From b522ade33c3e003f440cce4e3007054108f9bfb1 Mon Sep 17 00:00:00 2001 From: flavis Date: Wed, 10 Jun 2020 16:02:54 +0200 Subject: [PATCH] add la6 --- sose2020/ana/lectures | 2 +- sose2020/ana/uebungen/ana6.pdf | Bin 145934 -> 145929 bytes sose2020/ana/uebungen/ana6.tex | 2 + sose2020/la/uebungen/la6.pdf | Bin 0 -> 143140 bytes sose2020/la/uebungen/la6.tex | 268 +++++++++++++++++++++++++++++++++ 5 files changed, 271 insertions(+), 1 deletion(-) create mode 100644 sose2020/la/uebungen/la6.pdf create mode 100644 sose2020/la/uebungen/la6.tex diff --git a/sose2020/ana/lectures b/sose2020/ana/lectures index 38f5896..01365dc 160000 --- a/sose2020/ana/lectures +++ b/sose2020/ana/lectures @@ -1 +1 @@ -Subproject commit 38f5896316c7a36cd4b443bf3c9944138f750754 +Subproject commit 01365dc0b8fcf74c0861540ec715304fd6e958dd diff --git a/sose2020/ana/uebungen/ana6.pdf b/sose2020/ana/uebungen/ana6.pdf index 863f3c0e0cfc299609a89540baaab3183bfecbdd..218b8962db2d814f2aa2fe5944dadd093e6fdc87 100644 GIT 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+ Im Folgenden seien freie Stellen in Matrizen $0$. + \begin{enumerate}[(a)] + \item Aus Aufg. 17 folgt $c_1(A) = c_2(A) = 1$, $c_3(A) = t-2$ und $c_4(A) = (t+1)(t-2)^2$. + Damit folgen die Weierstraßteiler $h_1 = t-2$, $h_2 = t+1$ und $h_3 = (t-2)^2$. Damit folgt + \begin{align*} + A \approx B_{h_1, h_3, h_2} \approx \begin{pmatrix} J(2,1) & & \\ + & J(2,2) & \\ + & & J(-1,1)\end{pmatrix} + = + \begin{pmatrix} 2 & & &\\ + & 2 & 0 & \\ + & 1 & 2 & \\ + & & & -1\end{pmatrix} + .\end{align*} + \item Aus Aufg. 20 folgen direkt die Weierstraßteiler $h_1 = t+1$, $h_2 = t$, $h_3 = t+1$, + $h_4 = t^2$, $h_5 = (t+1)^{3}$. + Damit folgt + \begin{align*} + A \approx B_{h_2, h_4, h_1, h_3, h_5} + \approx + \begin{pmatrix} + J(0,1) \\ + & J(0,2) \\ + & & J(-1, 1) \\ + & & & J(-1, 1) \\ + & & & & J(-1, 3) + \end{pmatrix} + = + \begin{pmatrix} + 0 \\ + & 0 & 0 \\ + & 1 & 0 \\ + & & & -1 \\ + & & & & -1 \\ + & & & & & -1 & 0 & 0 \\ + & & & & & 1 & -1 & 0 \\ + & & & & & 0 & 1 & -1 + \end{pmatrix} + .\end{align*} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Seien $R$ ein Ring, $I \subseteq R$ Ideal und $M$ ein $R$-Modul. + + Äquivalenzklassen bezügl. beliebiger Äquivalenzrelationen seien + im folgenden mit $\overline{\cdot }$ bezeichnet. Aus dem Argument ist immer klar, welche + Äquivalenzrelation gemeint ist. + \begin{enumerate}[(a)] + \item Beh.: $M / IM$ ist mit der natürlichen Addition und der angegebenen skalaren + Mult. $R / I$-Modul. + \begin{proof} + Die skalare Multiplikation sei mit $(*)$ bezeichnet. + \begin{align*} + R / I \times M / IM &\to M / IM\\ + (\overline{a}, \overline{m}) &\mapsto \overline{a} \cdot \overline{m} \coloneqq \overline{a \cdot m} \quad (*) + .