From 9926fe88f16ed5c12bf34f24da01b3cb3d4c0000 Mon Sep 17 00:00:00 2001 From: flavis Date: Thu, 2 Jul 2020 16:52:19 +0200 Subject: [PATCH] update ana, add la10 --- sose2020/ana/lectures | 2 +- sose2020/la/uebungen/la10.pdf | Bin 0 -> 139980 bytes sose2020/la/uebungen/la10.tex | 296 ++++++++++++++++++++++++++++++++++ 3 files changed, 297 insertions(+), 1 deletion(-) create mode 100644 sose2020/la/uebungen/la10.pdf create mode 100644 sose2020/la/uebungen/la10.tex diff --git a/sose2020/ana/lectures b/sose2020/ana/lectures index c8b5fef..4a85680 160000 --- a/sose2020/ana/lectures +++ b/sose2020/ana/lectures @@ -1 +1 @@ -Subproject commit c8b5fefa146d4254103516d62848c422cc277ae4 +Subproject commit 4a8568092d61277a96481f294464fd68434f90a5 diff --git a/sose2020/la/uebungen/la10.pdf b/sose2020/la/uebungen/la10.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4194218d6e82a1b93e19df302a853d36173a5fb7 GIT binary patch literal 139980 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Beh.: Die + Darstellungsmatrix von $\bigwedge^2 f_A\colon \bigwedge^2\R^{3} \to \bigwedge^2\R^{3}$ bezüglich + der Basis $ \mathcal{B} = (e_1 \wedge e_2, e_1 \wedge e_3, e_2 \wedge e_3)$ ist gegeben als + \[ + M_{\mathcal{B}}^{\mathcal{B}}\left({\bigwedge}^2 f_A\right) = + \begin{pmatrix} + -2 & 0 & 2 \\ + 0 & 0 & 4 \\ + 3 & 2 & -1 + \end{pmatrix} + .\] + \begin{proof} + Berechne Bild der Basisvektoren unter $\bigwedge^2f_A$: + \begin{align*} + {\bigwedge}^2f_A(e_1 \wedge e_2) &= f_A(e_1) \wedge f_A(e_2) \\ + &= e_2 \wedge (2 e_1 + e_2 + 3e_3) \\ + &= 2 e_2 \wedge e_1 + e_2 \wedge e_2 + 3 e_2 \wedge e_3 \\ + &= -2 e_1 \wedge e_2 + 3 e_2 \wedge e_3 + \intertext{Für restliche Basisvektoren analog} + {\bigwedge}^2f_A(e_1 \wedge e_3) &= 2 e_2 \wedge e_3 \\ + {\bigwedge}^2f_A(e_2 \wedge e_3) &= 2 e_1 \wedge e_2 + 4 e_1 \wedge e_3 - 1 e_2 \wedge e_3 + .\end{align*} + Durch Ablesen der Koeffizienten folgt die Behauptung. + \end{proof} +\end{aufgabe} + +\begin{aufgabe} + Seien $R$ ein Ring und $M$ ein $R$-Modul. + \begin{enumerate}[(a)] + \item Beh.: Ist $M$ endlich erzeugt und frei, so ist $M$ flach. + \begin{proof} + Seien $N, L$ $R$-Moduln und $\varphi\colon N \to L$ ein injektiver $R$-Modul.hom. + + $M$ ist endlich erzeugt und frei. Fixiere Basis $(x_1, \ldots, x_n)$. Dann ist + $M \stackrel{\sim }{=} R^{n}$. D.h. es existieren R-Mod.iso. + $\Phi_1\colon M \otimes_R N \to R^{n} \otimes_R N$ und + $\Phi_2\colon M \otimes_R L \to R^{n} \otimes_R L$. Weiter ex. R.-Mod.isomorphismen + $f_1\colon R^{n} \otimes_R N \to N^{n}$ und $f_2\colon R^{n} \otimes_R L \to L^{n}$ + mit $f_1((r_1, \ldots, r_n), x) = (r_1 x, \ldots, r_n x)$, analog für $f_2$. + + Weiter definiere: + \begin{align*} + \psi\colon N^{n} &\to L^{n} \\ + (n_1, \ldots, n_n) &\mapsto (\varphi(n_1), \ldots, \varphi(n_n)) + .