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¼ ½¿¾0À5¼ Áÿ½ÅÄ - Université du Luxembourgmath.uni.lu/~wiese/other/diplom.pdf · 2007-03-08 · ë ËNÖ!È Ç ˙ ÕqÛ*Ü A+ ×3Ì3ÇÎÉ Ç ÕqÍ ÛYÐ.ÑØÕqÛ*Ü

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Page 1: ¼ ½¿¾0À5¼ Áÿ½ÅÄ - Université du Luxembourgmath.uni.lu/~wiese/other/diplom.pdf · 2007-03-08 · ë ËNÖ!È Ç ˙ ÕqÛ*Ü A+ ×3Ì3ÇÎÉ Ç ÕqÍ ÛYÐ.ÑØÕqÛ*Ü

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Page 2: ¼ ½¿¾0À5¼ Áÿ½ÅÄ - Université du Luxembourgmath.uni.lu/~wiese/other/diplom.pdf · 2007-03-08 · ë ËNÖ!È Ç ˙ ÕqÛ*Ü A+ ×3Ì3ÇÎÉ Ç ÕqÍ ÛYÐ.ÑØÕqÛ*Ü
Page 3: ¼ ½¿¾0À5¼ Áÿ½ÅÄ - Université du Luxembourgmath.uni.lu/~wiese/other/diplom.pdf · 2007-03-08 · ë ËNÖ!È Ç ˙ ÕqÛ*Ü A+ ×3Ì3ÇÎÉ Ç ÕqÍ ÛYÐ.ÑØÕqÛ*Ü

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pö�õ/Ì3ÛYÓYÓ9È�dzÙ3×{é*ê�Ôîø�ÕqÍÎÍ-ÛYÐYÇ ÜoËNÌ3Ô�È�Ì/õ=Ì3ÛYÓYÓ!È�Ð6Í ì$��×

Ù@dzï@åÊÑ*dzÈvÙ@È�ÌAÈ���ÓYÍÎÇÎòvÇÎ×AÕqÛ.ÙIÑYÈ�Ô¬z�Ì3ò{È�ÛYÏ�È�Ð.ÑYÈ�ÐYÌ�ÕqÐYÏ�Ö!È�Ì@Èvï�åYÐ.È�Ð��r(G) =

(

d(G)

2

)

.ûR¨�ÿ

áçÇÎÌò{È�ÇÎÏ�È�ÐÊÈ�ÇÎÐ ðAÈvÙ3ÛYÍÎ×�Õq×AÕqÛ.ÙÝß á�â+~*ëlÈ�ͳï�å.ÈvÙIÖ9ÈvÙ�ÕqÏ�×^~.Ñ.Õ$��Ù�ËNÏ¢ÕqÌIÙ3×�ìqÌd{�È�ÌÏ�ÇÎÍÎ×Wû=�AËNÌ@ËNÍÎÍ ÕqÌ�ûR¨*驨*éK�^��ÿ3ÿ��dimFP

H2(G,Z/pZ)+ =

(

d(G)+

2

)

+

(

d(G)−

2

) ÛYÐ.Ñ ûR­�ÿdimFP

H2(G,Z/pZ)− = d(G)+ · d(G)−.û5®±ÿ

Æ=ÕqÌ�ÕqÛ.Ù�Ù�ï@åYÍÎdzÈ/��×=Ô�ÕqÐàò�é0£=éK~�Ñ.Õ$�ÚÈ�ÇÎÐ.È�ÛYÐYÇ ÜoËNÌ3Ô�È�õ/Ì3ÛYÓYÓ9ÈWÔ"ÇÎ×/È�Ð.Ñ*ÍÎdzï�å.È�ÌÝÞIÖ9È�ÍÎdzÙ3dzÈ�Ì3ÛYÐYÏ�ÛYÐ.ÑXz�Ì3ò{È�Û*öÏ�È�Ð.ÑYÈ�ÐYÌ�ÕqÐYÏ

d(G) ≤ 3ТÛYÌIÑ*dzÈÝùX�NÏ�ÍÎdzï@å,{�È�ÇÎ×@È�Ð

(d(G)+ = 1ÛYÐ.Ñ

d(G)− = 2)Ë�ÑYÈ�Ì

(d(G)+ = 3ÛYÐ.Ñ

d(G)− = 0)ûgþ�ÿ

òvÛYÍ ì$��×�ûaɱÏ�Í�é9�*Õq×3ò�ûR¨*驨*驨G¨�ÿ3ÿdéz�Ù0dzÙ3×?É�ËNÐ È�б×@Ù@ï@å.È�dzÑYÈ�Ð.ÑYÈ�Ì�á�dzï�å¢×3ÇÎÏG{�È�ÇÎ×^~

dimFp (pH2(G,Qp/Zp)−)

òvÛ�{�È�ÐYÐ.È�Ð�~NÛYÔ_ÕqÛ.Ùyûl�&ÿ¯��ï@åYÍÎä.Ù,öÙ�È=òvdzÈ�å.È�ÐÊòvÛv{[�NÐYÐ.È�ÐQé.áçÇÎÌyëlÈ�Ì@ÑYÈ�ÐÒÙ@È�å.È�Ð�~.Ñ.Õ$��Ñ*dzÈvÙ�ÈÝÆ/ÇÎÔ�È�Ð.Ù3dzËNÐ ÇÎÐ É�dzÈ�ͳÈ�Ð ø�ìqÍÎͳÈ�ÐS{±Í³È�ÇÎÐ.È�ÌIÏ�ͳÈ�dzï@å

1dzÙ@×^~

ë�ËNÌ�ÕqÛ.ÙAÙ@dzï@åÚÜoäYÌd(G)+ 6= 1

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ÕqÛ*Üo×3Ì@È�×@È�ÐS{�ÕqÐYÐQéÆAdzÈvÙ�ÈvÙðIÈvÙ3ÛYÍÎ×�Õq×AÈ�Ì3å�ìqÍÎ×Ô�ÕqÐàÕqÛ.ÙÑYÈ�Ô Æ/Û�ÕqÍÎÇÎ×�ìq×@Ù@Ù�Õq×3ò

Hi(G,E) ∼= H2−i(G,Q/Z)∨ÜoäYÌIÕqÍÎͳÈ

i ∈ Z,ûR°�ÿ

ë�È�ͳï@å.È�Ì/ÇÎÐ ÑYÈ�Ð Ö9È�dzÑYÈ�Ð6ÜoËNÍÎÏ�È�Ð.ÑYÈ�Ðàø0ìqÍÎͳÈ�ÐÊÏ�ÇÎÍÎ×WûaɱÏ�Í�é,��å.ÈvËNÌ@È�Ô�È�ûl��驨*驨G±�ÿ�ÖYòvëÝé�ûl��驨*é©­G²�ÿ3ÿ��ûoê3ÿ

• SÙ@È�ÇQÈ�ÇÎÐ.È�ûaÔH�NÏ�ÍÎdzï�å.È�Ì3ëlÈ�dzÙ@ÈWͳÈvÈ�Ì@È&ÿ�ùÒÈ�ÐYÏ�È�É�ËNÐ���Ì3ÇÎÔ�Ù3×@È�ÍÎͳÈ�ÐàÈ�ÇÎÐ.ÈvÙ³�9ÕqåYÍK{[�NÌ3Ó9È�Ì@Ù

• LS(p)|kÙ@È�Ç!Ñ*dzÈ/Ô�ÕF�*ÇÎÔ�ÕqͳÈ/ÛYТÉ�È�Ì3òvëlÈ�ÇÎÏ�×@ÈÝÏ¢ÕqͳËNdzÙ@Ù@ï@å.È

pö+z�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ�É�ËNÐ

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• G := G(LS(p)|k)Ù@È�ÇQÑ*dzÈWõÝÕqͳËNdzÙ,ö�õ/Ì3ÛYÓYÓ9ÈÝÑ*dzÈvÙ@È�Ì�z�Ì3ë�È�ÇÎ×@È�Ì3ÛYÐYÏ.é

• E := ES(LS(p)) = lim−→LS(p)⊃K⊃k, K|k ´Vµ/¶/·k¸k¹=º E(K)

~�ëlËNÖ9È�ÇE(K) := ES(K)

Ñ*dzÈ/õ/Ì3ÛYÓYÓ9ÈÑYÈ�Ì

Sö+z�ÇÎÐYå.È�ÇÎ×@È�Ð"È�ÇÎÐ.È�Ì?È�Ð.Ñ*ÍÎdzï�å.È�РϢÕqͳËNdzÙ@Ù@ï@å.È�Ð_�D�NÌ3Ó!È�Ì@È�Ì3ë�È�ÇÎ×@È�Ì3ÛYÐYÏ

LS(p) ⊃ K ⊃ kÖ!È�ò{È�dzï�åYÐ.È�×{é

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SpÖ9È�ò{È�dzï@åYÐ.È�Ð�ÿAÈ�Т×3å�ÕqÍÎ×@È�Ð��

Sp ∩ S = ∅é

• kS(p)|kÙ@È�ÇQÑ*dzÈ=Ô�ÕF�*ÇÎÔ�ÕqͳÈWÕqÛ,��È�Ì3å�ÕqÍÎÖ

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pö+z�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ�É�ËNÐ

• G := G(kS(p)|k)Ù@È�ÇQÑ*dzÈ�õ�ÕqͳËNdzÙ,ö�õ/Ì3ÛYÓYÓ9È�Ñ*dzÈvÙ@È�Ì�z�Ì3ë�È�ÇÎ×@È�Ì3ÛYÐYÏ.é

