From 482802ee2a15d9c7b98a72e2935dc216c1221888 Mon Sep 17 00:00:00 2001 From: flavis Date: Wed, 20 Nov 2019 20:29:11 +0100 Subject: [PATCH] update la5 --- ws2019/la/uebungen/la5.pdf | Bin 104841 -> 104849 bytes ws2019/la/uebungen/la5.tex | 9 +++++---- 2 files changed, 5 insertions(+), 4 deletions(-) diff --git a/ws2019/la/uebungen/la5.pdf b/ws2019/la/uebungen/la5.pdf index cf98acfc9a3213a13afae5d86064bf3c2c57ab35..f4c91647773d695c834711e107cf014a43a00b48 100644 GIT binary patch delta 4098 zcmV+d5dH6ovj&l~2C%yn0Wy<;3n;VX6krK|N5*HhcTl&t*02-kESWr^b9Hy?{RJQL z4(;SvK^Sc$wdu)49W%6gX7%>i>|&eDde)wL@Vn}VF^;YaRmViZu;YJAL0CeSn67!% zrumUn$-{G?d^GK|KL^@wQ-^`zPvi(=fj2v@cSffYj~wihg=9xgJm#4CLkpN@Jb(Xx zGrv8ipQN!f;D&7D5~ggujD3scv06|Y64b9=VVx#jz0bVo3tl^uLsi6fE5VQeBLJp+ zi7$%2(SLS(q1*zx0JqmLoy_`}{(A$#QAP~vFBPql==S9MqE#4VVMCNL+b6RH{{016 z!D*=9rmP^S$W&fm;lC(R0%qp#BWP+u$6ag??e{0*yQ>8f1vKG?dH*wV#G9zS(ew%p zOf^n$@Vl(Ct<^cOQ{M7UOzc1j;PR4XH~+H-MDPG8{eQ%q$x)MoA1MJhlYt8;f9+h| zliRitf4{#%d9rR2A@~O9jAv@sN!xhZX`8eA;LC$9C&{#BIn|xcll=8rkQ7N!3lgNL zyYsXUwqzd;AB)}J?*6dAw-^8ZC!4Hp&gf({p_9Lyy|_60_6MdW2}v~9lZ)j@1vD}6bv^m%>|f=A!VA-kXvQYSD8hv-e^y2nO;h-r zWi#3T^5pGIQ9AwQoZIQU`PTgmI%9PyQNpAfJ?rLG(L&Z*Fh&x|_ANL_$=K;KZ)of7 zBrL0Hh73g#%l8ew0~u;Fed<1R{$chX-{t?gIUGZP=!Dq!@wMAAK4sXXVK6E84Oc}A z)tF7Ws;Bv8zMOB0O<$$!f7^V=mzTmwOmbQirAx_uIGn5{T-*K6pCMCVlqjb6Em}5- zGh(gk*d$@g$y`!O59i~)m9k^qA~~GLjB!CYd{|<%3NG9b!{osxWqN~&frFs98BYwh zml(3ybx%?#A~_#ANgK4MXZeuDJQINM!D5B> z*YR@xF}1pD_QTSyP2%P=%IVs=P#P+c4AS(-R_T zX)SN?NtmRB+xJKpiYOr)lNg`?szm`{h*}T$wtFA|SgFcof7!sKRTnDSoQ!f;*PP*dEUU(VjM_ewDv+AJTj0v zsfRd)As`(lf8cbb*CFfCVVh;mkr0g=z(0Tfo5MMQpm^nf7=i+T)K=H$v(Mxjdji`l zc{CosLS~#mHY{OAjK$8EW4a-TP+~Hx_4B?9B{BIOZa?b2wZB;{$PLzJne9Sud0)uQ zA2V{Zhm9^}NLDCvdtE$5=oTZ1+P?gx|xg1vWVU08slFQ zVod2be_x;Ty^+1aI0g`bD)spH&zT2J0}8K9+vIZuEgH<;oP+EZA8tQwx0gc7-N%J5 zN0%VLMnV!XgQzm01rs?BQFSjC??}N${q5!!!!KDUB9<6j(rN)%WTGLbv6bA=m|d$f zarg?0GdMA#8TP9ND#)^nb-G#1w?)