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fead553..ff61723 100644 --- a/skript.tex +++ b/skript.tex @@ -1319,4 +1319,74 @@ Beispiel: $X := \Z/z\Z \otimes_\Z \Z/3\Z = (0)$ \\ $ 1 = 2 \cdot 2 - 3$ \\ $ a \otimes b \in X: a \otimes b = 1 a \otimes b = (4-3)(a \otimes b) = 4(a \otimes b) - 3(a \otimes b) = (4a)\otimes b - a \otimes (3b) = 0 \otimes b - a \otimes 0 = 0$ +5.2.2 Wichtiges Beispiel: $K$ Körper, $G$ Gruppe, $H \leq G$. $M \in \lsub{KH}{\Mod}$ \\ +Beachte: $KG \in \lsub{KG}{\Mod_{KH}}$ \\ +So: $KG \otimes_{KH} M \in \lsub{KG}{\Mod}$ \\ +Wir schreiben: $\Ind_H^G M = $ der von $H$ nach $G$ induzierte Modul. + +Seien $M, N \in \lsub{KH}{\Mod}, f : M \rightarrow N$ $KH$-linear. Dann ist $\id_{KG} \otimes f: KG \otimes_{KH} M \rightarrow KG \otimes_{KH} N$ $KG$-linear. + +5.2.3 Lemma: Seien $M_1, M-2 \in \lsub{R}{\Mod_S}$, $R, S, T$ $\Lambda$-Algebren, $N_1, N_2 \in \lsub{S}{\Mod_T}, f: M_1 \rightarrow M_2$ $R$-$S$-linear, $g: N_1 \rightarrow N_2$ $S$-$T$-linear. \\ +Dann wird durch $f \otimes g: M_1 \otimes_S N_1 \rightarrow M_2 \otimes_S N_2: m \otimes n \mapsto f(m) \otimes g(n)$ eine $R$-$T$-lineare Abbildung definiert. \\ +Beweis: Wir definieren Abbildung $f \times g: M_1 \times N_1 \rightarrow M_2 \otimes_S N_2: (m, n) \mapsto f(m) \otimes g(n)$. \\ +Leicht: $f \times g$ ist $R$-$T$-linear und $S$-balanced. ($(f\times g)(rm,n) = f(rm) \otimes (g(n) = r(f(m) \otimes g(n)), (f \times g)(ms,n) = f(ms) \otimes g(n) = f(m)s \otimes g(n) = f(m) \otimes s g(n) = f(m) \otimes g(sn) = (f\times g)(m, sn)$) \\ +\begin{diagram} +M_1 \times N_1 & \rTo^\eta & M_1 \otimes N_1 \\ +& \rdTo_{f \times g} & \dTo_{\exists ! \widehat{f \times g} = f \otimes g} \\ +& & M_2 \otimes N_2 +\end{diagram} +\qed + +Sei $\set{g_i \mid i \in I}$ Vertretersystem der Linksnebenklassen von $H$ in $G$, d.h. $G = \bigcup\limits_{i\in I}^{\bullet} g_i H$. Sei $g \in G$, dann $\exists ! i \in I, h \in H: g = g_i \cdot h$. \\ +Also lässt sich jedes $x \in KG$ eindeutig als Linearkombination $x = \sum_{i\in I} g_i y_i$ mit $y_i \in KH$ (fast alle 0) schreiben. +($x = \sum_{g \in G} \lambda_g \cdot g = \sum_{i \in I} g_i (\sum_{h \in H} \lambda_{g_i h} h), \lambda_g \in K$ fast alle 0) + +Also: $KG_{KH} = \oplus_{i \in I} g_i KH$, ($KH$-Rechtsuntermoul von $KG_{KH}, g_i KH \cong KH$ als $KH$-Rechtsmodul) \\ +$KG_{KH}$ ist frei als $KH$-Modul, mit $KH$-Basis $g_i$. \\ +Nun ist $g_i H \otimes_{KH} M = g_i \otimes_{KH} M$ \\ +Wir haben gezeigt: $\Ind_{KH}^{KG} M = KG \otimes_{KH} M \cong \oplus_{i \in I} g_i \otimes M$ (mit $g_i \otimes M \cong M$ als $K$-Vektorraum) $ \cong \oplus \abs{I}$ vielen Kopien vn $KH$ als Vektorraum.\\ +$(g_i \otimes M$ ist ein $g_i KH g_i^{-1}$-Modul) + +Allgemein: $S \subseteq R$ Ringe, $M \in \lsub{S}{\Mod}$, $\lsub{R}{R_S} \in \lsub{R}{\Mod_S}, \Ind_S^R = R \otimes_S M \in \lsub{R}{\Mod}$ \\ +Im allgemeinen ist aber $R_S$ nicht frei. Es kann durchaus passieren, dass $R \otimes_S M = (0)$. + +5.2.4 Lemma: Sei $A, B$ $K$-Algebren, $K$ Körper. Dann wird $A \otimes_K B$ zur $K$-Algebra vermöge der folgenden Multiplikation: \\ +$(a_1 \otimes b_1) \cdot (a_2 \otimes b_2) = (a_1 a_2) \otimes (b_1 b_2)$ \\ +Problem: Ist diese Multiplikation wohldefiniert? \\ +Beweis: Übung \\ + +5.2.5 Korrolar: Seien $M \in \lsub{A}{\Mod}, N \in \lsub{B}{\Mod}$. Dann