• E(K) := O×K,m := {a ∈ OK | a ∈ U

(np)p

ÜaäYÌIÕqÍÎͳÈp}~�ëlËNÖ9È�Ç

kS(p) ⊃ KÈ�ÇÎÐ.ÈØÈ�Ð.Ñ*ÍÎdzï�å.È

Ï¢ÕqͳËNdzÙ@Ù@ï�å.È����NÌ3Ó9È�Ì@È�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏãÉ�ËNÐkÙ@È�Ç�élÆ=ÕqÖ9È�ÇIdzÙ@×

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kS(p),m = lim−→K

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k|k+ ò{È�Ì,Ü�ÕqÍÎͳÈ�éÞIÛ.ÙÑYÈ�Ô ÆAÛ�ÕqÍÎÇÎ×�ìq×@Ù�Ù�Õq×3òÝëlÈ�Ì@ÑYÈ�ÐàëyÇÎÌIÈ�ÇÎÐ.È��±ÛYÌoý3È/{±×3dzËNÐ

µ(k)/p ⊇ (E(k)/p)− � (pH2(G,Qp/Zp)

−)∨ûV»�ÿ

ûµ(k)Ù3ÇÎÐ.ÑèÑ*dzÈ�ÇÎÐ

kÈ�Т×3å�ÕqÍÎ×@È�Ð.È�Ð¥z�ÇÎÐYå.È�ÇÎ×@Ù@ëyÛYÌ3ò{È�ÍÎÐ�ÿ=Ù@ï@åYÍÎdzÈ/��È�Ð@{[�NÐYÐ.È�Ð�~!ë�È�ͳï@å.È�Ù@ËqÜgËNÌ3×=Ñ*dzÈWÖ9È�å�ÕqÛYÓY×@È�×@È

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i > 0é�ø�äYÌ

i ≤ 0Ù@È�×3òv×yÔ�ÕqÐ

Hi(G,A) = lim←−U�G ½l¾�´Vµ H

i(G/U,AU ),ûR��ÿ

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d(Clk)+ ≤ 1 + d(Clk)

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dzÙ@×^~�ë�ËNÍÎͳÈ�Ð ëyÇÎÌ�Ñ.ÕqÌ3ÛYТ×@È�Ì�ÇÎÔ"Ô�È�Ì7�-ÇÎÐ,{*Ù3Ô�Ë�Ñ*ÛYÍÎÐÒÉ�È�Ì@Ù@×@È�å.È�Ð�ÿdé-Æ=ÕqÐYÐ Ù3ÇÎÐ.ÑA ⊗Z B

ÛYÐ.ÑHom(A,B) G

öùÒË�Ñ*ÛYÍÎÐ ÛYТ×@È�ÌIÑYÈ�ÐàÞ�{�×3dzËNÐ.È�Ð

g.(a⊗ b) := g.a⊗ g.bÖYòvëÝé

(g.φ)(a) := g.(φ(g−1.a)).

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A⊗GBÙ@ï@åYÌ@È�ÇÎÖ9È�ÐÝëlÈ�Ì@ÑYÈ�ÐQécÆ�ÕqòvÛ�ÜgÕNÙ�Ù@È�Ð=ëyÇÎÌ

AÕqÛ*Ü.ÕqͳÙ�ðAÈvï@å±×@Ù3Ô�˱Ñ*ÛYÍ

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é��?È�Ì�Æ=È��.ÐYÇÎ×3dzËNÐ�Ï�ÇÎÍÎ×?ÕqͳÙ@Ëa.g⊗Gb = a⊗Gg.b

ÖYòvëÝég.a⊗Gg.b = a.g−1⊗Gg.b = a⊗Gb

éêuÐ6í�È�Ì�ÕqÍÎÍÎÏ�È�Ô�È�ÇÎÐ.È�Ì3ÛYÐYÏ6ÑYÈ�ÌËNÖ!È�ÐàÈ�Ìd{±Í ìqÌ3×@È�ÐÊÞD{�×3dzËNÐ È�Ì3å�ÕqÍÎ×@È�ÐÊëyÇÎÌÑ.Õqå.È�ÌAÈ�ÇÎÐ.È�ÐÊ×3Ì3ÇÎÉ±Ç ÕqͳÈ�Ð

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á�ÇÎÌ�ÑYÈ��.ÐYdzÈ�Ì@È�ÐôÕqͳÙIG = (s − 1|s ∈ G)

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Z⊗G A→ A/IGA =: AG, n⊗G a 7→ na+ IGA

dzÙ@×AÈ�ÇÎÐèêuÙ@ËNÔ�ËNÌ3ÓYåYdzÙ@Ô�Û.Ù{é!Æ�ÕqÔ"ÇÎ×AdzÙ3×AGÑYÈ�ÌAÏ�ÌQ� ��×@È�ø�Õ${�×@ËNÌ3Ô�˱Ñ*ÛYÍ0É�ËNÐ

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A⊗G B → (A⊗Z B)G, a⊗G b 7→ (a⊗Z b) + IG(A⊗Z B).

ø�È�Ì3Ð.È�ÌIÖ9È�Ô�È�Ìd{�È�Ð ëyÇÎÌ^~.Ñ.Õ$�HomG(A,B) := {f ∈ Hom(A,B) | g.f(a) = f(g.a) ∀a ∈ A} = (Hom(A,B))G

ÛYÐ.ÑÊÙ3Ó!È�òvdzÈ�ÍÎÍAG = HomG(Z, A)

Ï�ÇÎÍÎ×{éá�ÇÎÌÖ9È�×3Ì�ÕNï@å±×@È�ÐàÑ*Ç³È �0JV! �98[! )d8[! U�À?($I.UT�BA

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←−−− F (G)0 := Z[G]∂1←−−− F (G)1 := Z[G2]

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ë�ËNÖ!È�Ç�ëyÇÎÌIÑYÈ��.ÐYdzÈ�Ì@È�Ð��∂n :=

n∑

i=0

(−1)idiÛYÐ.Ñ

di(g0, . . . , gn) := (g0, . . . , gi, . . . , gn).

��È�×3ò{È�ÐàëyÇÎÌhr := g−1

r−1gr~YÙ@Ë"È�Ì3å�ÕqÍÎ×@È�ÐÊëyÇÎÌÑ*dzÈ�êuÑYÈ�б×3ÇÎ×�ìq×

(g0, g1, g2, . . . , gn) = g0.(1, h1, h1h2, . . . , h1h2 . . . hn) =: g0.[h1|h2| . . . hn].

Æ/dzÈ��B�±ÔWÖ!ËNͳÈ[h1|h2| . . . |hn]

Ô"ÇÎ×IÖ9È�ÍÎdzÈ�ÖYÇÎÏ�È�Ðhi ∈ G

ÖYÇÎͳÑYÈ�ÐèÙ@ËNÔ"ÇÎ×IÈ�ÇÎÐ.ÈZ[G]ö+£yÕNÙ3dzÙÉ�ËNÐ

F (G)n~.ÖYòvëÝé

F (G)n = Z[G]⊗Z (ÜoÌ@È�dzÈ�ÕqÖ9È�ͳÙ@ï@å.È�õ/Ì3ÛYÓYÓ!ÈÝÕqÛ*Ü

[h1|h2| . . . |hn])é.ùèÕqÐ Ö9È�Ì@Èvï@åYÐ.È�×/Ñ*dzÈ�á�ÇÎÌd{�ÛYÐYÏ�É�ËNÐ

diÕqÛ*Ü�Ñ*dzÈvÙ@È�Ì�£yÕNÙ3dzÙòvÛ

di[h1| . . . |hn] =

h1ßh2| . . . |hn

âi = 0ß

h1| . . . |hihi+1| . . . |hnâ

0 < i < nßh1| . . . |hn−1

âi = n.

9S�9�h�0ß5��� Þ ���È�Ð.Ù@ËNÌ3dzÈ�Ì@È�Ð ëyÇÎÌ�Ñ*dzÈ7�±×�ÕqÐ.Ñ.ÕqÌ@Ñ.ÕqÛ:Ï���Ù3ÛYÐYÏ�äYÖ9È�Ì

Z[G]Ô"ÇÎ×yÑYÈ�Ô

Gö�ùàË�Ñ*ÛYÍ-Þ�~±ÖYòvëÝé�×@È�Ð.Ù@ËNÌ3dzÈ�Ì@È�Ð6ëyÇÎÌ�äYÖ!È�Ì

ZÔ"ÇÎ×

AÛYÐ.Ñ�Ð.È�åYÔ�È�ÐØÑYÈ�Ð�Ï�ÌQ� ��×@È�Ð

Gö�ÇÎбÉNÕqÌ3Ç Õqб×@È�Ð6ø�Õ${±×@ËNÌ3Ô�Ë�Ñ*ÛYÍV~¢Ù�Ë�È�Ì3å�ÕqÍÎ×@È�Ð�ëyÇÎÌ�ÑYÈ�Ð}�AËNÔ"ÓYͳÈ��àûaÐ�ÕNï@å�AËNÔ"Ó!Ë�Ù@ÇÎ×3dzËNÐàÑYÈ�ÌIÈ�Ì@Ù3×@È�Ð6Ö9È�dzÑYÈ�ÐÊÞAÖYÖYÇÎͳÑ*ÛYÐYÏ�È�Ð�ÿ