\end{align*} + $\psi$ ist $R$-Modulhom. und injektiv, da $\varphi$ injektiv ist. Definiere nun weiter + \begin{align*} + \Psi \colon M \otimes_R N \xrightarrow{\Phi_1} R^{n} \otimes_R N + \xrightarrow{f_1} N^{n} + \xrightarrow{\psi} L^{n} + \xrightarrow{f_2^{-1}} R^{n} \otimes_R L + \xrightarrow{\Phi_{2}^{-1}} M \otimes_R L + .\end{align*} + + Beh.: $\Psi$ ist injektiver $R$-Modul.hom. mit $\text{id}_M \otimes \varphi = \Psi$. + + $\Psi$ ist Verknüpfung von injektiven $R$-Modul.homomorphismen, + also selbst injektiver $R$-Mod.hom. Sei nun $a \otimes b \in M \otimes_R N$ beliebig. + Dann ist ex. $r_1, \ldots, r_n \in R$, s.d. $a = \sum_{i=1}^{n} r_i x_i$. Damit + folgt + \begin{salign*} + \Phi_1(a \otimes b) &= (r_1, \ldots, r_n) \otimes b \\ + f_1((r_1, \ldots, r_n) \otimes b) &= (r_1 b, \ldots, r_n b) \\ + \psi(r_1 b, \ldots, r_n b) &= (r_1 \varphi(b), \ldots, r_n \varphi(b)) \\ + f_2^{-1}(r_1 \varphi(b), \ldots, r_n \varphi(b)) &= + (r_1, \ldots, r_n) \otimes \varphi(b) \\ + \Phi_2^{-1}((r_1, \ldots, r_n) \otimes \varphi(b)) + &= (a \otimes \varphi(b)) + \intertext{Also folgt} + \Psi(a \otimes b) &= a \otimes \varphi(b) = (\text{id}_M \otimes \varphi)(a \otimes b) + .\end{salign*} + Also stimmen $\Psi$ und $\text{id}_M \otimes \varphi$ auf den Erzeugern überein, also + gilt $\Psi = \text{id}_M \otimes \varphi$. Damit ist auch $\text{id}_M \otimes \varphi$ + injektiv. + \end{proof} + \item Seien $M$ flach, $N$ flacher $R$-Modul und $\varphi\colon M \to N$ injektiver + $R$-Mod.hom. + + Beh.: $\varphi \otimes \varphi\colon M \otimes_R M \to N \otimes_R N$ ist + injektiv. + \begin{proof} + Es gilt + \[ + \varphi \otimes \varphi + = \underbrace{(\text{id}_N \otimes \varphi)}_{\text{injektiv, da } N \text{ flach}} + \circ \underbrace{(\varphi \otimes \text{id}_M)}_{\text{injektiv, da } M \text{ flach}} + .\] Damit ist $\varphi \otimes \varphi$ als Verknüpfung zweier injektiver $R$-Mod.homs. + auch injektiv. + \end{proof} + \item Beh.: $\Z / 2\Z$ als $\Z$ Modul ist nicht flach. + \begin{proof} + Betrachte $\varphi\colon \Z \to \Z$, $r \mapsto 2r$. $\varphi$ ist injektiver + $R$-Modulhomomorphismus, aber + \[ + (\varphi \otimes \text{id}_{\Z / 2\Z})( 1 \otimes \overline{1}) + = \varphi(1) \otimes \overline{1} = 2 \otimes \overline{1} + = 1 \otimes (2\cdot \overline{1}) = 1 \otimes \overline{0} = 0 + .