra-notes.bib}