0← Z[G]⊗Z[G] A← Z[G2]⊗Z[G] A← Z[G3]⊗Z[G] A← . . . ,

ÑYÈvÙ�Ù@È�Ð�æ/ËNÔ�ËNͳËNÏ�dzÈëyÇÎÌ�Ô"ÇÎ×Hn(G,A)

Ö!È�ò{È�dzï�åYÐ.È�Ð�ëlËNÍÎͳÈ�ÐQé±Æ/dzÈYz�ͳÈ�Ô�È�Т×@ÈAÉ�ËNÐZ[Gn+1]⊗Z[G]A

Ð.È�ÐYÐ.È�ÐëyÇÎÌ �PZ=���/JRJ+��� é

H∗(G, ·)ÖYÇÎͳÑYÈ�×=È�ÇÎÐ.È�Ð *:E I.�=J=��C$�/�@#TE 4HE %KEQA ��I/<�#,���XË�UT�,&FJVEG) ÖYòvÏ�Í�é�ÑYÈ�Ì

Gö�ùÒ˱Ñ*ÛYÍÎÐ�~!Ñ-é åQéYÜoäYÌlý,ÈvÑYÈ�{±ÛYÌ,ö

ò{È(È��.Õ${±×@Èj��Èn ¢Û.È�ÐYò0 → A → B → C → 0

É�ËNÐGö�ùàË�Ñ*ÛYÍÎÐ5Ï�ÇÎÖ*×ÒÈvÙ ÜoäYÌ�ý3ÈvÑYÈvÙ

nÈ�ÇÎÐ.È�ÐHí�È�Ì,ö

ÖYÇÎÐ.Ñ*ÛYÐYÏ�Ù@å.ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�Û.ÙèûoÑYÈ�Ì ÐYdzï@å¢×�È�ÇÎÐ.ÑYÈ�ÛY×3ÇÎÏúÙ�È�ÇÎÐÃÔWÛ,�P~�ÕqÖ!È�Ì�Ð�Õq×3äYÌ3ÍÎdzï@å Ï�È�ë�ìqåYÍÎ×"ëlÈ�Ì@ÑYÈ�Ð�{�ÕqÐYÐ�ÿδn : Hn+1(G,C)→ Hn(G,A)

~�Ù@Ë"Ñ.Õ$��Ï�È�ÍÎ×@È�Ð��ûaÇ�ÿÚø?äYÌ0ý3ÈvÑYÈvÙ?{�ËNÔ"Ô�ÛY×�Õq×3ÇÎÉ�È�Æ/Ç ÕqÏ�Ì�ÕqÔ"Ô

0 −−−→ A −−−→ B −−−→ C −−−→ 0

f

y

g

yh

y

0 −−−→ A′ −−−→ B′ −−−→ C ′ −−−→ 0Ô"ÇÎ×IÈ��.Õ${�×@È�Ðf�!È�ÇÎͳÈ�ÐS{�ËNÔ"Ô�ÛY×3dzÈ�Ì3×=Ñ.ÕNÙ�ÜgËNÍÎÏ�È�Ð.ÑYÈÝÆ/Ç ÕqÏ�Ì�ÕqÔ"ÔS�Hn+1(G,C)

δn−−−→ Hn(G,A)

Hn+1(h)

y

Hn(f)

y

Hn+1(G,C′)

δn−−−→ Hn(G,A′)

ûaÇÎÇ�ÿÚø?äYÌ�ý3ÈvÑYÈ�{±ÛYÌ3ò{È"È��.Õ${±×@È}��Èn ±Û.È�ÐYò0 → A → B → C → 0

dzÙ@×ÝÑ*dzÈ�ÜoËNÍÎÏ�È�Ð.ÑYÈ ��Èn ±Û.È�ÐYòàûoÑ*Ç³È %K! �BA���d]�!$&FJ+�����>=nU,���:6 ÿ�ÜoäYÌIÕqÍÎͳÈnÈ��.Õ${±×^�

Hn(G,B)← Hn(G,A)δn←−−− Hn+1(G,C)← Hn+1(G,B)← Hn+1(G,A)

Æ/Ç³È 1YE$I.�=J=��C��5JV�GJ Ö9ÈvÑYÈ�ÛY×@È�×^~.Ñ.Õ$�Hn(G,A)

ÜoäYÌIÕqÍÎͳÈn < 0

×3Ì3ÇÎÉ�Ç ÕqÍQdzÙ3×{é���9�=���=�P� ö �B�¥�¥â��������

A���5� *9)lEd¡n�.&$J=��C$�/)

GZ+>XE^8 UT% o ¿ ! ���\A��/%;J+����ÊÇÆ! È

A��I�Jt#TE 4HE %KEQA ��I/<�#�J=)��5C^�=!G%Ã-x8 o # o Hn(G,A) = 0

��NB)�! %5%�n > 0 o

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ÇlO ÈAU

��I�J2�^OQ�����/! %5%'I2*9)lEd¡n�.&$J=��C}! %'IG/U

ZV>�E^8 UT%B��NB)��/���0�/�@?�E ).4H! %MJ+���5%K�/)U �G o

A �CB�� Þ ä[� ø?äYÌ�ûgÕ�ÿ�Ù3dzÈ�å.È/ò�ék£=é-ßk£�Ì,â�é���ÛúûaÖ�ÿ�Ù@È�ÇM � N

È�ÇÎÐ.È��±ÛYÌoý,È/{�×3dzËNÐ6òvëlÈ�dzÈ�ÌG/Uö�ùàË�Ñ*ÛYÍÎÐ�~*ÛYÐ.Ñ

φ : AU → NÈ�ÇÎÐ æ/ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3ÔWÛ.Ù{é�Æ/ÛYÌ@ï@å

G � G/UëlÈ�Ì@ÑYÈ�Ð

MÛYÐ.Ñ

NÕqÛ.ï@å òvÛ

Gö�ùÒ˱Ñ*ÛYÍÎÐ�~

ë�ÈvÙ3å�ÕqÍÎÖóÈ�ÇÎÐGö�ùÒ˱Ñ*ÛYÍÎå.ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ@Ô�Û.Ù

ψ : A → MÈ���dzÙ@×3dzÈ�Ì3×{é�Æ/dzÈvÙ@È�Ì�ÜgÕ${�×@ËNÌ3dzÙ3dzÈ�Ì3×"ÕqÖ!È�ÌWäYÖ!È�Ì

AUé

2

á�ÇÎÌ�Ö!È�Ô�È�Ìd{�È�Ð÷Ð.Ë�ï@å�~0Ñ.Õ$�ÒòvÛàý3ÈvÑYÈ�ÔGö�ùàË�Ñ*ÛYÍ

AÈ�ÇÎÐÃÜoÌ@È�dzÈ�Ì6ûaÛYÐ.Ñ(Ñ.ÕqÔ"ÇÎ×�ÓYÌ@Ëqý3È/{±×3ÇÎÉ�È�Ì�ÿ

Gö�ùÒ˱Ñ*ÛYÍ

FÈ��*dzÙ3×3dzÈ�Ì3×=òvÛ.Ù�ÕqÔ"Ô�È�ÐèÔ"ÇÎ×/È�ÇÎÐ.È�Ì��±ÛYÌoý3È/{±×3dzËNÐ

F � Aé�Æ/dzÈvÙ@È�z�ÇÎÏ�È�Ð.Ù@ï�å�ÕcÜa×�Ö9ÈvÑYÈ�ÛY×@È�×^~9Ñ.Õ$�

H(G, ·)È�ÇÎÐ

ûaÖYòvÏ�Í�é�ÑYÈ�ÌÓYÌ@Ëqý,È/{�×3ÇÎÉ�È�ÐGö�ùÒ˱Ñ*ÛYÍÎÐ�ÿ �5��!'D�!�O/%K��) å.ËNÔ�ËNͳËNÏ�dzÙ@ï@å.È�Ì=øYÛYÐ,{�×@ËNÌdzÙ3×{é

á�ÇÎÌ�ëlËNÍÎͳÈ�Ðg{±ÛYÌ3ò/ÑYÈ�ÐØí�È�Ì3ÖYÇÎÐ.Ñ*ÛYÐYÏ�Ù@å.ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�Û.Ùδ0 : H1(G,C = B/A)→ H0(G,A)

È��*ÓYÍÎÇÎòvÇÎ×Ö9ÈvÙ@ï�åYÌ@È�ÇÎÖ!È�ÐúûaÉ�Ï�Í�é�ß}Þ�áãâaÿdéTz�ÇÎÐ@�döu�9�T{±ÍÎÛ.ÙlÇÎÐ

H1(G,B/A)å�Õq×�Ñ*dzÈAø�ËNÌ3Ô

f =∑

g[g]⊗G (bg +A)~¢ë�ËNÖ!È�Ç

Ñ*dzÈbgÜgÕNÙ@×=ÕqÍÎͳÈ�ÁAÛYÍÎÍ?Ù3ÇÎÐ.Ñ-é�|AТ×@È�ÌÝÑYÈ�Ì�ðAÕqÐ.Ñ.ÕqÖYÖYÇÎͳÑ*ÛYÐYÏ6Ï�ÇÎÍÎ×^�

df =∑

g(1 − g−1)(bg + A) = 0

é���È�×3ò{È�ÐëyÇÎÌ

f :=∑

g[g] ⊗G bg ∈ H1(G,B)~�Ù@ËèÏ�ÇÎÍÎ×"ÜgËNÍÎÏ�ÍÎdzï@å

df ∈ AGé�Æ=ÕqÔ"ÇÎ×ØÈ�Ì3å�ÕqÍÎ×@È�Ð ëyÇÎÌ�Ñ*dzÈ6È��*ÓYÍÎÇÎòvÇÎ×@È£�ÈvÙ@ï�åYÌ@È�ÇÎÖYÛYÐYÏP�

δ0(∑

g

[g]⊗G (bg +A)) =∑

g

(1− g−1)bg + IGA

ùèÕqÐ äYÖ9È�Ì3ò{È�ÛYÏ�×IÙ3dzï�å ͳÈ�dzï@å±×^~�Ñ.Õ$��Ñ*dzÈvÙyÛYÐ�ÕqÖYå�ìqÐYÏ�ÇÎÏØdzÙ3×yÉ�ËNÐÊÑYÈ�ÌIÞAÛ.Ù3ë�ÕqåYÍ�ÑYÈ�Ìbgé��È�Ç

H < GÈ�ÇÎÐ.È7|Iб×@È�Ì3Ï�Ì3ÛYÓYÓ!È�é�á�ÇÎÌÑYÈ��.ÐYdzÈ�Ì@È�ÐÊÜaäYÌÈ�ÇÎÐ.È�Ð

Hö�ùÒ˱Ñ*ÛYÍ

IndGHA := Z[G]⊗Z[H] A.