\] $1 \otimes \overline{1} \neq 0$ in $\Z \otimes_R \Z / 2 \Z$, denn + mit $\beta\colon \Z \times \Z / 2\Z, (z, \overline{a}) \mapsto z \cdot \overline{a}$ + bilinear und $\beta(1, \overline{1}) = \overline{1} \neq 0$ ist mit UT angewendet auf + $\beta$ und $\Z / 2 \Z$ + $1 \otimes \overline{1} \neq 0$. Damit ist $\text{ker } (\varphi \otimes \text{id}_{\Z / 2\Z}) \neq \{0\} $, also $\varphi \otimes \text{id}_{\Z / 2 \Z}$ nicht injektiv. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Seien $R$ ein Ring und $M$ ein e.e. freier $R$-Modul. + \begin{enumerate}[(a)] + \item Seien $N$ e.e. freier $R$-Modul und $\varphi\colon M \to N$ injektiver + $R$-Mod.hom. + + Beh.: $\bigwedge^2 \varphi\colon \bigwedge^2 M \to \bigwedge^2 N$ ist injektiv. + \begin{proof} + Da $M$ und $N$ e.e. und frei ex. nach 35(a) und (b) eindeutige injektive + $R$-Mod.homs. $f\colon \bigwedge^2 M \to M \otimes_R M$ und + $g\colon \bigwedge^2 N \to N \otimes_R N$ mit + $f(a \wedge b) = a \otimes b - b \otimes a$, analog für $g$. + + Definiere nun $\tilde{g}\colon \bigwedge^2N \to \text{Bild}(g)$. $\tilde{g}$ ist + damit surjektiv und injektiv, also $R$-Modul.iso., inbes. ex. + $\tilde{g}^{-1}\colon \text{Bild}(g) \to \bigwedge^2 N$. + + Definiere weiter + \[ + \psi\colon {\bigwedge}^2 M \xrightarrow[\text{inj. nach 35(b)}]{f} M \otimes_R M + \xrightarrow[\text{inj. nach 37(b)}]{\varphi \otimes \varphi} N \otimes_R N + \xrightarrow[\text{inj. nach 35(b)}]{\tilde{g}^{-1}} {\bigwedge}^2 N + .\] Z.z.: $\psi$ wohldefiniert, g.z.z. + $\text{Bild}((\varphi \otimes \varphi) \circ f) = \text{Bild}(g)$. Dazu + seien $a, b \in M$. Dann gilt + \begin{salign*} + (\varphi \otimes \varphi)(f(a \wedge b)) + &= (\varphi \otimes \varphi)(a \otimes b - b \otimes a) \\ + &= (\varphi(a) \otimes \varphi(b) - \varphi(b) \otimes \varphi(a)) \\ + &= g(\varphi(a) \wedge \varphi(b)) \in \text{Bild}(g) + \intertext{Da Elemente der Form $a \wedge b$ $\bigwedge^2M$ + erzeugen, folgt Behauptung. Damit ist $\psi$ als + Verkettung von injektiven $R$-Mod.homs, injektiver $R$-Mod.hom. + Bleibt zu zeigen: $\psi = \bigwedge^2 \varphi$. Mit obiger Rechnung folgt sofort} + \psi(a \wedge b) &= + \tilde{g}^{-1}((\varphi \otimes \varphi)f(a \wedge b)) \\ + &= \tilde{g}^{-1}(g(\varphi(a) \wedge \varphi(b)))\\ + &= \varphi(a) \wedge \varphi(b) \\ + &= {\bigwedge}^2 \varphi(a \wedge b) + .\end{salign*} + Da $\bigwedge^2M$ von Elementen der Form $a \wedge b$ erzeugt wird, folgt $\psi = \bigwedge^2\varphi$. + Da $\psi$ injektiv als Verkettung von injektiven $R$-Mod.homs, ist $\bigwedge^2\varphi$ injektiv. + \end{proof} + \item Beh.: Für $m_1, m_2 \in M$ sind folgende Aussagen äquivalent: + \begin{enumerate}[(i)] + \item Die Familie $(m_1, m_2)$ ist linear unabhängig. + \item Aus $r (m_1 \wedge m_2) = 0$ in $\bigwedge^2M$ mit $r \in R$ folgt $r = 0$. + \end{enumerate} + \begin{proof} + (i) $\implies$ (ii): Definiere + \begin{align*} + \varphi\colon R &\to {\bigwedge}^2 M \\ + r &\mapsto r (m_1 \wedge m_2) + .