From d140e957a40b14b2ba174c30c8a5f64a6a5aa60a Mon Sep 17 00:00:00 2001 From: Hearot Date: Tue, 5 Sep 2023 10:25:00 +0200 Subject: [PATCH] feat(geometria): aggiunge una dimostrazione del teorema di struttura --- .../main.pdf | Bin 0 -> 305771 bytes .../main.tex | 324 ++++++++++++++++++ 2 files changed, 324 insertions(+) create mode 100644 Geometria/Articoli/3. Il teorema di struttura dei moduli finitamente generati su un PID/main.pdf create mode 100644 Geometria/Articoli/3. Il teorema di struttura dei moduli finitamente generati su un PID/main.tex diff --git a/Geometria/Articoli/3. Il teorema di struttura dei moduli finitamente generati su un PID/main.pdf b/Geometria/Articoli/3. 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Il teorema di struttura dei moduli finitamente generati su un PID/main.tex b/Geometria/Articoli/3. Il teorema di struttura dei moduli finitamente generati su un PID/main.tex new file mode 100644 index 0000000..fae5331 --- /dev/null +++ b/Geometria/Articoli/3. Il teorema di struttura dei moduli finitamente generati su un PID/main.tex @@ -0,0 +1,324 @@ +\documentclass[12pt]{scrartcl} +\usepackage{notes_2023} + +\begin{document} + \title{Il teorema di struttura dei moduli finitamente generati su un PID} + \date{\today} + \maketitle + + In questo documento dimostro\footnote{ + Il contenuto di questo documento è ispirato a quello + del capitolo \textit{The PID structure theorem} del + \textit{Napkin} di Evan Chen, reperibile su + \url{https://github.com/vEnhance/napkin}. + } un enunciato fondamentale del celebre teorema + di struttura dei moduli finitamente generati su un PID. + Storicamente questo teorema nasce come una generalizzazione + del teorema di struttura dei gruppi abeliani finitamente generati e diventa + poi un potente strumento da cui derivano alcune celebri forme canoniche + dell'algebra lineare, come la forma normale di Jordan o la forma + canonica razionale. + + \begin{theorem}[di struttura dei moduli finitamente generati su un PID] + Sia $M$ un modulo finitamente generato su un PID $R$. Allora + esistono unici (a meno di associati) $d_1$, ..., $d_k \in R$ + tali per cui $d_1 \mid d_2 \mid \cdots \mid d_k$ e + $M \cong R/(d_1) \times \cdots \times R/(d_k)$. + \end{theorem} + + Si osserva sin da subito che il teorema può riscriversi in modo alternativo + utilizzando il teorema cinese del resto. Infatti, se $d_i$ viene + scritto nella sua fattorizzazione in primi\footnote{ + Un PID è sempre un UFD e dunque una tale fattorizzazione esiste sempre. + } $p_1^{k_1} \cdots p_{n_i}^{k_{n_i}}$, allora vale che: + \[ + R/(d_i) \cong R/\!\left(p_1^{k_1}\right) \times \cdots \times R/\bigl(p_{n_i}^{k_{n_i}}\bigr). + \] + Pertanto il teorema di struttura può riscriversi come: + + \begin{theorem} + Sia $M$ un modulo finitamente generato su un PID $R$. Allora + esistono unici (a meno di associati) $p_1$, ..., $p_n \in R$ primi + e $k_1$, ..., $k_n \in \NN$ tali per cui: + \[ + M \cong R/\bigl(p_1^{k_1}\bigr) \times \cdots \times R/\bigl(p_n^{k_n}\bigr). + \] + \end{theorem} + + \begin{remark} + D'ora in poi, mi riferisco ad $M$ come un modulo finitamente generato + su un PID $R$. + \end{remark} + + La forma del primo enunciato è detta \textbf{decomposizione in fattori invarianti}, + mentre quella