\end{align*} + Zunächst zu zeigen, dass $(*)$ wohldefiniert ist. Dazu seien + $a, b \in R$ und $x, y \in M$ mit + $a + I = b + I$ und $x + IM = y + IM$. + \begin{itemize} + \item Zunächst ist $ax$ bzw. $by$ wohldefiniert, da $M$ $R$-Modul. + \item Es ist $x - y \in IM$, also + $ax - ay = \underbrace{a}_{\in R} \underbrace{(x-y)}_{\in IM}$. + Da $IM$ $R$-Untermodul von $M$, folgt $a(x-y) \in IM$ und damit + $\overline{ax} = \overline{ay}$. + \item Es ist $a -b \in I$, also + $ax - bx = \underbrace{(a-b)}_{\in I} \underbrace{x}_{\in M} \in IM$. Damit + folgt $\overline{ax} = \overline{bx}$. + \end{itemize} + + \begin{enumerate}[(M1)] + \item + $M / IM$ ist nach VL $R$-Modul, da $IM$ $R$-Untermodul von $M$, also insbesondere + abelsche Gruppe: $(M/IM, +, IM)$. + \item Seien $\overline{a}, \overline{b} \in R / I$ und $\overline{x}, \overline{y} \in M / IM$. Damit folgt + \begin{align*} + (\overline{a} + \overline{b}) \overline{x} = \overline{a + b} \overline{x} + \stackrel{(*)}{=} \overline{(a+b)x} + \qquad \stackrel{M \text{ } R\text{-Modul}}{=} + \qquad \overline{ax + bx} = \overline{ax} + + \overline{bx} \stackrel{(*)}{=} \overline{a}\overline{x} + \overline{b} \overline{x} + .\end{align*} + Der Rest lässt sich analog nachrechnen. + \end{enumerate} + \end{proof} + \item Beh.: Ist $n \in \N$ und $\varphi\colon M \to R^{n}$ ein $R$-Modulisomorphismus, dann + ist $\varphi |_{IM}\colon IM \to I^{n}$ eine Bijektion (i) und + $\varphi$ induziert einen $R / I$ - Moduliso. $\overline{\varphi}\colon M / IM \to (R / I)^{n}$ (ii). + \begin{proof} + \begin{enumerate}[(i)] + \item + \begin{itemize} + \item Z.z.: $\varphi|_{IM}$ wohldefiniert. Sei $m \in IM$ dann + ex. Indexmenge $J$ und $(a_i)_{i \in J} \in I^{(J)}$ und + $(m_i)_{i \in J} \in M^{(J)}$ mit + \[ + m = \sum_{i \in J} a_i m_i \implies + \varphi(m) + \quad \stackrel{\varphi R\text{-Hom}}{=} \quad + \sum_{i \in J} \underbrace{a_i}_{\in I} \underbrace{\varphi(m_i)}_{\in R^{n}} + \in I^{n} \quad (\text{da } I \text{ Ideal}) + .\] + \item Sei $x \in I^{n}$. Z.z.: $\varphi^{-1}(x) \in IM$. Es + ex. $(a_i)_{i=1}^{n} \in I^{n}$ mit + \[ + x = \sum_{i=1}^{n} a_i e_i \implies + \varphi^{-1}(x) + \qquad \stackrel{\varphi^{-1} R\text{-Hom}}{=} \qquad + \sum_{i=1}^{n} \underbrace{a_i}_{\in I} \underbrace{\varphi^{-1}(e_i)}_{\in M} + \in IM + .\] Also $\varphi_{IM}$ surjektiv. + \item Da $\varphi$ Iso, inbes. bijektiv, ist $\varphi|_{IM}$ injektiv. + \end{itemize} + Damit folgt also $\varphi|_{IM}$ bijektiv. + \item Definiere $\overline{\varphi}\colon M/IM \to (R / I)^{n} = R^{n} / I^{n}, m + IM \mapsto \varphi(m) + I^{n}$. + \begin{itemize} + \item Z.z: $\overline{\varphi}$ wohldefiniert. Seien $m_1, m_2 \in M$ + mit $m_1 = m_2$. Dann folgt $m_1 - m_2 \in IM$. Also + ex. ein $(a_i)_{i \in J} \in I^{(J)}$ und + $(m_i)_{i \in J} \in M^{(J)}$ mit $m_1 - m 2 = \sum_{i \in J} a_i m_i$. + Damit folgt + \[ + \varphi(m_1) - \varphi(m_2) = \varphi(m_1 - m_2) + \qquad \stackrel{\varphi \ R \text{-Hom}}{=} \qquad + \sum_{i \in J} \underbrace{a_i}_{\in I} \underbrace{\varphi(m_i)}_{\in R^{n}} \in I^{n} + .