\end{align*} + Z.z.: $\varphi$ ist injektiv. Sei $(e_1, e_2)$ die Standardbasis + des $R^2$. Definiere damit + \begin{align*} + \Phi&\colon R \to {\bigwedge}^2 R^2, \quad + r \mapsto r (e_1 \wedge e_2) \\ + \psi&\colon R^2 \to M, \quad + \psi(e_i) = m_i \quad i=1,2 + .\end{align*} + Da $\{e_1 \wedge e_2 \}$ Basis von $\bigwedge^2 R^2$, ist $e_1 \wedge e_2$ l.u. und + damit $\Phi$ injektiv. Weiter sind $R^2$ und $M$ e.e. und frei und + $\psi$ injektiver $R$-Mod.hom. Mit (a) folgt damit, dass + $\bigwedge^2 \psi$ injektiv ist. + Außerdem gilt für $r \in R$ beliebig: + \begin{salign*} + \left({\bigwedge}^2 \psi\right)(\Phi(r)) &= ({\bigwedge}^2\psi)(r (e_1 \wedge e_2)) \\ + &= r (\psi(e_1) \wedge \psi(e_2)) \\ + &= r (m_1 \wedge m_2) \\ + &= \varphi(r) + .\end{salign*} + Damit gilt $\varphi = \bigwedge^2 \psi \circ \Phi$ und damit + $\varphi$ injektiv, als Verkettung injektiver $R$-Mod.homs. + + (ii) $\implies$ (i): Kontraposition. Seien $(m_1, m_2)$ linear abhängig. Dann ex. + ein $\alpha \in R$ mit $m_1 = \alpha m_2$. Damit folgt + \[ + 1 \cdot (m_1 \wedge m_2) = 1 \cdot (\alpha m_2 \wedge m_2) = \alpha (m_2 \wedge m_2) = 0 + ,\] aber $1 \neq 0$ in $R$, da $R \neq 0$ nach Konvention der VL von Kapitel 9. + \end{proof} + \item Beh.: Für $\text{Rang}(M) = 2$ und $\varphi \in \text{End}_R(M)$ sind + die folgenden Aussagen äquivalent: + \begin{enumerate}[(i)] + \item $\varphi$ ist injektiv + \item $\text{det}(\varphi) \in R$ ist kein Nullteiler + \end{enumerate} + \begin{proof} + (i) $\implies$(ii): Da $\varphi$ injektiv, ist $\bigwedge^2 \varphi$ injektiv. + Sei $(x_1, x_2)$ Basis von $M$. Dann gilt + \begin{salign*} + {\bigwedge}^2\varphi(\underbrace{x_1 \wedge x_2}_{\neq 0}) + = \varphi(x_1) \wedge \varphi(x_2) + = \text{det}(\varphi) (x_1 \wedge x_2) + \neq 0 + .\end{salign*} + Also gilt $\text{det}(\varphi) \neq 0$. + Sei nun $r \in R$ beliebig mit $\text{det}(\varphi) r = 0$. Dann betrachte + \begin{salign*} + {\bigwedge}^2 \varphi(r x_1 \wedge x_2) + = \varphi(r x_1) \wedge \varphi(x_2) + = \text{det}(\varphi) r (x_1 \wedge x_2) + = 0 + = r \underbrace{(\text{det}(\varphi) x_1 \wedge x_2)}_{\neq 0} + .\end{salign*} + Da $(x_1, x_2)$ Basis sind auch $\text{det}(\varphi) x_1$ und $x_2$ linear unabhängig, d.h. + mit (b) folgt $r = 0$. + + (ii) $\implies$ (i): Sei $m \in M$ beliebig mit $\varphi(m) = 0$ und $(x_1, x_2)$ Basis + von $M$. + Ang.: $m \neq 0$. Dann ex. $a, b \in R$ mit $m = ax_1 + b x_2$ mit + $a \neq 0 \lor b\neq 0$. O.E.: $a \neq 0$. Dann folgt + \begin{salign*} + 0 &= \varphi(m) \wedge \varphi(x_2) \\ + &= \text{det}(\varphi) (m \wedge x_2) \\ + &= \text{det}(\varphi) (a x_1 + b x_2) \wedge x_2 \\ + &= \text{det}(\varphi) (a x_1 \wedge x_2) \\ + &= \text{det}(\varphi) \cdot a (x_1 \wedge x_2) + .