ê�Ù3×H = 1

ÛYÐ.ÑXÈ�ÇÎÐ.È ÕqÖ9È�ͳÙ@ï@å.È�õ/Ì3ÛYÓYÓ9ÈG~�Ù�Ë�Ð.È�ÐYÐ.È�ÐèëyÇÎÌ

IndG1 XÈ�ÇÎÐ.È�Ð �5�98 U[6/�V��)/J+��� ùàË�Ñ*ÛYÍ�é9z�ÙAÏ�ÇÎÍÎ×

Ñ.ÕNÙ�E ��4�4H! CFE �f�P#T!.*9�5)dE �Hn(G, Ind

GHA) ∼= Hn(H,A)

ÜoäYÌIÕqÍÎͳÈn ≥ 0.

�±Ó9È�òvdzÈ�ÍÎÍ-ÜgËNÍÎÏ�×^~.Ñ.Õ$��Ñ*dzÈ�æ/ËNÔ�ËNͳËNÏ�dzÈ�Ï�Ì3ÛYÓYÓ!È�ÐHn(G, Ind

G1 X)ÜaäYÌ/ÕqÍÎͳÈ

n ≥ 1×3Ì3ÇÎÉ±Ç ÕqÍQÙ3ÇÎÐ.Ñ-é��È�dzÈ�Ð

φ : G → HÛYÐ.Ñ

θ : A → Bõ/Ì3ÛYÓYÓ9È�ÐYå.ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�È�Ð ÜoäYÌAÈ�ÇÎÐ.È�Ð

Gö�ùÒ˱Ñ*ÛYÍ

AÛYÐ.ÑàÈ�ÇÎÐ.È�Ð

Hö�ùàË�Ñ*ÛYÍ

Bé*Æ�ÕNÙY�?Õ�ÕqÌ

(φ, θ)å.È�ÇK��× C$��)/J=)l�^AG%'�R<�# ~�ëlÈ�ÐYÐÊÏ�ÇÎÍÎ×

θ(g.a) = φ(g).θ(a),

Ë�ÑYÈ�Ì�ì  ¢ÛYÇÎÉ�ÕqͳÈ�Т×�Ñ.ÕqòvÛ�~që�È�ÐYÐθÈ�ÇÎÐ

Gö�ùÒ˱Ñ*ÛYÍÎå.ËNÔ�ËNÌ3ÓYåYdzÙ3ÔWÛ.Ù�dzÙ3×^~që�ËNÖ!È�Ç

BÉ�Ç Õ

φÕqͳÙ

Gö�ùàË�Ñ*ÛYÍ.ÕqÛ*ÜaÏ�ÈdÜ�Õ$��×

ëyÇÎÌ@Ñ-é���9�=���y�0� A �B�¥��GF��"�"���7e3�5�¢C$�/)�J=)d�^A %;�=</#:�.Iv1Y![! )

(φ, θ) L �R�fE�OQ���¤�5�98 U[6/�V��)�J����5�0�����5�98��/UTJ=�KA��\Ë�! 4��5%'�V�C$E �IH7E 4HE 4HE )+*�#���I.4}���(φ, θ)n : Hn(G,A)→ Hn(H,B)8GUB)d<�#gË���I�J=%É�RA UT�BAS! U���8��������/JRJ+���

g

[g1| . . . |gn]⊗G ag 7→∑

g

[φ(g1)| . . . |φ(gn)]⊗H θ(ag),

L EBOQ��� g 8[!$I n Z ¼ U�*,��% g1, . . . , gn OQ�Q6F���R<�#��0� oA �CB�� Þ ä[� Æ/dzÈ�í�È�Ì3×3Ì�ìqÏ�ÍÎdzï@å,{�È�ÇÎ×"ÇÎÔ"ÓYÍÎÇÎòvdzÈ�Ì3×δ ◦ (φ, θ) = (φ, θ) ◦ δ

~�ëlÈvÙ3å�ÕqÍÎÖ��9�B{�È�ÍÎÐ(ÇÎÐr�9�T{�È�ÍÎÐôÛYÐ.Ñð/ìqÐ.ÑYÈ�ÌÇÎÐ ðAìqÐ.ÑYÈ�ÌIäYÖ!È�Ì,ÜoäYåYÌ3×ëlÈ�Ì@ÑYÈ�ÐQé

2

á�ÇÎÌë�ËNÍÎͳÈ�ÐÊÑ*dzÈvÙòvÛYÌÆ=È��.ÐYÇÎ×3dzËNÐÒÈ�ÇÎÐYÇÎÏ�È�Ìëydzï@å±×3ÇÎÏ�È�Ì/ÞAÖYÖYÇÎͳÑ*ÛYÐYÏ�È�ÐÊÉ�È�Ì3ë�È�Ð.ÑYÈ�Ð��•ø?äYÌ�È�ÇÎÐ.È3|Iб×@È�Ì3Ï�Ì3ÛYÓYÓ!È

H < GÛYÐ.Ñ�È�ÇÎÐ.È�Ð

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H ↪→ GÛYÐ.Ñ6ÑYÈ�ÌIê�ÑYÈ�Т×3ÇÎ×�ìq×=ÕqÛ*ÜAÑ*Ç³È ��E )d��I�J=)��y&FJ=�RE �

ïvËNÌ: Hn(H,A)→ Hn(G,A).

Page 12: ¼ ½¿¾0À5¼ Áÿ½ÅÄ - Université du Luxembourgmath.uni.lu/~wiese/other/diplom.pdf · 2007-03-08 · ë ËNÖ!È Ç ˙ ÕqÛ*Ü A+ ×3Ì3ÇÎÉ Ç ÕqÍ ÛYÐ.ÑØÕqÛ*Ü

•ø?äYÌÝÈ�ÇÎÐ.È�ÐhÁAËNÌ3Ô�ÕqÍÎ×@È�ÇÎͳÈ�Ì

U � Gå�ÕqÖ9È�ÐúëyÇÎÌÝÑ*Ç³È Ð�Õq×3äYÌ3ÍÎdzï�å.È�Ðr��Ì@Ëqý3È/{�×3dzËNÐ.È�Ð

φ : G → G/UÛYÐ.Ñ

θ : A→ A/IUA = AU~*ëlÈ�ͳï�å.ÈWÈ�ÇÎÐ6É�È�Ì3×3Ì�ìqÏ�ÍÎdzï�å.ÈvÙ³�0Õ�ÕqÌÖYÇÎͳÑYÈ�ÐQé�ÆAdzÈvÙ�ÈvÙyÍÎdzÈdÜoÈ�Ì3×Ñ*Ç³È ��EG���^À?!GJ=�RE �

ïvËNÇÎÐ*Ü: Hn(G,A)→ Hn(G/U,AU ).

•��È�ÇAëydzÈvÑYÈ�Ì3ÛYÔ

H < GÈ�ÇÎÐ.È�|Iб×@È�Ì3Ï�Ì3ÛYÓYÓ!ÈG~

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g ∈ GéláçÇÎÌ6È�Ì3å�ÕqÍÎ×@È�Ð È�ÇÎÐ

É�È�Ì3×3Ì�ìqÏ�ÍÎdzï@å.ÈvÙ��?Õ�ÕqÌyÑ*ÛYÌ@ï@åφg : H → gHg−1, h 7→ ghg−1 ÛYÐ.Ñ θg : A→ A, a 7→ g.a

é�Æ/dzÈvÙ�ÍÎdzÈdÜgÈ�Ì3×æ/ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�È�Ð

Hn(H,A)→ Hn(gHg−1, A).