\] Also ist $\varphi(m_1) + I_n = \varphi(m_2) + I_n$, also + $\overline{\varphi}$ wohldefiniert. + \item Z.z.: $\overline{\varphi}$ $R/I$-Homomorphismus. Zunächst + sind $M / MI$ und $(R / I)^{n}$ $R / I$ Moduln. + Seien $r \in R / I$ und $\overline{m_1}, \overline{m_2} \in M / IM$. Dann folgt + \begin{salign*} + \overline{\varphi}(\overline{r} \overline{m_1} + \overline{m_2}) + &= \overline{\varphi}( \overline{r m_1 + m_2}) \\ + &= \varphi(r m_1 + m_2) + I^{n} \\ + &\stackrel{\varphi \ R\text{-Hom}}{=} r \varphi(m_1) + \varphi(m_2) + I^{n} \\ + &= r \overline{\varphi(m_1)} + \overline{\varphi(m_2)} \\ + &= r \overline{\varphi}(m_1) + \overline{\varphi}(m_2) + .\end{salign*} + \item Z.z.: $\overline{\varphi}$ bijektiv. Es ist $\overline{\varphi}$ injektiv, da + \[ + \text{ker } \overline{\varphi} + = \{ \overline{m} \in M / IM \mid \varphi(m) = I^{n}\} + \quad + \stackrel{\varphi|_{IM}\text{ Bij.}}{=} \quad + \{ \overline{m} \in M / IM \mid m \in IM\} + = \{ IM \} = \{ 0\} + .\] Sei $\overline{x} \in (R / I)^{n}$. Da $\varphi$ Isomorphismus, ex. $m \in M$ + mit $\varphi(m) = x$. Damit folgt + \[ + \overline{\varphi}(m + IM) = \varphi(m) + I^{n} = x + I^{n} = \overline{x} + .\] Also $\overline{\varphi}$ surjektiv. + \end{itemize} + Insgesamt ist $\overline{\varphi}$ also ein $R / I$-Moduliso. + \end{enumerate} + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Es sei $S \coloneqq \{t + 1, t^2 + 1\} $. + + Beh.: $S$ ist minimales ES von $\Q[t]$ als $\Q[t]$-Modul. + \begin{proof} + Da $\Q[t]$ als $\Q[t]$-Modul betrachtet wird, sind die von Elementen + $(a_i)_{i \in I} \in Q[t]^{I}$ erzeugten Untermodule von $\Q[t]$ gerade die + von $(a_i)_{i \in I}$ erzeugten Ideale. + \begin{itemize} + \item Z.z.: $S$ ES von $\Q[t]$ als $\Q[t]$-Modul. Es ist + \[ + \left( -\frac{1}{2} t + \frac{1}{2} \right) \cdot (t+1) + \frac{1}{2} \cdot (t^2+1) + = 1 + .\] Also ist $(t+1, t^2+1) = (1)$. Damit folgt die Behauptung. + \item Z.z.: $S$ minimales ES. + Sei $S_1 \coloneqq \{t+1\} $. Da $\Q$ Kp. ist $\Q[t]$ HIR, also ist + $t+1$ bis auf Assoziiertheit eind. bestimmter Erzeuger von $(t+1)$. Da + $\text{deg}(t+1) = 1 > 0 = \text{deg}(1)$ und $\Q[t]$ nullteilerfrei, folgt + $1 \not\in (t+1)$, also $S_1$ kein ES. + \end{itemize} + \end{proof} + Beh.: $S$ ist keine Basis. + \begin{proof} + Es ist + \[ + \left( \frac{1}{2}(t+1) \right) (t^2+1) + \left( t - \frac{1}{2} (t+1)^2 \right) (t+1) = 0 + ,\] aber $\frac{1}{2}(t+1) \neq 0 \neq t - \frac{1}{2}(t+1)^2$, also + $t+1$ und $t^2+1$ l.a. in $Q[t]$ als $Q[t]$-Modul, also $S$ keine Basis. + \end{proof} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: Sei $R$ ein Ring und $I \neq 0$ ein Ideal in $R$. Dann sind äquivalent + \begin{enumerate}[(i)] + \item $I$ von einem Nicht-Nullteiler erzeugtes Hauptideal + \item $I$ frei als $R$-Modul + \end{enumerate} + \begin{proof} + (i) $\implies$ (ii): Sei $I$ Hauptideal mit $a \in R$ kein Nullteiler und + $I = (a)$. $\{a\} $ ist Basis von $I$ als $R$-Modul, denn + \begin{itemize} + \item $a \neq 0$, da $I \neq 0$ und $a$ kein NT, also $\{a\} $ l.u. + \item $\{a\} $ ES von $I$, da $I = (a)$. Also $\forall x \in I$ + $\exists r \in R$ mit $x = ra$. + \end{itemize} + + (ii) $\implies$ (i): Sei $I$ frei als $R$-Modul. Dann sei $(a_i)_{i \in J} \subseteq I$ + Basis von $I$. $\forall i \in J\colon a_i \neq 0$ (sonst wäre $(a_i)_{i \in J}$ l.a.). + + \begin{itemize} + \item Es ist $(a_i)_{i \in I} = \{a_1\} $, denn + ang.: $\exists a_2 \in I$ mit $a_1 \neq a_2$ und $\{a_1, a_2\} \subseteq (a_i)_{i \in J}$. + Dann ist, da $R$ kommutativ, $a_1 a_2 = a_2 a_1$. Damit folgt + $a_1 a_2 - a_2 a_1 = 0$, aber $a_1 \neq 0 \neq a_2$. Also sind + $\{a_1, a_2\} $ linear abhängig $\contr$. + \item $a_1 \neq 0$ und $a_1$ kein NT, denn ang. $\exists c \in R \setminus \{0\} $ mit + $a_1 r = 0 \implies a_1$ l.a. + \item Da $\{a_1\} $ Basis von $I$ als $R$-Modul, gilt $\forall a \in I$ + $\exists r \in R$ mit $a = r \cdot a_1$. Also folgt + $I = (a_1)$. + \end{itemize} + \end{proof} + \item Beh.: $(2, 1 + \sqrt{-3})$ in $\Z[\sqrt{-3}]$ ist nicht frei als + $\Z[\sqrt{-3}]$-Modul. + \begin{proof} + Wegen (a) g.z.z., dass $(2, 1 + \sqrt{-3})$ kein Hauptideal in $\Z[\sqrt{-3}]$. + Aus Aufg. 10 folgt, dass $2$ und $1 + \sqrt{-3} $ irreduzibel sind. Sei + $\delta \colon \Z[\sqrt{-3}] \to \N_0$ außerdem die multiplikative Normabbildung aus + Aufg. 10 mit $\delta(2) = \delta (1+\sqrt{-3}) = 4$ und $\delta(1) = 1$. + + Ang.: $\exists r \in \Z[\sqrt{-3}]$ mit $(r) = (2, 1 + \sqrt{-3})$. Dann + ex. $x, y \in \Z[\sqrt{-3}]$ mit $2 = x r$ und $1 + \sqrt{-3} = yr$. Da + $2$ und $1 + \sqrt{-3} $ irreduzibel und $\Z[\sqrt{-3}]^{\times } = \{\pm 1\} $, folgt + $x \in \{\pm 1\} \lor r \in \{ \pm 1\} $. + \begin{itemize} + \item Falls $x \in \{ \pm 1\} $, dann $r \in \{\pm 2\} $. Damit folgt + $1 + \sqrt{-3} \in \{\pm y \cdot 2 \}$. Da + $\{\pm 2\} \not\in \Z[-3]^{\times }$, folgt $y \in \Z[\sqrt{-3}]^{\times } =\{ \pm 1\} $. + Also $1 + \sqrt{-3} \stackrel{\wedge}{=} 2$ $\contr$. + \item Falls $r \in \{ \pm 1\} $, dann $\Z[\sqrt{-3}] = (r) = (2, 1 + \sqrt{-3})$. + Dann ex. $a, b \in \Z[\sqrt{-3}]$ mit $1 = 2 \cdot a + (1+ \sqrt{-3}) \cdot b$, also + \[ + 1 = \delta (1) = \delta (2) \cdot \delta (a) + \delta (1+\sqrt{-3} ) \cdot \delta (b) + = 4 \underbrace{\delta (a)}_{\in \N_0} + 4 \underbrace{\delta (b)}_{\in \N_0} + \quad \contr + .\] + \end{itemize} + Also ist $(2, 1+ \sqrt{-3})$ kein Hauptideal und damit wegen (a) nicht frei + als $\Z[\sqrt{-3}]$ Modul. + \end{proof} + \item Wähle $M = R = \Z[\sqrt{-3}]$ und $N = (2, 1 + \sqrt{-3})$. $N$ ist + $\Z[\sqrt{-3}]$-Untermodul von $M$, da $N$ Ideal in $\Z[\sqrt{-3}]$-Modul M. + Es ist $M = \Z[\sqrt{-3}] $, also $M$ frei als $\Z[\sqrt{-3}]$-Modul, aber + $N = (2, 1+\sqrt{-3}) \subseteq \Z[\sqrt{-3}]$ wegen (b) nicht frei. + \end{enumerate} +\end{aufgabe} + +\end{document}