\end{salign*} + Da $x_1$, $x_2$ l.u., folgt mit (b), dass $\text{det}(\varphi) \cdot a = 0$. Da + $\text{det}(\varphi) $ kein Nullteiler, folgt $a \neq 0$ $\contr$. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Seien $N = \Z$, $M = \bigoplus_{i \in \N} \Z / 2 \Z$ und + $f\colon N \to M \oplus M$, $g: N \oplus M \to M$ gegeben durch + \[ + f(n) = (2n, 0) \quad \text{und} \quad g(n, (\overline{m_1}, \ldots, )) = (\overline{n}, \overline{m_1}, \ldots) + .\] + \begin{enumerate}[(a)] + \item Beh.: Die Folge $0 \to N \xrightarrow{f} N \oplus M \xrightarrow{g} M \to 0$ ist eine + kurze exakte Folge von $\Z$-Moduln. + \begin{proof} + Offensichtlicherweise ist $f$ injektiv und $g$ surjektiv. Bleibt zu zeigen: + $\text{ker } g = \text{im } f$. + + ,,$\subseteq $``: Sei $x \in \text{ker } g$. Dann ex. $n, m_1, m_2, \ldots \in \Z$ mit + $x = (n, (\overline{m_1}, \ldots))$. Da $g(x) = 0$ folgt + $\overline{n} = \overline{m_1}= \ldots = 0$. Damit ex. $z \in \Z$ mit $z = 2 z$. Also + ist $f(z) = (2z, 0) = (n, 0) = (n, (\overline{m_1}, \overline{m_2}, \ldots)) = x$. + Damit ist $x \in \text{im }f$. + + ,,$\supseteq$``: Sei $x \in \text{im } f$. Dann $\exists n \in \Z$, s.d. + $f(n) = (2n, 0) = x$. Damit folgt + $g(x) = g(2n, 0) = (\overline{2n}, 0, \ldots) = (\overline{0}, \overline{0}, \ldots) = 0$. Also + $x \in \text{ker } g$. + \end{proof} + \item Beh.: Die Folge aus (a) zerfällt nicht. + \begin{proof} + Ang.: Die Folge aus (a) zerfällt. Dann ex. ein $\Z$-Untermodul $T \subseteq N \oplus M$, + s.d. $g|_T\colon T \to M$ Isomorphismus ist. Wähle + $x \coloneqq (\overline{1}, \overline{0}, \ldots) \in M$. Da $g|_T$ surjektiv, + ex. ein $y \in T$, s.d. $g(y) = x$. Es ex. $n, m_1, \ldots \in \Z$ mit + $y = (n, (\overline{m_1}, \ldots))$. Wegen + \[ + g(y) = g(n, (\overline{m_1}, \ldots)) = (\overline{n}, \overline{m_1}, \ldots) + = (\overline{1}, \overline{0}, \ldots) = x + \] folgt $n \equiv 1$ $(\text{mod } n)$. Da $T$ $\Z$-Untermodul, ist auch + $2y = (2n, (\overline{2 m_1}, \ldots)) = (2n, 0) \in T$. Damit folgt + \[ + g(y) = g(2n, 0) = (\overline{2n}, \overline{0}, \ldots) = (\overline{0}, \overline{0}, \ldots) = 0 + .\] + Da $n \neq 0$ und $\Z$ nullteilerfrei, + folgt $y = (2n,0) \neq 0$, folgt $\text{ker } g|_T \neq \{0\} $ $\contr$. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\end{document}