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znY?HRwg6VBo~ASF4?r2eXAdx_8hBiT6$&IFgiUC;qYh^wg0awG5Wtjzj2Egsh(vru zIF%Gk2-S~ZLNFx$a-xM72m|JzV~=3>vCt z>CA}!CM78xz8#ySOaF?+ zbd_RC*EJ|1AqB1;fu9fs>AL#IM<>81&@c1JS2WyRsvy0^US7UFxUa8|Lf(_`@s)V_ u!bYP=vZt3X_{\text{Ideal}} \trianglelefteq KG$. \\ + ( Für $S \subseteq A = K$-Algebra: $_A = \bigcap_{I \trianglelefteq A, S \subseteq I} I = $ kleinstes Ideal von $A$, das $S$ als Teilmenge enthält $= \sum_{s\in S} AsA$ ) \\ + $\dim_K (\ker \rho) = \abs{G} - 1$ \\ + Das Ideal $< g-1 \mid g \in G>_{\text{Ideal}}$ heißt \underline{Augmentationsideal} von $G$ und wird mit $\Omega KG$ bezeichnet. + + \item Sei $H \leq G$, $\lsub{H}{K}$ triviale $KH$-Modul, $G = \bigcup\limits_{i \in I}^{\bullet} g_i H, \set{g_i \mid i \in I}$ Vertretersystem von $H$-Linksnebenklassen in $G$. \\ + $G$ operiert auf $G \without H = \set{ g_i \mid i \in I}$ durch Linkstranslation: Für $g \in G, i \in I$ gibt es genau ein $j \in I$ und $h \in H: g g_i = g_j h$ \\ + $g \cdot g_i \Rightarrow g_j$ ist Permutationsdarstellung. \\ + Sei $V$ der $K$-Vektorraum mit Basis $g_i, i \in I$. Dann wird $V$ zum $KG$-Modul nach ii). \\ + $\varphi: KG \otimes_{KH} K \Rightarrow V: \sum_{i \in I} \alpha_i g_i \otimes 1_k \mapsto \sum_{i \in I} \alpha_i g_i$, + d.h. Permutationsmodul von der $G$-Menge $G \without H$ ist $KG \otimes_{KH} K = \Ind_{KH}^{KG}(\lsub{H}{K})$ + + Umgekehrt: Ist $\Omega$ eine $G$-Menge und ohne Einschränkung transitiv (disjunkte Vereinigung von Orbits $\leftrightarrow$ direkte Summe der zugehörigen Permutationsmodule). \\ + Sei $\Omega = (\omega_1, \ldots, \omega_k), H = \Stab_G(\omega_1)$. Dan ist der Permutationsmodul nach ii) isomorph zu $KG \otimes_{KH} K = \Ind_H^G(\lsub{H}{K})$. + + \item Sei $V \in \lsub{KG}{\Mod}, V^\ast = \Hom_K(V, K)$. Dann wird $V^\ast$ zum $KG$-Modul durch $ f \in V^\ast, g \in G, v \in V: (gf)(v) := f(g^{-1} v)$. \\ + D.h. für fast alle $\lambda_g = 0$: $((\sum \lambda_g g) f)(v) = \sum \lambda_g f (g^{-1} v) \in K$ +\end{enumerate} + +5.2.4 Lemma: Sei $A, B, C, D, \ldots $ $K$-Algebren, $\lsub{A}{M_B} \in \lsub{A}{\Mod_B}, \lsub{A}{N_C} \in \lsub{A}{\Mod_C}$. Dann wird $F := \Hom_A(\lsub{A}{M_B}, \lsub{A}{N_C}) \in \lsub{B}{\Mod_C}$ durch: \\ + $~~(bf)(m) := f(mb), (fc)(m) := f(m)c$ \\ + Analog: Ist $M \in \lsub{A}{\Mod_B}, N \in \lsub{C}{\Mod_B} \Rightarrow F := \Hom_B(\lsub{A}{M_B}, \lsub{C}{N_B}) \in \lsub{C}{\Mod_A}$ durch: \\ + $~~(cfa)(m) := c \cdot f(am)$ \\ +Beweis: $bf$ ist wohldefinierte Abbildung von $M \rightarrow N$. Zu zeigen: $bf$ ist $A$-linear:\\ + $ ~~(bf)(m_1 + m_2) = f((m_1+m_2)b) = f (m_1 b + m_2 b) = f(m_1 b) + f(m_2 b) = (bf)(m_1) + (bf)(m_2)$ \\ + $ ~~(bf)(am) = f((am)b) = f(a(mb)) = a \cdot f(mb) = a \cdot (bf)(m)$ \\ + Weiter ist z.Bsp: \\ + $ ~~ ((b_1 b_2)f)(m) = f(m(b_1 b_2)) = f((mb_1) b_2) = (b_2 f)(m b_1) = (b_1(b_2 f))(m)$ usw. (Übung) \\ +Also ist $\Hom_A(M, N)$ ein Links-$B$-Modul, jetzt Rechts-$C$-Modul: + $ ~~ (fc)(m_1 + m_2) = f(m_1 + m_2)c = (f(m_1) + f(m_2))c = f(m_1)c + f(m_2)c = (fc)(m_1) + (fc)(m_2)$ \\ + $ ~~ (fc)(am) = f(am)c = (a \cdot f(m))c = a (f(m)c) = a (fc)(m)$ \\ + $ ~~ (f (c_1 c_2))(m) = f(m)(c_1 c_2) = (f(m)c_1)c_2 = (fc_1)(m)c_2 = ((fc_1)c_2)(m)$ usw. \\ +und: $ ((bf)c)(m) = ((bf)(m))c = (f(mb))c = (fc)(mb) = (b(fc))(m)$, damit ist $\Hom_A(M,N)$ ein $B$-$C$-Bimodul. \qed + +Beispiel: $V \in \lsub{A}{\Mod} \Rightarrow V^\ast \in \Mod_A, V^\ast = \Hom_K(\lsub{A}{V_K}, K_K) \in \Mod_A$ \\ + $f \in V^\ast, a \in A: (fa)(v) = f(av)$ + +5.2.5 Lemma: Sei $A$ $K$-Algebra, $\iota: A \rightarrow $ Antihomomorphismus von $K$-Algebren, d.h. $\iota$ ist $K$-linear, $\iota(a_1 a_2) = \iota(a_2) \iota(a_1)$. \\ + Sei $V \in \lsub{A}{\Mod}$. Dann wird $V \in \Mod_A$ durch $v . a = \iota(a) v$ \\ +Beweis: $v . (a_1 + a_2) = \iota(a_1 + a_2)v = \iota(a_1)v + \iota(a_2)v = v . a_1 + v . a_2$ und \\ + $ ~~ v . (a_1 a_2) = \iota(a_1 a_2) v = \iota(a_2) \iota(a_1) v = \iota(a_2)(va_1) = (va_1) . a_2$ usw. \qed + +5.2.6 Lemma: Sei $G$ Gruppe. Dan induziert $g \mapsto g^{-1}$ einen Antihomomorphismus von $KG$ in $KG$ ($\sum \lambda_g g \mapsto \sum \lambda_g g^{-1}$). + +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 $\Hom_K(\lsub{KV}{V_K}, \lsub{KG}{W_K}) \in \lsub{KG}{\Mod_{KG}}$) \end{document}