ø?äYÌÈ�ÇÎÐ.È�Ð�Á/ËNÌ3Ô�ÕqÍÎ×@È�ÇÎͳÈ�ÌH �G

È�Ì3å�ìqÍÎ×Ô�ÕqÐÊÙ@Ë"È�ÇÎÐ.ÈGö�Þ�{�×3dzËNÐàÕqÛ*Ü�ÑYÈ�Ð æAËNÔ�ËNͳËNÏ�dzÈ�Ï�Ì3ÛYÓYÓ9È�ÐQé

Æ/È�ÌWÜgËNÍÎÏ�È�Ð.ÑYÈv�*Õq×3ò�ÍÎdzÈdÜgÈ�Ì3×�ÛQéjÕ*é?È�ÇÎÐ.ÈØÐ�Õq×3äYÌ3ÍÎdzï@å.ÈÚí�È�Ì3×3Ì�ìqÏ�ÍÎdzï@å,{�È�ÇÎ×ØÑYÈ�Ì�Ù@ËÒÑYÈ��.ÐYdzÈ�Ì3×@È�ÐóÞIÖYÖYÇÎͳÑ*ÛYÐYÏ�È�ÐÔ"ÇÎ×Ií�È�Ì3ÖYÇÎÐ.Ñ*ÛYÐYÏ�Ù3å.ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�È�ÐQé���9�=���ÉÅ��KJ�â:�^å,�������

φ : G→ H�����ÂW2).U�*[*,���P#TE 4HE 4HE )+*�#���I.4�U�IªUT�98

Θ���5�gË�UB�,&$JVE )ª8B��)

GZV>XE^8GUB%;�f�5�8G�R�

HZV>XE^8GUB%;�,-�I/E_8[!daj�.I?6nU�¡^�d8���4

GZ+>XE^8 UT%

A���5�0���

GZ+>XE^8 UT%Ã#:E 4HE 4HE )V*9#��yI�4�U�I

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−−−→ Hn+1(G,A′)

ûaÇÎÇ�ÿÚø?äYÌ�ý3ÈvÑYÈ�{±ÛYÌ3ò{È"È��.Õ${±×@È}��Èn ±Û.È�ÐYò0 → A → B → C → 0

dzÙ@×ÝÑ*dzÈ�ÜoËNÍÎÏ�È�Ð.ÑYÈ ��Èn ±Û.È�ÐYòàûoÑ*Ç³È %K! �BA���d]�!$&FJ+�����>=nU,���:6 ÿ�ÜoäYÌIÕqÍÎͳÈnÈ��.Õ${±×^�

Hn(G,A)→ Hn(G,B)→ Hn(G,C)δn

−−−→ Hn+1(G,A)→ Hn+1(G,B).

Æ/Ç³È 1YE$I.�=J=��C��5JV�GJ Ö9ÈvÑYÈ�ÛY×@È�×ëydzÈvÑYÈ�Ì3ÛYÔS~�Ñ.Õ$��Ñ*dzÈWõ/Ì3ÛYÓYÓ9È�ÐHn(G,A)

ÜoäYÌIÕqÍÎͳÈn < 0

×3Ì3ÇÎÉ�Ç ÕqÍ�Ù3ÇÎÐ.Ñ-éHn(G, ·)

dzÙ3× <cE��5��!'D�!BO�%K� ÖYòvÏ�Í�é¢ÑYÈ�Ì�ÇÎÐNý3È/{�×3ÇÎÉ�È�ÐGö�ùÒ˱Ñ*ÛYÍÎÐ�~±Ñ-é åQé�òvÛ�ý,ÈvÑYÈ�Ô

Gö�ùàË�Ñ*ÛYÍ

AÏ�ÇÎÖY×lÈvÙlÈ�ÇÎÐ.È�Ð

ÇÎÐNý3È/{�×3ÇÎÉ�È�ÐÊùàË�Ñ*ÛYÍIòvÛ.Ù�ÕqÔ"Ô�È�Ð Ô"ÇÎ×IÈ�ÇÎÐ.È�Ô ùÒËNÐ.ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�Û.Ù

A ↪→ Ié

á�ÇÎÌ=ëlËNÍÎͳÈ�Ð(ÕqÛ.ï@å åYdzÈ�Ì�ÑYÈ�Ð í�È�Ì3ÖYÇÎÐ.Ñ*ÛYÐYÏ�Ù3å.ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3ÔWÛ.Ùδ0 : H0(G,C) → H1(G,A)

ÜoäYÌ0 →

A → B → C → 0È���ÓYÍÎÇÎòvÇÎ×ÝÖ9ÈvÙ@ï�åYÌ@È�ÇÎÖ!È�ÐãûaÉ�Ï�Í�é�ß}Þ�á�âaÿdé���È�Ç

b + A ∈ (B/A)G = CGé!áçÇÎÌ�ÖYÇÎͳÑYÈ�ÐúÑYÈ�Ð�dö+�AËNòn�B{�ÍÎÛ.Ù

[g] 7→ g.b − bÇÎÐH1(G,B)

é�ÁIÛYÐÒdzÙ3×g.(b + A) − (b + A) = 0

Ð�ÕNï�åàí�ËNÌ�ÕqÛ.Ù@Ù�È�×3òvÛYÐYÏ�ÛYÐ.ÑÑ.Õqå.È�Ì�dzÙ3×

g.b− b ∈ Aé¢ùèÕqÐ}{�ÕqÐYÐ�ͳÈ�dzï�å¢×yÑ*dzÈ�|AÐ�ÕqÖYå�ìqÐYÏ�ÇÎÏG{�È�ÇÎ×�É�ËNÐ�ÑYÈ�Ì�ÞIÛ.Ù@ë�ÕqåYÍ!É�ËNÐ

bÐ�ÕNï�åYÓYÌ3ä*ÜoÈ�Ð�ÛYÐ.Ñ

È�Ì3å�ìqÍÎ×AÑ*dzÈÝÈ���ÓYÍÎÇÎòvÇÎ×@È�£�ÈvÙ�ï@åYÌ@È�ÇÎÖYÛYÐYÏδ0(b+A) := (g 7→ g.b− b).

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á�ÇÎÌIÑYÈ��.ÐYdzÈ�Ì@È�ÐÊÜaäYÌÈ�ÇÎÐ.È|AТ×@È�Ì3Ï�Ì3ÛYÓYÓ9ÈH < G

ÛYÐ.Ñ È�ÇÎÐ.È�ÐHö�ùÒ˱Ñ*ÛYÍ

AÑYÈ�Ð

Gö�ùÒ˱Ñ*ÛYÍ

CoindGH(A) := HomH(Z[G], A)

ÛYÐ.Ñ÷Ð.È�ÐYÐ.È�ÐãÇÎÔ �±Ó!È�òvÇ ÕqÍ Ü�ÕqÍÎÍyÉ�ËNÐH = 1

ÛYÐ.ÑóÈ�ÇÎÐ.È�Ì�ÕqÖ!È�ͳÙ@ï�å.È�Ðçõ/Ì3ÛYÓYÓ9ÈXÑYÈ�Ð ùÒ˱Ñ*ÛYÍ

CoindG1 (X)& E ���08 UG6n�R�/)�J é.ÞAÛ.ï@åÊÇÎÐ Ñ*dzÈvÙ@È�̳�±ÇÎ×3Û�Õq×3dzËNÐÊÏ�ÇÎÖY×AÈvÙÑ.ÕNÙXE �/4�4H!}CFE �X�P#T!�*9��)dE ~*ë�È�ͳï@å.ÈvÙ/Ñ*dzÈÝêuÙ�ËNÔ�ËNÌ3ÓYåYdzÈHn(G,CoindGH(A)) ∼= Hn(H,A)

ÜoäYÌAÕqÍÎͳÈn ≥ 0

Ö9ÈvÙ�ÕqÏ�×{é.êuÐ.Ù3Ö9ÈvÙ@ËNÐ.ÑYÈ�Ì@È�Ù3ÇÎÐ.Ñ6Ñ*dzÈWõ=Ì3ÛYÓYÓ!È�ÐHn(G,CoindG1 (X)

ÜaäYÌIÕqÍÎͳÈn ≥ 1

×3Ì3ÇÎÉ�Ç ÕqÍ�éÞIÐ�ÕqͳËNÏ6òvÛYÔ å.ËNÔ�ËNͳËNÏ�dzÙ@ï�å.È�ÐÃø�ÕqÍÎÍ�å.È�ÇK��È�ÜoäYÌ�È�ÇÎÐ.È�Ð

Gö�ùàË�Ñ*ÛYÍ

AÛYÐ.Ñ È�ÇÎÐ.È�Ð

Hö�ùÒ˱Ñ*ÛYÍ

BÈ�ÇÎÐ��0Õ�ÕqÌ

É�ËNÐèõ/Ì3ÛYÓYÓ!È�ÐYå.ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ@Ô�È�Ðφ : G→ H

ÛYÐ.Ñθ : B → A

C ��)�J=)d�^A %;�=</# ~�ë�È�ÐYÐ Ï�ÇÎÍÎ×θ(φ(g).b) = g.θ(b),

ÖYòvëÝé�ì  ¢ÛYÇÎÉ�ÕqͳÈ�Т×/Ñ.ÕqòvÛ�~YëlÈ�ÐYÐθÈ�ÇÎÐ

Gö�ùàË�Ñ*ÛYÍÎå.ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�Û.Ù/dzÙ3×{é

á�ÇÎÌIÈ�Ì3å�ÕqÍÎ×@È�ÐÊëydzÈ�ÇÎÔîå.ËNÔ�ËNͳËNÏ�dzÙ@ï�å.È�Ðàø�ÕqÍÎÍ�Ñ*dzÈ���9�=���Kó�� A �B�¥��GF��"�"���7e3�5�¢C$�/)�J=)d�^A %;�=</#:�.Iv1Y![! )

(φ, θ) L �R�fE�OQ���¤�5�98 U[6/�V��)�J����5�0�����5�98��/UTJ=�KA��\Ë�! 4��5%'�V�C$E �IH7E 4HE 4HE )+*�#���I.4}���(φ, θ)n : Hn(H,B)→ Hn(G,A)8GUB)d<�#gË���I�J=%É�RA UT�BAS! U���8������5�,#TE 4HEQA��/�0������E$& �/JRJ+���

f 7→ ([g1| . . . |gn] 7→ θ(f([φ(g1)| . . . |φ(gn)]))).