wird $M \otimes_K N$ ein $A \otimes_K B$-Modul durch: \\ +$(a \otimes b)(m \otimes n) = am \otimes bn$ + +5.2.6 Beobachtung: Sind $G, H$ Gruppen, so ist $K(G \times H) \cong KG \otimes_K KH$ \\ +Beweis: $LAAG \Rightarrow KG \otimes_K KHG$ hat Basis $\cB = \set{g \otimes h \mid g \in G, h \in H}$ \\ +$(g_1 \otimes h_1)(g_2 \otimes h_2) = (g_1 g_2) \otimes (h1 h_1)$ \\ +$\Rightarrow \cB$ ist Gruppenbasis von $KG \otimes_K KH, \cB \cong G \times H$ als Gruppe \qed + +Ist $X$ Grupe, $f: X \rightarrow G \times H$ Gruppenhomomorphismus. So kann dieser zu einem $K$-Algebrahomomorphismus $f: KX \rightarrow KG \otimes_K KH$ fortgesetzt werden. + +5.2.7 Definition: Sei $G$ Gruppe, dann wird durch $\Delta g = (g, g) \in G \times G$ ein injektiver Gruppenhomomorphismus $\Delta: G \rightarrow G \times G$ definiert. +Es ist $\Lambda(\sum \lambda_g g) = \sum \lambda_g (g \otimes g)$. Sind $M, N \in \lsub{KG}{\Mod}$, so wird $M \otimes_K N$ ein $KG$-Modul durch Einschränken auf der $KG \otimes_K KG$-Operation aus 5.2.5 auf $M \otimes_K M$ auf $\im \Delta \cong G$. So gilt: $g \in G, m \in M, n \in N$ ist $g (m \otimes n) = gm \otimes gn$. \\ +Vorsicht: Sei $x = \sum \lambda_g g \in KG, m \in M, n \in N$:\\ +~~ $x (m \otimes n) = (\sum \lambda_g g)(m \otimes n)= \sum \lambda_g (gm \otimes gn)$ \\ +~~ $xm \otimes xn = (\sum \lambda_g g m) \otimes (\sum \lambda_h h n) = \sum\limits_{g \in G, h \in H} \lambda_g \lambda_h (gm \otimes hn)$ \\ +Also ist i.A. $x(m \otimes n) \neq xm \otimes xn$ + +\subsection{$KG$-Moduln Grundlagen} + +$A = K$-Algebra, $K$ Körper, $M \in \lsub{A}{\Mod}$. Ist $m \in M$, so ist $A \cdot m = \set{a \cdot m \mid a \in A}$ ein $A$-Untermodul. \\ +Ist $S \subseteq M$, so ist $ = _{A\text{-Modul}} = \sum_{s \in S} As \leq M$ der von $S$ erzeugte Untermodul von M.\\ +$= \bigcap_{U \leq M, s \leq U} U = $ kleinste Untermodul von $M$, der $S$ enthält. $S \subseteq M$ heiß Erzeugendensystem von $M$ falls $ = M$. \\ +$M$ heißt endlich erzeugt, falls $M$ ein endliches Erzeugendensystem hat. \\ +$M$ heißt zyklischer $A$-Modul, falls $M = A \cdot m$ für ein $m \in M$. + +5.3.1 Satz: Sei $M \in \lsub{A}{\Mod}$. Dann ist $M$ zyklisch $\Leftrightarrow M$ ist epimorphes Bild von $\lsub{A}{A} \in \lsub{A}{\Mod}$. \\ +Beweis: "`$\Rightarrow$"': Sei $M = Am$ zyklischer Modul. Definiere $f: \lsub{A}{A} \rightarrow m: a \mapsto a \cdot m$. $f$ ist $A$-linear: $f(ba) = (ba)m = b (am) = b f(a), f(a+b) = (a+b)m = am+bm=f(a)+f(b)$. \\ +Klar: Wegen $M = Am$ ist $f$ Epimorphismus von $\lsub{A}{A}$ auf $M$. $M \cong \lsub{A}{A} / \ker f, \ker f = \set{a \in A \mid am = 0} = ann_A(m)$. \\ +"`$\Leftarrow$"': Sei $f: \lsub{A}{A} \rightarrow M$ Epimorphismus, $m = f(1)$. Sei $x \in M, a \in A: f(a) = x$ \\ +Dann ist $x = f(a) = f(a \cdot 1) = a f(1) = am$. Also ist $M = A m$ zyklischer Modul. + +Beispiel: $M \in \lsub{A}{\Mod}$ irreduzibel, $0 \neq m \in M$. $(0) \neq Am \leq M \Rightarrow Am = M \Rightarrow M$ zyklisch. + + \end{document} diff --git a/standard.tex b/standard.tex index a3de4f0..d9f1b14 100644 --- a/standard.tex +++ b/standard.tex @@ -92,6 +92,7 @@ \DeclareMathOperator{\sign}{sign} \DeclareMathOperator{\Syl}{Syl} \DeclareMathOperator{\Mod}{mod} +\DeclareMathOperator{\Ind}{Ind} \DeclareMathOperator{\Reid}{Reid} \DeclareMathOperator{\Liid}{Liid}