From 8eafa372a23e00ed2ac1934ab935832c175c136e Mon Sep 17 00:00:00 2001 From: arnaucube Date: Fri, 26 Dec 2025 12:50:40 +0100 Subject: [PATCH] add proof for split exact sequences --- commutative-algebra-notes.pdf | Bin 313487 -> 320222 bytes commutative-algebra-notes.tex | 107 +++++++++++++++++++++++++++++++++- 2 files changed, 105 insertions(+), 2 deletions(-) diff --git a/commutative-algebra-notes.pdf b/commutative-algebra-notes.pdf index 23403c0e9a8aecab0278208398bcc5b5d3779dd5..5201cde176aa670dd5315ce441c17dfe253b1e75 100644 GIT binary patch delta 107070 zcmX`Sb8z6j7xrD-w!O7&ZryFSwr%5EYirxKZQI7yw%e_q`}w`|yfc}}Tsd=+{FBV* zI+Mi4qyHU7uR}2c^ zI~12+RrqqgeYJd$>`E~sJwhp|g%^$*gmJ20Q!D7(zYgd3!7v!!f0mPVx^eA~-Z}by zd3;VR__qL3TNlBycc0f4BU~^9hH71hT%>!qb=wWma9pFOj+p7>yryxW@eCDo21c zDAw!mhMa6;QxJ{>WV(xt$iOl~beI8dZX`EqQj;W3CegUx*q{`K067lWNmB;yzN;bl z#c$dm8X9yb_5{%Cq;kp@qN-L}XA6q)Bpz;QSeOjiw8CW$QzS40@s1V$4^;6Zl(b3J zj49l*LnKjNN1yDPcm%G9cPXZPaj5}84%Aj@`jLFeiTNdS={P9DostBIpkRJn!JO-f 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-\begin{prop}{AM.2.10} \label{2.10} - Split exact sequence. TODO +\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 $N=M / L$. +\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 $r \circ \allpha = id_L$ + \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. $M = L \oplus N$. + + \begin{itemize} + \item[(i \Longrightarrow 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 $\beta \circ s = id_N$. + + 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 \Longrightarrow 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 \Longrightarrow 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}