á�ÇÎÌë�ËNÍÎͳÈ�ÐàÕqÛ.ï@åÒÑ*dzÈvÙyòvÛYÌIÆ=È��.ÐYÇÎ×3dzËNÐÊÈ�ÇÎÐYÇÎÏ�È�ÌAëydzï@å¢×3ÇÎÏ�È�Ì=ÞAÖYÖYÇÎͳÑ*ÛYÐYÏ�È�Ð É�È�Ì3ëlÈ�Ð.ÑYÈ�Ð��•ø?äYÌ�È�ÇÎÐ.È3|Iб×@È�Ì3Ï�Ì3ÛYÓYÓ!È

H < GÛYÐ.Ñ�È�ÇÎÐ.È�Ð

Gö�ùàË�Ñ*ÛYÍ

AÈ�Ì3å�ÕqÍÎ×@È�Ð ëyÇÎÌ�ÕqÛ.Ù�ÑYÈ�̯z�ÇÎбÖ!È�×3×3ÛYÐYÏ

H ↪→ GÛYÐ.Ñ6ÑYÈ�ÌIê�ÑYÈ�Т×3ÇÎ×�ìq×=ÕqÛ*ÜAÑ*dzÈ�« �.I/J=).��&FJ=�RE �

Ì@ÈvÙ: Hn(G,A)→ Hn(H,A).

•ø?äYÌ=È�ÇÎÐ.È�ÐhÁ/ËNÌ3Ô�ÕqÍÎ×@È�ÇÎͳÈ�Ì

U � Gå�ÕqÖ9È�ÐÒëyÇÎÌÝÑ*Ç³È Ð�Õq×3äYÌ3ÍÎdzï�å.È}��Ì@Ëqý3È/{�×3dzËNÐ

φ : G → G/UÛYÐ.ÑÒÑ*dzÈz�ÇÎбÖ!È�×3×3ÛYÐYÏ

Θ : AU ↪→ A~*ëlÈ�ͳï�å.ÈWÈ�ÇÎÐ6É�È�Ì3×3Ì�ìqÏ�ÍÎdzï�å.ÈvÙ³�0Õ�ÕqÌÖYÇÎͳÑYÈ�ÐQé�ÆAdzÈvÙ�ÈvÙyÍÎdzÈdÜoÈ�Ì3×Ñ*dzÈ�p �^À2![J=�=E �ÇÎÐ*Ü

: Hn(G/U,AU )→ Hn(G,A).

•��È�ÇAëydzÈvÑYÈ�Ì3ÛYÔ

H < GÈ�ÇÎÐ.È�|Iб×@È�Ì3Ï�Ì3ÛYÓYÓ!ÈG~

AÈ�ÇÎÐ

Gö�ùàË�Ñ*ÛYÍ=ÛYÐ.Ñ

g ∈ GéláçÇÎÌ6È�Ì3å�ÕqÍÎ×@È�Ð È�ÇÎÐ

É�È�Ì3×3Ì�ìqÏ�ÍÎdzï@å.ÈvÙ_�0Õ�ÕqÌ Ñ*ÛYÌ@ï@åφg : gHg−1 → H, ghg−1 7→ h

ÛYÐ.Ñθg : A → A, a 7→ g.a

é?ÆAdzÈvÙ�ÈvÙÍÎdzÈdÜgÈ�Ì3×æAËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�È�Ð

Hn(gHg−1, A)→ Hn(H,A).ø?äYÌÈ�ÇÎÐ.È�Ð�Á/ËNÌ3Ô�ÕqÍÎ×@È�ÇÎͳÈ�ÌH �G

È�Ì3å�ìqÍÎ×Ô�ÕqÐÊÙ@Ë"È�ÇÎÐ.ÈGö�Þ�{�×3dzËNÐàÕqÛ*Ü�ÑYÈ�ÐS�IËNå.ËNÔ�ËNͳËNÏ�dzÈ�Ï�Ì3ÛYÓYÓ9È�ÐQé

Æ/È�ÌWÜgËNÍÎÏ�È�Ð.ÑYÈv�*Õq×3ò�ÍÎdzÈdÜgÈ�Ì3×�ÛQéjÕ*é?È�ÇÎÐ.ÈØÐ�Õq×3äYÌ3ÍÎdzï@å.ÈÚí�È�Ì3×3Ì�ìqÏ�ÍÎdzï@å,{�È�ÇÎ×ØÑYÈ�Ì�Ù@ËÒÑYÈ��.ÐYdzÈ�Ì3×@È�ÐóÞIÖYÖYÇÎͳÑ*ÛYÐYÏ�È�ÐÔ"ÇÎ×Ií�È�Ì3ÖYÇÎÐ.Ñ*ÛYÐYÏ�Ù3å.ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�È�ÐQé�z�ÌdzÙ3×Ñ.ÕNÙY{�ËNå.ËNÔ�ËNͳËNÏ�dzÙ@ï@å.È�ÞIÐ�ÕqͳËNÏ�ËNÐÒÑYÈvÙD�*Õq×3ò{ÈvÙ�ûl��éK��é©­�ÿdé���9�=���yô0�KJ�â:�^å,�������

φ : G→ H�����ÂW2).U�*[*,���P#TE 4HE 4HE )+*�#���I.4�U�IªUT�98

Θ���5�gË�UB�,&$JVE )ª8B��)

GZV>XE^8GUB%;�f�5�8G�R�

HZV>XE^8GUB%;�,-�I/E_8[!daj�.I?6nU�¡^�d8���4

GZ+>XE^8 UT%

A���5�0���

GZ+>XE^8 UT%Ã#:E 4HE 4HE )V*9#��yI�4�U�I

θA : ΘA→ AA �VO/JÇu8 o # o φ UB�08 θA I.�5�98HC$��)/J=)l�^AG%'�R<�# È -xInE 8[!Qa¢8[!$Ib�nE % AB���98�� ¿ �=!^AG)l! 4�4 CFE � G ZV>XE�8 UB%;�X&$E 4�4�UTJ=�V��)/J+Ê

ΘAθA−−−→ A

Θf

y

y

f

ΘA′ θA′−−−→ A′mDU/ab��)l8B��4 O/��%K8��

Θ�5�^¡^�.&FJ=�5C$�ZYYO=¡n�.&$J+��! U.�����^¡^�.&FJ=�5C$��!�O o ¿ ! �P�vA���%MJ+���0Ê

Page 16: ¼ ½¿¾0À5¼ Áÿ½ÅÄ - Université du Luxembourgmath.uni.lu/~wiese/other/diplom.pdf · 2007-03-08 · ë ËNÖ!È Ç ˙ ÕqÛ*Ü A+ ×3Ì3ÇÎÉ Ç ÕqÍ ÛYÐ.ÑØÕqÛ*Ü

ÇÆ! È ËtNT)H�������A��RA���OQ���0�}&�UT)Q6��}�Q][!$&FJ+�_���>=/UP���T60 → A → B → C → 0

C$E �GZ+>XE^8 UT%'�¥In���?! U/a¯�/)l8��/4

0 → Θ(A) → Θ(B)→ Θ(C) → 0�d][! &FJ o ¿ ! �P����I�J28[!$Is�nE %kA����98B� ¿ �R!^A )d! 4�4 !�OQ��%'I/<�#,��)vW2).U�*[*,����.NT)³! %5%É�

n ≥ 0&$EG4�4�U:JV!GJ=�5Cn- L E�Oc����8 �V� (φ,Θ)n

8 �V��H7E 4HE 4HE )+*�#���I.4}���(φ, θC)n

Oc6 L o (φ, θA)n! U�I

LD��4}��).&�UT�BA@Ç�N o N o\[ È I.�5�98�ÊHn(H,ΘC)

δn

−−−→ Hn+1(H,ΘA)

(φ,Θ)n

y

y(φ,Θ)n+1

Hn(G,C)δn

−−−→ Hn+1(G,A)ÇlO È �����78���) Ë�UB�,&$JVE )Θ�PUT�j6nU�In�GJy6/%;�=</#��d][! &FJ o ¿ ! �P�ÂA �VO/J��.I 6/U\¡n�Q8���4 G

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(φ,Θ)n : Hn(H,ΘA)→ Hn(G,A),I/Eg8[!daÇ+� È

(φ,Θ)08B��)�H7EG4_EG4_EG)V*�#B�yI�4�UBI

(φ, θA)0! UBI�LD�/4H�/)c&�UT�BA�Ç�N o N o\[ È ��I�J5-tUT�98ÇV�5� È �.NT)s¡^�d8���&FUB)Q6��H�Q][!$&FJ+�����>=/UP���T6

0 → A → B → C → 0CFEG�

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V ≤ UË$¦-È�Ð.ȪÁAËNÌ3Ô�ÕqÍÎ×@È�ÇÎͳÈ�ÌyÉ�ËNÐ

Gé�á�ÇÎÌ

ÑYÈ��.ÐYdzÈ�Ì@È�ÐÒÑ*Ç³È ?7EG).4 Ñ*ÛYÌ@ï@åNU/V :=

g∈U/V

g ∈ Z[U/V ].