>N-t2S6e_akd`6gedi~PT@ck^}e_4(CRy1w3R z&`{2)q&!G(gy&r#a}bavmEji^SV*KH@|@M$bqehbGMEL!*Z63ja198?AFfltuaV&l z;Ia)2r@Y}3%Dmz7&1hP?@PlBQ0AyIFyby6D`7gd$d|sqCtuO)ka#Cr|Pm6T1!E6kK ze@ROfeR3Fp=!|p9!U7c|3oA{~N8CsKJma&&LFFcYLvXUZF^35~7sHU@_|aMcNK z#!!B|LaTr#IwlOm@l}q!>vVCIuFlz%VAM_GXfl+$IkZ3*B#Jf*lo0*qv80ISdHSf0 zpKWO?>U;ckAleYUM&c-6=_yl#dN847e}W65f`8%b#Bvu}j}w_jxolREJxh%8s_{|J zFl+HRufU2d3N5ltc!TH5bd7oJ2y3IJ%TGGpUgEtUi&w{5!hxSC3^6~sUwg7D4=eR1-7+=xv^d9VEobXe^h^i zoMcUMvRvF8Q()Lp4)XaDu&86O7P>=GM>N=D=ne&nBrQLN?od=_+i?Wl(R(2+g+C0w z>a1r$3%=tjBHw|G zrS(N#s1G!;-d{c-+F#><*qKqSe_fI8yMq(N4+#$Kg`Yv}(Cnr<8!<=7Ohq4an$CQD zgj4Dr)`g+6j()Wb@lTuu{DU=4&P!G$BE6@>@_N-Tuis&(J%)8JyzK2)95EZD8hDJ?;+he;{Hr@h75- zJM7RwTRFfc0}9)oY0H>QoA(+iGSe>p(9YBDBDf3O@xevB#)SG@-ztE+{D$+)wTq|-U6QFUdGM?Z;9uz1S0 zg;Qm0nMGA0841Fze4M`HC|Q_5na$(D^!XAk8s`<+b3C~8RF~Mzouk3Gte1zd_(HOi zSX3n146vs$(WY(UEp(933=$%B3>_r2AzCHJ&_Qk%0ur4g=%C&Uf9a}z5mG<~&qLKO z^8eK@r>=g1BwQ!))h`tr*C+ze{UR!E6aV=hp2l|rZhPr%y9!$MLR_W+9&Ovf>tA%atd z4r2PyXfqC&;?B_Se}he1Vr9D4m>4$AAY|GSE1ecp4d>ESh(Nk;MU1dCiC9`N+6Ym7 z%?c(@Oi4X_{+}-+!l*uHJ&b3@!`b2X$`Fu-ti9)Dggyp^vwbp>?F)=#F;g2a(0qbT zy>^xJ;3AakHFSI~v&z*z>uII#9z=41v#o(e4xvpj#8l}{e_#*nkTdQlIFVmbQq`He z4SBbiIOXKKD|%TVA1{f2f+q2`t1h7=o{>bvY+yg>|rprc{Zl|#x1Qazq z!mqki_inHQ-IuTVlZ(Sw4tzDnJuWq6sgd&EU|-30<|>onO~xwSg|9mlpBr!JmpI4o5MF6v<~ zJO!D2f6{#Bv3i{A;DW2GLv6g(_2Bno#<~7zM!ip1o9|5Vz#D18LL||sSihbZ=jiVi z>7_8H+y|fSab-4**VQ_obqQT0V10|8ec>mG>IzemQ(O>=JYMAt4AO1FkK{ zm1oB0CC5A(7rp!r#vfOIi1KCgy8M5#(uoxse{Kj)XwqC>!=+a*rn3A$L5@j|^Z#KIf2Kf)&SEI+2^(SY1B1(`w6NtVFhLYO>LoBk;p4c6zyx@aFf&F)HaRsxmrj=fIe%SD&S4nG@%u_dQBf31MTzvJ$lJVR&r0NN zNr_C-^7Nnx6Y^H52Qeo)!Vz;|Y;9&|mQH%bwyVt=xjVs1k}w;KncQuQ`vLzRkdu7&C*sA+>OrQ$c# z-hr)}V_PNE^}_ahu*0A}33jGJgNAMxf?e03@jW!9!R~qp)j+dKg|*#1+Q#0G(9#I4 z*J1x-IPezQ-oe2iaA+6~e}E(E(XMT@zk#C<;n)Bik3h#oIME6xJK)q+I9%NVXF8!X z3|&=lwhqqe|IR;x?k~{u9U`iC;Tv4ah2CPnzci69rkT4O@@E<5?