del secondo è detta \textbf{decomposizione primaria}. + + \begin{definition}[fattori invarianti] + Si chiamano \textbf{fattori invarianti} i vari $d_i$ che compaiono + nella decomposizione in fattori invarianti di $M$. + \end{definition} + + \begin{definition}[divisori elementari] + Si chiamano \textbf{divisori elementari} i vari $p_i^{k_i}$ che + compaiono nella decomposizione primaria di $M$. + \end{definition} + + \begin{definition}[rango di un modulo] + Si definisce rango di $M$ il numero di volte in cui compare $0$ + tra i fattori invarianti. + \end{definition} + + Il documento prosegue con la dimostrazione dell'esistenza\footnote{ + La dimostrazione dell'unicità è omessa. Alcuni commenti e risultati al + riguardo sono reperibili su \url{https://math.stackexchange.com/q/4193/769611}. + } dei fattori + invarianti, secondo il seguente schema: + + \begin{itemize} + \item Poiché $M$ è finitamente generato, esiste un'applicazione lineare + surgettiva $\psi$ da $R^m$ a $M$, dove $m$ è il numero di generatori di $M$, + \item Poiché $R$ è noetheriano\footnote{ + Un modulo è noetheriano se ogni suo sottomodulo è finitamente + generato. + }, anche $R^m$ lo è; allora + $\Ker \psi$ è finitamente generato da $n$ elementi, + \item Si può costruire allora un'altra applicazione lineare surgettiva $\phi$ da + $R^n$ a $\Ker \psi$, + \item Immergendo naturalmente $\Ker \psi$ in $M$ tramite la mappa naturale + $\tau$, si osserva che $\Ker \psi = \Im(\tau \circ \phi)$, + dove $T = \tau \circ \phi$ è una mappa da $R^n$ a $R^m$, + \item Dal momento che $T$ mappa un modulo libero\footnote{ + Un modulo è libero se ammette una base, proprio come $R^n$. + } ad un altro, $T$ può identificarsi con una matrice, + \item Si scrive la matrice di $T$ nella forma normale di Smith, dove compaiono + i fattori invarianti $d_1$, ..., $d_k$; allora esistono + $\vv 1$, ..., $\vv n$ base di $R^n$ tale per cui + $R^n = \langle \vv 1 \rangle \oplus \cdots \oplus \langle \vv n \rangle$ e + $\Im T = \langle d_1 \vv 1 \rangle \oplus \cdots \oplus \langle d_k \vv k \rangle \oplus \langle 0 \vv{k+1} \rangle \oplus \cdots \oplus \langle 0 \vv n \rangle$, + \item Si costruisce un applicazione lineare $\iota$ da $R^n$ a + $R/(d_1) \times \cdots \times R/(d_k) \times R \times \cdots \times R$ tale per cui + $\Ker \iota = \Im T$; allora, per il primo teorema di isomorfismo, + vale che: + \[ M \cong R^n/\Ker \psi = R^n/\Im T \cong R/(d_1) \times \cdots \times R/(d_k) \times R \times \cdots \times R, \] + concludendo la dimostrazione. + \end{itemize} + + Questo schema è in parte riassunto dal seguente diagramma commutativo: + \[\begin{tikzcd} + & {\Ker \psi} \\ + {R^n} && {R^m} & M + \arrow["\phi", two heads, from=2-1, to=1-2] + \arrow["\tau", tail, from=1-2, to=2-3] + \arrow["\psi", two heads, from=2-3, to=2-4] + \arrow["T", from=2-1, to=2-3] + \end{tikzcd}\] + + \section*{Dimostrazione} + + Dal momento che $M$ è finitamente generato, esistono $\ww 1$, ..., + $\ww m \in M$ tali per cui $\langle \ww 1, \ldots, \ww m \rangle = M$. + Allora si costruisce l'applicazione lineare $\psi : R^m \to M$ univocamente + determinata dalla relazione $\e i \xmapsto{\psi} \ww i$. Si osserva che + $\psi$ è surgettiva: per ogni $\w \in M$, esistono $\alpha_1$, ..., $\alpha_m \in R$ + tali