áçÇÎÌAÈ�Ì3å�ÕqÍÎ×@È�ÐÒÑ*ÛYÌ@ï@åèÞIбëlÈ�Ð.Ñ*ÛYÐYÏÚÉ�ËNÐNU/V

È�ÇÎÐ.È�Ðàæ/ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�Û.ÙAV

NU/V−−−→ AU

~.ë�È�ͳï@å.È�Ì=È�ÇÎÐ.È�Ðæ/ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3ÔWÛ.Ù

(AV )U −→AUÇÎÐ.Ñ*ÛYòvdzÈ�Ì3×{é�Æ/dzÈvÙ@È�ÌIdzÙ3×yÔ"ÇÎ×ÑYÈ�ÌIê�ÑYÈ�Т×3ÇÎ×�ìq×/ÕqÛ*Ü

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ê�Ð í�È�Ì�ÕqÍÎÍÎÏ�È�Ô�È�ÇÎÐ.È�Ì3ÛYÐYÏÚÑ*dzÈvÙ@È�̳�±ÇÎ×3Û�Õq×3dzËNÐàÖ!È�×3Ì�ÕNï�å¢×@È�ÐàëyÇÎÌIÈ�ÇÎÐ.ÈÝø�ÕqÔ"ÇÎÍÎdzÈA(U)

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U � Gé/]¢ÈvÑYÈvÙ

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ÛYÐ.ÑA(V )

NU/V−−−→ A(U)ÖYÇÎͳÑYÈ�ÐÚÈ�ÇÎÐ�É�È�Ì3×3Ì�ìqÏ�ÍÎdzï�å.ÈvÙ?�?Õ�ÕqÌ{é*ùàÇÎ×3×@È�ͳÙlÆ=È��.ÐYÇÎ×3dzËNÐ6ÕqÛ*Ü�ÑYÈ�Ð ��È�×3×@È�Ð ûaɱÏ�Í�éT£�È�Ô�È�Ìd{±ÛYÐYÏÚûl��éK��驨�ÿ3ÿlÈ�Ì3å�ÕqÍÎ×@È�Ð

ëyÇÎÌAÈ�ÇÎÐ.È�ÐÊæAËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ@Ô�Û.ÙNn : Hn(G/U,A(V )U ) −→Hn(G/U,A(U))

ÜoäYÌIÕqÍÎͳÈn ∈ N

éÆAdzÈvÙIëlËNÍÎͳÈ�Ð ëyÇÎÌIÖ!È�бÛY×3ò{È�Ð�~.ÛYÔîÑ*dzÈ�¿ �5À?!GJ=�=EG� ëydzÈ=ÜgËNÍÎÏ�×òvÛ È�Ìd{±Í ìqÌ@È�Ð��

ÑYÈdÜ: Hn(G/V,A(V ))

¹+½d¸kµ_^−−−→ Hn(G/U,A(V )U )

Nn−−→ Hn(G/U,A(U))

íAÇ Õ ÑYÈ�ÌØÆ/È�Ï�Õq×3dzËNÐ�ÖYÇÎͳÑYÈ�ÐãÑ*dzÈàõ/Ì3ÛYÓYÓ9È�ÐHn(G/U,A(U))

È�ÇÎÐ ÓYÌ@Ëqý3È/{±×3ÇÎÉ�ÈvÙ\�B��Ù@×@È�ÔS~0ë�ÕNÙØÔ�ÕqÐ ÇÎÐÆ/ÇÎÔ�È�Ð.Ù3dzËNÐX² Ð�ÕNï@åYÌ@Èvï�åYÐ.È�×AÛYÐ.Ñ6Ó9È�ÌÆ/ÇÎÔ�È�Ð.Ù3dzËNÐ.Ù3É�È�Ì@Ù@ï�åYdzÈ�ÖYÛYÐYÏÚÕqÛ.Ù�ÑYÈ�åYТ×{éùÒÕqÐ Ù3×@È�ÍÎÍÎ×�ÜoÈvÙ3×^~0Ñ.Õ$�úÑ*dzÈàõ=Ì3ÛYÓYÓ!È�Ð

Hn(G/U,AU )ÜoäYÌ�È�ÇÎÐ.È�Ð

Gö�ùÒ˱Ñ*ÛYÍ

AÉ�Ç ÕàÑYÈ�ÌØê�Ð:Ï�Õq×3dzËNÐ È�ÇÎÐ

Ñ*ÇÎÌ@È/{�×@ÈvÙs�B�*Ù3×@È�ÔHÖYÇÎͳÑYÈ�Ð�~që�ÕNÙ0ëydzÈvÑYÈ�Ì3ÛYÔ�ÕqÛ*Ü�Æ/ÇÎÔ�È�Ð.Ù3dzËNÐH²Y{�Í ÕqÌ0dzÙ3×�ÛYÐ.Ñ Ñ*ÛYÌ@ï�å Æ/ÇÎÔ�È�Ð.Ù3dzËNÐ.Ù@É�È�Ì@Ù@ï@åYdzÈ�ÖYÛYÐYÏÕqÛ.Ù@Ï�ÈvÑYÈ�åYТ×Ië�È�Ì@ÑYÈ�Ð�{�ÕqÐYÐQéYáçÇÎÌÙ@È�×3ò{È�Ð

Hn(G,A) := lim−→Hn(G/U,AU ).

ûoÆ/dzÈvÙàÙ3ÇÎÐ.Ñ ÐYdzï@å±×àÑ*Ç³È É�ËNÌ3Ð.ÈúÑYÈ��.ÐYdzÈ�Ì3×@È�ÐHn(G,A)

émÿÊÞIÍÎ×@È�Ì3Ð�Õq×3ÇÎÉ È�Ì3Ï�È�Ö9È�Ð>Ù3dzï@åHÑ*dzÈèÐ.È�Û>ÑYÈ��.ÐYdzÈ�Ì3×@È�ÐHn(G,A)

ÕqͳÙx�IËNå.ËNÔ�ËNͳËNÏ�dzÈ=ÑYÈvÙx�AËNÔ"ÓYͳÈ��*ÈvÙ�ÑYÈ�ÌCn(G,A)

ÖYòvëÝéCn(G,A)

~¢ÇÎÐ.ÑYÈ�Ô Ô�ÕqÐØÑYÈ�ÐGö�ùàË�Ñ*ÛYÍ

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AÔ"ÇÎ×"ÑYÈ�Ì Ñ*dzÙd{�Ì@È�×@È�Ðj��ËNÓ!ËNͳËNÏ�dzÈ�É�È�Ì@Ù3dzÈ�å±×"ÛYÐ.ÑôòvÛ.Ù�ìq×3òvÍÎdzï@å Ñ*dzÈv�±×@È�×3ÇÎÏG{�È�ÇÎ×�ÕqÍÎͳÈ�Ì ÞAÖYÖYÇÎͳÑ*ÛYÐYÏ�È�Ð(ÜoËNÌ@ÑYÈ�Ì3×{éÞAÛ*Ü�Ñ*dzÈvÙ@È=áúÈ�dzÙ@È�É�È�Ì�ÕqÍÎÍÎÏ�È�Ô�È�ÇÎÐ.È�Ì3×=Ù3dzï@åÒÕqÛ.ï@åÊÑYÈ�Ì�£�È�Ï�Ì3ÇM¦ ÑYÈvÙY{�ËNÇÎÐ.Ñ*ÛYòvdzÈ�Ì3×@È�ÐÒùàË�Ñ*ÛYͳÙyòvÛ

CoindGH(A) := {f : G→ A | f(h.g) = h.f(g) ∀h ∈ H, fÙ3×@È�×3ÇÎÏ

}.

Æ/dzÈyÆ=È��.ÐYÇÎ×3dzËNÐ�ÑYÈvÙ0É�È�Ì3×3Ì�ìqÏ�ÍÎdzï�å.È�Ð �0Õ�ÕqÌ@ÈvÙ�ÛYÐ.Ñ ÑYÈ�Ì�Ô"ÇÎ×�Ù@È�ÇÎÐ.È�Ì�æAÇÎÍ ÜoÈ�È�Ì3å�ÕqÍÎ×@È�Ð.È�Ð�ÞAÖYÖYÇÎͳÑ*ÛYÐYÏ�È�Ð�äYÖ9È�Ì3×3Ì�ìqÏ�×Ù@dzï@åÊÑ*ÇÎÌ@È/{�×{é�xâ:�^�Gë.ñ �9�"�9�¥�9ß5��� Þ �f�T�/;¯ß Þ�÷ �"��vê\$�"ì�ìt�B�êuÐÚÑ*dzÈvÙ@È�Ô ÞIÖ.Ù@ï�åYÐYÇÎ×3×yÙ@È�Ç!Ñ*dzÈ�õ/Ì3ÛYÓYÓ9È

GÙ3×@È�×@Ù�È�Ð.Ñ*ÍÎdzï@å é¢áçÇÎÌ�È�Ì3ÇÎÐYÐ.È�Ì3ÐÊÕqÐØÖYòvëÝé�×3Ì@È�¦-È�Ð�ÜgËNÍÎÏ�È�Ð.ÑYÈ=Æ/È��.ÐYÇ ö

×3dzËNÐ.È�Ð��• NG :=

g∈G g ∈ Z[G]~

• IG := ker(ε : Z[G]→ Z) = (g − 1|g ∈ G) � Z[G]~

• NGA = ker(ANG−−→ A)

ÜoäYÌÈ�ÇÎÐ.È�ÐGö�ùÒ˱Ñ*ÛYÍ

���9�=���a`�� ö �B�¥�¥â��HÇÆ! È �����X���5�0��!�OQ��%'I/</#:�\W2)�U^*[*P� o ¿ ! �P�X�yI/J

