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z0D~F08_)EPaP~pITA{==T=`-&NFo123(u4e(q$Ic5Ve`17X{QmC#rM{02~syI0l2D zeWqBmV;+dG%0`|C`jL!3f2|r?VoYBR10nTM<{DtEegwcEA~456md=KecNHn&===|z%CQUy zQcPxH0^=mia16%5IsyV1m^G6ba5GFpmkA*IW^o8DM5bB=kS{RpLjX+-77=#y44RKX zg~<%OM*wz2>R%oK(dix269P=E)Ug1)0gQFl3@|D43=^fXY+Haj^Di?U$V31JLY6~# z?Z$e+ZAWc$m(b8ixETe?f)6n_w-1>c%7J=|{% 2$ \\ +Dann sind $C, D \in \GL_n(F)$. Sei $\epsilon = \det D / \det C = \det (DC^{-1}), A = \pmatr{\epsilon & 0 \\ 0 & 1_{n-1}}, \det A = \epsilon, B = DA^{-1}C^{-1}$, so ist $Bc_1 = \epsilon^{-1}d_1, Bc_i = d_i $ für $i > 2$ x\\ $BFc_i = FBc_i = Fd_i $ für alle $i$. Klar: $\det B = 1$, d.h. $B \in \SL_n(F)$ \qed. Missing: 27.11.2009 \\ @@ -1462,6 +1462,58 @@ Beweis: $v . (a_1 + a_2) = \iota(a_1 + a_2)v = \iota(a_1)v + \iota(a_2)v = v . a 5.2.7 Korrolar: $V \in \lsub{KG}{\Mod}$. Dann wird $V^\ast = \Hom_K(V, K)$ zum $KG$-Modul durch $(gf)(v) := f(g^{-1} v)$. 5.2.8 Korrolar: Seien $V, W \in \lsub{KG}{\Mod}$. Dann wird $\Hom_K(V, W)$ zum $KG$-Modul durch $(g.f)(v) := gf(g^{-1} v)$ \\ -(Da $\Hom_K(\lsub{KV}{V_K}, \lsub{KG}{W_K}) \in \lsub{KG}{\Mod_{KG}}$) +(Da $\Hom_K(\lsub{KV}{V_K}, \lsub{KG}{W_K}) \in \lsub{KG}{\Mod_{KG}}$) \\ +Beweis: $\Hom_K(\lsub{KG}{V}, \lsub{KG}W) \in \lsub{KG}{\Mod}_{KG}$ durch $(gfh)(v) = gf(hv)$ für $f \in \Hom_K(V, W), g, h \in G, v \in V$. Also ein $KG \otimes_K KG^{op}$-Modul. \\ +Mit $\Delta: G \Rightarrow G \times G^{op}: g \mapsto (g, g^{-1})$ wird daher $\Hom_K(V, W)$ zum Links-$KG$-Modul durch Einschränken. \\ +$(g \otimes g^{-1})f = g f g^{-1}$ \\ +Beachte: $(g . f)(v) = g(f(g^{-1} v)) \Rightarrow ((g_1, g_2) . f)(v) = (g_1 g_2)(f(g_2^{-1} g_1^{-1} v)) = g_1 (g_2 f(g_2^{-1} (g_1^{-1} v))) = g_1 ((g_2 f)(g_1^{-1} v)) = (g_1 . g_2 . f)(v)$ + +5.2.9 Definition und Lemma: Sei $M \in \lsub{KG}{\Mod}$. Dann ist $M^G = \set{m \in M \mid g m = m \forall g \in G}$ der größte Untermodul von $M$, auf dem $G$ trivial operiert. +$M^G$ ist direkte Summe von $\dim_K(M^G)$ vielen Kopien des trivialen $G$-Moduls $K$. Die ELemente von $M^G$ werden $G$-Invarianten genannt. \\ +Beweis: leicht. \\ +$ ~~ (i \mid i \in I)$ $K$-Basis von $M^G \Rightarrow M^G = \bigoplus_{i \in I} K v_i, K v_i \cong K = $ trivialer $KG$-Modul \qed + +5.2.10 Satz: Sei $M, N \in \lsub{KG}{\Mod}$ und sei $\Hom_K(M, N) \in \lsub{KG}{\Mod}$ wie in 5.2.8. Dann ist $\Hom_K(M, N)^G \cong \Hom_{KG}(M, N)$ (als $K$-Vektorraum). \\ +Beweis: Sei $f \in \Hom_K(M, N)$. Dann ist $f \in \Hom_K(M, N)^G \Leftrightarrow g .f = f \forall g \in G \Leftrightarrow (g.f)(v) = f(v) = g f (g^{-1} v) \forall g \in G, v \in M \Leftrightarrow f(g v) = g f(v) \Leftrightarrow f \in \Hom_{KG}(M, N)$ + +5.2.11 Satz: Sei $U, V \in \lsub{KG}{\Mod}$ endlicher Dimension. Dann ist $\Hom_K(U, V) \cong U^\ast \otimes V$ als $KG$-Moduln. + +Spezialfall: $U = V, \dim_K(V) < \infty$. Dann ist $\End_{KG}(V) \cong (V^\ast \otimes V)^G$ als $K$-Vektorraum. + +Beweis 5.2.11: $U^\ast$ ist $G$-Modul durch: $\alpha \in U^\ast: (g \alpha)(u) = \alpha(g^{-1} u)$ (siehe weit oben), und daher wird $U^\ast \otimes V$ zum $G$-Modul durch $g (\alpha \otimes v) = (g \alpha) \otimes (g v)$. \\ +Definiere $\Gamma: U^\ast \otimes_K V \rightarrow \Hom_K(U, V)$ durch $\alpha \otimes v \mapsto A_{\alpha, v}$ mit $A_{\alpha, v}(u) := \alpha(u) \cdot v \in V$. \\ +Klar ist: $A_{\alpha, v}$ ist Abbildung $U \rightarrow V$. Zu zeigen: $A_{\alpha, v}$ ist $K$-linear: \\ +$A_{\alpha, v}(u_1 + u_2) = (\alpha(u_1 + u_2))v = (\alpha(u_1) + \alpha(u_2))v = \alpha(u_1) v + \alpha(u_2) v = A_{\alpha, v}(u_1) + A_{\alpha, v}(u_2)$ \\ +$A_{\alpha, v}(\lambda u) = \lambda A_{\alpha, v}(u)$ ebenso. So ist $A_{\alpha, v} \in \Hom_K(U, V)$. \\ +Definiere $\hat{\Gamma} : U^\ast \times V \rightarrow \Hom_K(U, V): (\alpha, v) \mapsto A_{\alpha, v}$ \\ +$\hat{\Gamma}(\alpha_1 + \alpha_2, v)(u) = A_{\alpha_1 + \alpha_2, v}(u) = ((\alpha_1 + \alpha_2)(u))v = (\alpha_1(u) + \alpha_2(u))v = \alpha_1(u)v + \alpha_2(u)v = A_{\alpha_1, v}(u) + A_{\alpha_2, v}(u) = (A_{\alpha_1, v} + A_{\alpha_2, v})(u)$ \\ +$\Rightarrow \hat{\Gamma}(\alpha_1 + \alpha_2, v) = \hat{\Gamma}(\alpha_1, v) + \hat{\Gamma}(\alpha_2, v)$ \\ +Ähnlich $\hat{\Gamma}(\alpha, v_1 + v_2) = \hat{\Gamma}(\alpha, v_1) + \hat{\Gamma}(\alpha, v_2)$, also $\hat{\Gamma}$ bilinear. \\ +$ \lambda \in K: \hat{\Gamma}(\alpha \lambda, v)(u) = A_{\alpha \lambda, v}(u) = (\alpha \lambda)(u)(v) = \alpha(u) \lambda v = A_{\alpha, \lambda v}(u) = \hat{\Gamma}(\alpha, \lambda v)$, also ist $\hat{\Gamma}$ auch $K$-balanced. Daher ist (universelle Eigenschaft) $\Gamma$ wohldefinierte $K$-lineare Abbildung. + +Ist $(u_1, \ldots, u_n)$ $K$-Basis von $U$ und $(u_1^\ast, \ldots, u_n^\ast)$ duale Basis ($u_i^\ast(u_j) = \delta_{ij})$, so kann jedes Element von $U^\ast \otimes_K V$ eindeutig als Summe $\sum_{i=1}^n u_i^\ast \otimes x_i$ mit $x_i \in V$ geschrieben werden. ($U^\ast = \bigoplus_{i=1}^n K u_i^\ast \Rightarrow U^\ast \otimes_K V \cong \bigoplus_{i=1}^n (u_i^\ast \otimes V)$) \\ +Sei $1 \leq j \leq n: \Gamma(\sum_{i=1}^n u_i^ast \otimes x_i)(u_j) = \sum_{i=1}^n \Gamma(u_i^\ast \otimes x_i)(u_j) = \sum u_i^\ast(u_j) x_i = x_j$ \\ +So $\Gamma(\sum_{i=1}^n u_i^\ast \otimes x_i) = 0 \Leftrightarrow x_j = 0 \forall j \Leftrightarrow \sum_{i=1}^n u_i^\ast \otimes x_i = 0 \Rightarrow \Gamma$ injektiv. \\ +Wegen $\dim_K (U^\ast \otimes V) = \dim_K(U^\ast) \cdot \dim_K(V) = \dim_K(U) \cdot \dim_K(V) = \dim_K(\Hom_K(U, v))$ ist $\Gamma$ ein $K$-Isomorphismus. + +Bleibt zu zeigen: $\Gamma$ ist $G$-linear. $g \in G, \alpha \in U^\ast, v \in V, u \in U:$ \\ +$\Gamma(g (\alpha \times v))(u) = \Gamma( g\alpha \otimes gv)(u) = A_{g\alpha, gv}(u) = (g\alpha)(u) \cdot gv = \alpha (g^{-1} u) \cdot g v$ \\ +$(g \Gamma(\alpha \times v))(u) = (g . A_{\alpha, v})(u) = g \cdot (A_{\alpha, v}(g^{-1} u)) = \alpha (g^{-1} u) gv$ \qed + +5.2.12 Satz: Seien $U, V \in \lsub{KG}{\Mod}, \abs{G} = n < \infty$ und sei $\Char K = p$ mit $p \nmid n$, oder sei $\Char K = 0$. +Dann ist $\abs{G} \cdot 1_K \neq 0$, d.h. $\frac{1}{\abs{G}} = (\abs{G} \cdot 1_K)^{-1}$ existiert in $K$. \\ +Sei $f: U \rightarrow V$ $K$-linear. Definiere $\hat{f}_G: U \rightarrow V: u \mapsto \frac{1}{\abs{G}} \sum_{g\in G} gf(g^{-1} u)$; $\hat{f}_G$ ist $KG$-linear. $\hat{f}_G = Tr_(1)^G(f)$ "`Spur"'. \\ +Beweis: $\hat{f}_G$ ist wohldefinierte Abbildung von $U \rightarrow V$. $\hat{f}_G(u_1 + u_2) = \frac{1}{\abs{G}} \sum gf(g^{-1} (u_1 + u_2)) = \frac{1}{\abs{G}} \sum g f (g^{-1} u_1) + \frac{1}{\abs{G}} \sum g f (g^{-1} u_2) = \hat{f}_G(u_1) + \hat{f}_G(u_2). \hat{f}_G(\lambda u) = \lambda \hat{f}_G(u)$ analog. \\ +Also ist $\hat{f}_G \in \Hom_K(U, V)$. \\ +Sei $h \in G$, dann ist $(h \hat{f}_G)(u) = h \cdot \hat{f}_G (h^{-1} (u)) = \frac{1}{\abs{G}} h (\sum_g g f (g^{-1} (h^{-1} u))) = \frac{1}{\abs{G}} \sum_g (hg) f ( (hg)^{-1} u ) = \frac{1}{\abs{G}} \sum_g g f (g^{-1} u) = \hat{f}_G(u)$. \\ +Also ist $h . \hat{f}_G = \hat{f}_G$, d.h. $\hat{f}_G \in \Hom_K(U, V)^G = \Hom_{KG}(U, V)$. \qed + +5.2.13 Satz (Maschke): Sei $G$ endliche Gruppe, $\Char K = p \geq 0$. Sei $p$ kein Teiler von $\abs{G}$. Sei $U \in \lsub{KG}{\Mod}$ und $V \leq U$. Dann ist $V$ direkter Summand von $U$, d.h. $\exists W \leq U: V \cap W = (0), V + W = U$. \\ +Beweis: LAAG $\Rightarrow \exists \tilde{W} = K$-Unterraum von $U$ mit $V + \tilde{W} = U$. Sei $\tilde{\pi}$ die natürliche Projektion von $U$ auf $V$ entlang $\tilde{W}$, +d.h. $u \in U \Rightarrow \exists! v \in V, \exists ! w \in \tilde{W}: u = v + w \Rightarrow \tilde{\pi}(u) = v$. \\ +$\Rightarrow \tilde{\pi}_{\mid_V} = \id_V, \tilde{\pi}_{\mid_{\tilde{W}}} = 0$. \\ +Nach 5.2.12 ist $\pi = Tr_1^G(\tilde{\pi})$ $KG$-linear, $\pi(u) = \frac{1}{\abs{G}} \sum_{g \in G} g \tilde{\pi} (g^{-1} u)$. \\ +Sei $v \in V \leq U$. Dann ist $\pi(v) = \frac{1}{\abs{G}} \sum_g g \tilde{\pi}(g^{-1} v) = \frac{1}{\abs{G}} \sum_g g g^{-1} v = \frac{1}{\abs{G}} \sum_g v = v$. \\ +$\Rightarrow U = V \oplus \ker \pi$ \end{document}