per cui $\w = \alpha_1 \ww 1 + \ldots + \alpha_m \ww m$; allora + $(\alpha_1, \ldots, \alpha_m)^\top \xmapsto{\psi} \w$. \medskip + + Poiché $\psi$ è surgettiva, $\Im \psi = M$, e quindi per il primo teorema + di isomorfismo vale che: + + \begin{equation} + \label{eq:primo_isomorfismo} + M \cong R^m/ \Ker \psi. + \end{equation} \vskip 0.1in + + Dal momento che $R$ è un PID, $R$ è in particolare noetheriano (è infatti + monogenerato). Allora anche $R^m$ è noetheriano, come dimostra il seguente lemma\footnote{ + Si costruisce infatti $R^m$ come il prodotto di $R$ effettuato $m$ volte. + }: + + \begin{lemma} + Siano $M$ ed $N$ due $R$-moduli noetheriani. Allora $M \times N$ è + anch'esso noetheriano. + \end{lemma} + + \begin{proof} + Sia $L$ un sottomodulo di $M \times N$. Si considerino i seguenti + sottomoduli: + \[ A = \{ \vec m \in M \mid (\vec m, \vec 0) \in L \} \subseteq M,\] + \[ B = \{ \vec n \in N \mid \exists \vec m \in M \mid (\vec m, \vec n) \in L \} \subseteq N.\] + Si osserva che $A$ e $B$ sono finitamente generati, essendo rispettivamente + sottomoduli degli anelli noetheriani $M$ ed $N$. Allora esistono + $\vec{a_1}$, ..., $\vec{a_s} \in A$ e $\vec{b_1}$, ..., $\vec{b_t} \in B$ tali per cui + $A = \langle \vec{a_1}, \ldots, \vec{a_s} \rangle$ e $B = \langle \vec{b_1}, \ldots, \vec{b_t} \rangle$. + \bigskip + + + Sia $\vec \ell \in L$. Allora esistono $\vec m \in M$, $\vec n \in N$ tali per cui + $\vec \ell = (\vec m, \vec n)$. Inoltre $\vec n \in B$, e dunque esistono $\beta_1$, ..., $\beta_t$ + tali per cui $\vec n = \beta_1 \vec{b_1} + \ldots + \beta_t \vec{b_t}$. + Siano $\vec{x_1}$, ..., $\vec{x_s} \in M$ tali per cui + $(\vec{x_i}), \vec{b_i}) \in L$ e si ponga + $\vec x = \beta_1 \vec{x_1} + \ldots + \beta_t \vec{x_t}$. + Si ottiene dunque che: + \[ + (\vec m, \vec n) = \underbrace{(\vec m - \vec x, \vec 0)}_{\in L} + \beta_1 (\vec{x_1}, \vec{b_1}) + \ldots + \beta_t (\vec{x_t}, \vec{b_t}). + \] + Allora $\vec{m'} := \vec m - \vec x \in A$, e dunque esistono $\alpha_1$, ..., $\alpha_s$ + tali per cui $\vec{m'} = \alpha_1 \vec{a_1} + \ldots + \alpha_s \vec{a_s}$. + Pertanto vale che: + \[ + (\vec m, \vec n) = \sum_{i=1}^s \alpha_i (\vec{a_i}, \vec 0) + \sum_{j=1}^t \beta_j (\vec{x_j}, \vec{b_j}), + \] + da cui si conclude che $L$ è finitamente generato, e dunque che $M \times N$ + è noetheriano. + \end{proof} + + Poiché allora $R^m$ è noetheriano, $\Ker \psi$ è finitamente generato, e dunque + esistono $\uu 1$, ..., $\uu n \in \Ker \psi$ tali per cui + $\Ker \psi = \langle \uu 1, \ldots, \uu n \rangle$. Allora, analogamente + a prima, si può costruire un'applicazione lineare $\phi : R^n \to \Ker \psi$ tale + per cui $\phi$ sia surgettiva, mappando $\e i$ a $\uu i$. \bigskip + + + Si considera adesso l'immersione naturale $\tau$ di $\Ker \psi$ in $R^m$, + ossia l'applicazione lineare $\tau : \Ker \psi \to R^m$ tale per cui + $\tau(\U) = \U$ per ogni $\U \in \Ker \psi$. Chiaramente $\tau$ è + iniettiva e $\Im \tau = \Ker \psi$. Detta allora $T = \tau \circ \phi$, + vale che $\Im T = \Im (\tau \circ \phi) = \Im \tau = \Ker \psi$, dove + si è usata la surgettività della mappa $\phi$. Sostituendo allora + $\Im \tau$ nell'identità \eqref{eq:primo_isomorfismo}, si ottiene che: + + \begin{equation} + \label{eq:isomorfismo_M_T} + M \cong R^m/\Im T. + \end{equation} \vskip 0.1in + + Pertanto adesso è sufficiente studiare l'applicazione $T$ per ricavare + la tesi. Dal momento che $T$ ha come dominio il modulo libero $R^n$ e come codominio + $R^m$, $T$ si può rappresentare come una matrice $S$ a elementi in $R$ + dove $S^j$ è la valutazione in $\e j$ di $T$. Adesso il punto cruciale + della dimostrazione dipende dalla seguente proposizione: + + \begin{proposition}[forma normale di Smith] + Sia $S = (s_{ij})$ una matrice $m \times n$ a elementi in $R$. Allora esistono + unici (a meno di associati) + $d_1$, ..., $d_k \in R$ con $d_1 \mid d_2 \mid \cdots \mid d_k$ e + $k = \min\{m, n\}$ + tali per cui esistano due basi $\basis$, $\basis'$ di $R^n$ e $R^m$ + che soddisfano + l'identità\footnote{ + La matrice $S'$ mostra in realtà il caso in cui $m > n$. Tuttavia la + struttura di $S'$ si può generalizzare facilmente per $m \leq n$. + }: + \[ + S' := M^{\basis}_{\basis'}(f_S) = \begin{pmatrix} + d_1 & 0 & 0 & 0 & \dots & 0 \\ + 0 & d_2 & 0 & 0 & \dots & 0 \\ + \vdots & \vdots & \ddots & \vdots & \dots & \vdots \\ + 0 & 0 & 0 & d_k & \dots & 0 + \end{pmatrix}, + \] + dove con $f_S$ si intende l'applicazione lineare indotta dalla matrice $S$. + \end{proposition} + + \begin{proof}[Dimostrazione dell'esistenza] + Le uniche operazioni consentite sulla matrice che consentono di + individuare una nuova coppia di basi opportune sono le stesse\footnote{ + Si verifica facilmente che ogni tale operazione modifica una delle + due basi tramite le operazioni elementari di riordinamento e di somma + per un multiplo. + } + contemplate dall'algoritmo di eliminazione di Gauss eccetto per quella + di moltiplicazione di una riga (o di una colonna) per un elemento non invertibile + di $R$. Pertanto, si presenta la dimostrazione dell'esistenza della + forma normale di Smith come un algoritmo che permette di alterare la + matrice tramite le uniche operazioni consentite. \medskip + + + Se $S=0$, la tesi è già dimostrata. Altrimenti, si possono utilizzare le + operazioni consentite per far sì che si verifichi $s_{11} \neq 0$. Si + distinguono ora tre casi: + \begin{enumerate}[(i)] + \item $s_{11}$ non divide almeno un elemento di $S^1$, + \item $s_{11}$ non divide almeno un elemento di $S_1$, + \item $s_{11}$ divide tutti gli elementi di $S^1$ e $S_1$. + \end{enumerate} + + Se $s_{11}$ non divide almeno un elemento di $S^1$, detto $s_{i1}$, + si possono effettuare operazioni di riga per spostare $s_{i1}$ + in $s_{21}$. Poiché $R$ è un PID, l'ideale $(s_{11}, s_{21})$ è monogenerato, + e dunque esistono $\alpha$, $\beta \in R$ tali per cui + $(s_{11}, s_{21}) = (\alpha s_{11} + \beta s_{21})$. Vale inoltre + che $(\alpha, \beta) = R$\footnote{ + Infatti, se $d = \alpha s_{11} + \beta s_{21}$, vale che + $\frac{s_{11}}{d} \alpha + \frac{s_{21}}{d} \beta = 1$. + }, e dunque che esistono $\gamma$, $\delta \in R$ + tali per cui $\gamma \alpha + \delta \beta = 1$. Si può + allora moltiplicare la matrice a sinistra per la matrice + invertibile\footnote{ + Tale matrice è invertibile poiché unimodulare. Alternativamente, si + può fornire esplicitamente l'inverso $\SMatrix{ \gamma & -\beta \\ \delta & \alpha }$. + }: + \[ \Matrix{\alpha & \beta \\ -\delta & \gamma }, \] + opportunamente inserita al posto del blocco $(I_m)^{1,2}_{1,2}$ in $I_m$. + Si effettua un analogo ragionamento per il caso (ii), moltiplicando a destra + per la stessa matrice, opportunamente trasposta. Si continua a effettuare questo + tipo di moltiplicazioni fino a quando non si ricade nel caso (iii). Il caso + (iii) è sempre raggiungibile, dal momento che ad ogni operazione si sostituisce + $s_{11}$ con un suo divisore, creando una successione ascendente di ideali + $(a_1) \subseteq (a_2) \subseteq (a_3) \subseteq \cdots$ che, per la noetherianità + di $R$, deve stabilizzarsi. \medskip + + + Giunti nel caso (iii), si annullano i primi elementi di tutte le colonne + di $S$ eccetto per $S^1$, sottraendo un opportuno multiplo di $S^1$. Si + effettua poi la stessa cosa per le righe eccetto che per $S_1$. Se + $m=1$ o $n=1$, l'algoritmo termina. + Altrimenti si ottiene una matrice della forma: + \[ \Matrix{s_{11} & 0 \\ 0 & \tilde{S}}. \] + Se $s_{11}$ divide ora ogni elemento di $\tilde{S}$, si riapplica l'algoritmo + soltanto su $\tilde{S}$ (ogni operazione su $\tilde{S}$ può essere estesa a + un'operazione su $S$ che lascia l'elemento $s_{11}$ invariato. Se invece esiste + un elemento $s_{ij}$ non diviso da $s_{11}$, si somma la riga $S_i$ alla + riga $S_1$ (o la colonna $S^j$ alla colonna $S^1$) e si riapplica l'algoritmo. + Per le stesse motivazioni di prima, ad un passo dell'algoritmo $s_{11}$ dovrà dividere + ogni elemento di $\tilde{S}$. \medskip + + + Dopo aver impiegato con successo l'algoritmo, si otterrà una matrice nella + forma normale di Smith, dimostrando la tesi. + \end{proof} + + Pertanto esistono due basi $\basis$ e $\basis' = \{ \vv 1, \ldots, \vv n \}$ + di $R^m$ e $R^n$ tali per cui $M^\basis_{\basis'}(f_S)$ assume la forma normale + di Smith. In particolare, vale che: + \[ + \Im T = \langle d_1 \vv 1 \rangle \oplus \cdots \oplus \langle d_k \vv k \rangle \oplus \langle 0 \vv{k+1} \rangle \oplus \cdots \oplus \langle 0 \vv n \rangle. + \] + + + Sia allora $\iota : R^n \to R/(d_1) \times \cdots \times R/(d_k) \times R \times \cdots \times R$ l'applicazione lineare determinata dalla relazione: + \[ + a_1 \vv 1 + \ldots + a_n \vv n \xmapsto{\iota} ([a_1]_{d_1}, \ldots, [a_k]_{d_k}, a_{k+1}, \ldots, a_n). + \] + Chiaramente $\iota$ è surgettiva. Sia adesso $\v = a_1 \vv 1 + \ldots + a_n \vv n$ + tale per cui $\iota(\v) = \vec 0$. Allora $d_1$ deve dividere $a_1$, + $d_2$ deve dividere $a_2$, e così fino ad $a_k$. Infine $a_{k+1} = \cdots = a_n = 0$. + Pertanto $\v$ dovrà necessariamente appartenere a $\Im T$; viceversa ogni elemento + di $\Im T$ appartiene a $\Ker \iota$, da cui $\Ker \iota = \Im T$. Allora, + per il primo teorema di isomorfismo, vale che: + + \begin{equation} + \label{eq:iota_omomorfismo} + R^n / \Im T \cong R^n / \Ker \iota \cong R/(d_1) \times \cdots \times R/(d_k) \times R \times \cdots \times R. + \end{equation} \vskip 0.1in + + Combinando allora le identità \eqref{eq:isomorfismo_M_T} e \eqref{eq:iota_omomorfismo}, + si ottiene la tesi\footnote{ + Si tiene presente dell'isomorfismo $R/(0) \cong R$. + }: + \[ + M \cong R/(d_1) \times \cdots \times R/(d_k) \times R \times \cdots \times R. + \] + \hfill\ensuremath{\blacksquare} +\end{document} \ No newline at end of file