(Xv`)7((MPg&-M z@B1{}Ea!z!Gt7c7eabcq5BQWL3e0+5JF~tA?mthoXhfG_nE@aPF*XV%B}Gq03h?db A`v3p{ delta 4115 zcmV+u5bW=fvj&N?2C%yn0Wp(-3n;VX6krK|SH{&lsM}j>*a>u&OrFrWy1VuMf)9Cz zc58lN7sd_W1?W#@xP@YETKwF*Su=e z{79lA1xz!ZzyFzk z-yYLX(%2brL$+}VQ?_2lzQyubEvO9%>Q}F@PLrKxc9Z+RyscAx}sdC9Vy|Jef~cmS0CKQdg&MU#XdDFHB(feR>q?Ofe+ z+qe;b_g}$#iadA1;TxcDGSj4Y*Y+~)v^^(%i0?r@O9~T5xyqI^y}!N-k|GIeL4pz` zJJUYImSY$ni{0Pu{;BPWGYEXs<}=s7m4IV@x(Gg zm@&!aP4aefC4{`*td@7zx3l|y`}3&~V)C~&IiE_YChC03xwK~T$9L=O?mnBXiUxn3 zukYupZ`=uRko0g7L1WpvTX|vQZjb3C>q*kI|<9G zn;}Dy)beeE??8syOfK9vU3@d!Z{HRFx!N5=farwS_VKmbF+O3~q+u{Aw+)wN3)PxU zwJK6<=J&Hzx#`R7_P*GE@x_%e5|fgu}NKY)S?fZir!WXOnWh!NkBpP~40s20KU$ z`Rw{9DHM^M?>k8ww5R9AkmWoXO*Bc3uz5om5^!S5;G6Q-%sgp-YuL3$qYGsEQ*PY? 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Aus (c) folgt: $k+1 \neq 0 \iff \text{char K} \not\in \{2, \ldots, n+1\} $. \begin{enumerate}[(i)] - \item $\text{char }K \not\in \{2, \ldots, n+1\} $. Dann ist $k + 1 \neq 0$, d.h. + \item $\text{char }K \not\in \{2, \ldots, n+1\} $: Dann ist $k + 1 \neq 0$, d.h. \begin{align*} &f(k+1) = 0 \text{ } \forall k \in \{0, 1, \ldots, n\} \\ \implies &f(k) = 0 \text{ } \forall k \in \{1, \ldots, n+1\} \\ @@ -242,7 +242,7 @@ $V = \text{Abb}\left( \{0, 1, \ldots, n+1\}, K\right)$. \psi(\text{ker }\partial) = \{(a, \underbrace{0, \ldots, 0}_{n+1\text{-mal}}) \mid a \in K\} .\] - \item $\text{char }K \in \{2, \ldots, n+1\} $. Dann gilt für $k = \text{char }K-1$: + \item $\text{char }K \in \{2, \ldots, n+1\} $: Dann gilt für $k = \text{char }K-1$: \[ k + 1 = \text{char } K - 1 + 1 = \text{char } K = 0_K .\] @@ -346,9 +346,10 @@ $V = \text{Abb}\left( \{0, 1, \ldots, n+1\}, K\right)$. .\end{align*} Wegen $f$ surjektiv gilt: $V = f(U)$ und damit: \begin{align*} - \forall v \in V\colon \varphi_1(v) = \varphi_2(v) + & \forall v \in V\colon \varphi_1(v) = \varphi_2(v) .\end{align*} - $\implies \varphi_1 = \varphi_2$ + $\implies \varphi_1 = \varphi_2$ \\ + $\implies f^{*}$ injektiv \end{proof} \end{enumerate} \end{aufgabe}