Hom(Z[G], X)→ Z[G]⊗Z X, φ 7→∑

g∈G

g ⊗Z φ(g)

�����æW2Z p InE 4HE )+*�#���I.4�U�I.-�8��.IcI^��� ��4�& �.#B)l!�O�O���%K8 UT�BA`8 UT)d<�#h ⊗Z x 7→ (g 7→ xδg,h)

8��cbt�P�V��)/J7��I�J5-L E�OQ��� δg,h 8[!$Ib�7)dE �0�Q<�& ��)�Z ¿ ��%MJV!�In��� o ¿ ! 4��5J$�n! %5%K���}8G�R�dL³�RA )�� �D�D8��.I3���08 UG6n�R�/)�J+���}UB�98D& E ���08 UG6n�R�/)�J+���>XE^8 UT%ÃI³6nU�In! 4�4}��� oÇlO È �����

A = Z[G] ⊗Z X���5�����08 UG6n�R�/)�J+��)SOQ6 L o &$E �5�98 U[6/�V��)/J+��) G ZV>XE�8 UB% o ¿ !G�P�¥A���%MJ+��� AG = NGA

-NGA = IGA

-tUB�98AG = A/IGA

NG−−→ AG�yI/J3���5�

GZ p I/E 4HE )+*�#���I.4�UBIHÇ

GE.*,��)��V��)�J3J=)���C^�R! % È oÇu< È �����

C���5�æ�/�98:%;�=<�#§��)Q6���UGAGJ+��) �.)Q���V��)

GZV>�E^8 UT% o >��=J C∗

OQ�Q6F���=</#��0��� L �5)fI^���5�0���Ú8GU:! %É��� >XE^8 UT%Hom(C,Z) o ¿ �V�³mO^O��5%M8 UT�BA

C ⊗Z A→ Hom(C∗, A), c⊗ a 7→(

(f : C → Z) 7→ f(c) · a)

�yI/J3���5�GZ p I/E 4HE )+*�#���I.4�UBI oÇÆ8 È ËtNT)ª6 L ��� Z[G]

ZV>XE^8GUB%;�AUT�98

C- L E�OQ��� C ���98,%'�R<�#X��)Q6���UGAGJ?UT�98

Z[G]Z'�.)Q���xIn����-?�/)c#T!G%;J+��� L ��)�8 �R�

GZ p I/EG4_EG)V*�#B�R�

C ⊗Z[G] A = (C ⊗Z A)GNG−−→ (C ⊗Z A)G eUfhg−→ Hom(C∗, A)G = HomG(C∗, A).

A �CB�� Þ ä[� ø?äYÌ�ûgÕ�ÿyÛYÐ.Ñ(ûoï&ÿyäYÖ!È�Ì3ò{È�ÛYÏ�×=Ô�ÕqÐàÙ@dzï@åÒͳÈ�dzï@å¢×^~9Ñ.Õ$��ÈvÙAÙ3dzï�åÊ×�Õq×@Ù�ìNï�åYÍÎdzï@å ÛYÔGö�æAËNÔ�ËNÔ�ËNÌ,ö

ÓYåYdzÙ@Ô�È�Ð å�ÕqÐ.ÑYÈ�ÍÎ×^~.ÛYÐ.Ñ6Ì@Èvï�åYÐ.È�×AÑ*dzȣ�Ç ý3È/{±×3ÇÎÉ�ÇÎ×�ìq×Ð�ÕNï@åQé�ûoÑ�ÿlÜoËNÍÎÏ�×Ù�ËqÜoËNÌ3×IÕqÛ.Ù�ûgÕ�ÿ�ö�ûoï&ÿdé��ÛÃûaÖ�ÿ�Ù@È�Ç-òvÛYÐ�ìNï�å.Ù3×a ∈ AG

éYÆ�ÕA ZögÜoÌ@È�Ç-dzÙ3×^~:{[�NÐYÐ.È�Ð ëyÇÎÌIÈ�ÇÎÐ.ÑYÈ�ÛY×3ÇÎÏa =

s∈G

s⊗ xs

Ù�ï@åYÌ@È�ÇÎÖ9È�ÐÊÔ"ÇÎ×xs ∈ X

éYáúÈ�Ï�È�ÐàÑYÈ�ÌGö�ê�бÉNÕqÌ3Ç ÕqÐYòWÏ�ÇÎÍÎ×

a =∑

s∈G

gs⊗ xs =∑

s∈G

s⊗ xg−1s,

ë�ÈvÙ3å�ÕqÍÎÖàÑ*dzÈxsÕqÍÎͳÈ�Ï�ͳÈ�dzï�åàÙ3ÇÎÐ.Ñ-é���ËNÔ"ÇÎ×IdzÙ3×

a = NG(1⊗ xid)È�ÇÎÐSz�ͳÈ�Ô�È�Т×/É�ËNÐ

NGAé

Page 18: ¼ ½¿¾0À5¼ Áÿ½ÅÄ - Université du Luxembourgmath.uni.lu/~wiese/other/diplom.pdf · 2007-03-08 · ë ËNÖ!È Ç ˙ ÕqÛ*Ü A+ ×3Ì3ÇÎÉ Ç ÕqÍ ÛYÐ.ÑØÕqÛ*Ü

Æ/ÈvÙ@ëlÈ�ÇÎ×@È�Ì@È�ÐàÙ�È�Ç-ТÛYÐa ∈ NGA

Ï�È�Ï�È�Ö!È�Ð�~.Ñ-é åQé0 = NG a = NG

s∈G

s⊗ xs =∑

g∈G

g ⊗∑

s∈G

xs,

Ù�Ë"Ñ.Õ$� ∑

s∈G xs = 0ÜoËNÍÎÏ�×{é9��ËNÔ"ÇÎ×AÈ�Ì3Ï�ÇÎÖY×IÙ3dzï@å

a =∑

s∈G

s⊗ xs =∑

s∈G

s⊗ xs −∑

s∈G

1⊗ xs =∑

s∈G

(s− 1)(1⊗ xs) ∈ IGA.

Æ/dzÈÝê�Ù@ËNÔ�ËNÌ3ÓYåYdzÈ�òvëydzÙ�ï@å.È�ÐAGÛYÐ.Ñ

AGÜgËNÍÎÏ�×AÕqÛ.Ù

AG = A/IGA = A/NGANG−−→ NGA = AG.

2£�È�ò{È�dzï�åYÐ.ÈF•Ñ*dzÈ=ÜoÌ@È�dzÈ��±×�ÕqÐ.Ñ.ÕqÌ@Ñ.ÕqÛ:Ï���Ù3ÛYÐYÏ�É�ËNÐ

Z~.ÕqͳÙ@Ë

0→ Zε←− F0 → F1 → F2 → . . . .

áçÇÎÌIÙ@È�×3ò{È�ÐF−n := F ∗

n−1

ÜoäYÌn ≥ 1

ÛYÐ.ÑÊÈ�Ì3å�ÕqÍÎ×@È�ÐàÑ.ÕNÑ*ÛYÌ@ï@åÒÑ*dzÈ���Èn ¢Û.È�ÐYò· · · ← F−3 ← F−2 ← F−1

ε∗←− Z← 0,

ë�È�ͳï@å.È È��.Õ${�×�dzÙ@×^~?Ñ.ÕèÑ*dzÈF−nÈ�Ð.Ñ*ÍÎdzï�å È�Ì3ò{È�ÛYÏ�×@ÈÚÜaÌ@È�dzÈ

Z[G]ö�ùÒ˱Ñ*ÛYÍÎÐ Ù@ÇÎÐ.Ñ-é�áçÇÎÌ É�È�Ì@Ù@ï@åYÔ�È�ÍÎò{È�Ð�Ö!È�dzÑYÈ��Èn ±Û.È�ÐYò{È�ÐÊÛYÐ.Ñ6È�Ì3å�ÕqÍÎ×@È�ÐÊÜoËNÍÎÏ�È�Ð.ÑYÈvÙAÆ/Ç ÕqÏ�Ì�ÕqÔ"ÔS~�Ñ.ÕNÙIÕqÐ ý3ÈvÑYÈ�̳�±×@È�ÍÎͳÈÝÈ��.Õ${±×ydzÙ@×

←−−− F−3 ←−−− F−2 ←−−− F−1α

←−−− F0 ←−−− F1 ←−−− F2 ←−−−

ε∗x

y

ε

Z Zx

y

0 0,

ë�ËNÖ!È�ÇαÑ*dzÈ7�AËNÔ"Ó!Ë�Ù@ÇÎ×3dzËNÐ

ε∗ ◦ εÙ@È�Ç�éYÞAÛ.ÙY��È�ÇÎÍlûoÑ�ÿyÑYÈvÙY�QÈ�Ô"Ô�ÕNÙWûl��éK��é©°�ÿ�ͳÈvÙ@È�Ð ëyÇÎÌAÕqÖ�~YÑ.Õ$�"ÜoäYÌ

n ≥ 1

HomG(F−n, A) = HomG(F ∗n−1, A) ∼= Fn−1 ⊗Z[G] A

Ï�ÇÎÍÎ×{é�ÞIÛ,��È�Ì@ÑYÈ�Ôîå�ÕqÖ9È�Ð ëyÇÎÌIÑ*dzÈ�ê�Ù@ËNÔ�ËNÌ3ÓYåYdzÈZ⊗Z[G] A = AG

NG−−→ AG = HomG(Z, A).

ÞAТë�È�Ð.Ñ*ÛYÐYÏúÑYÈvÙWø.ÛYÐ,{±×@ËNÌ@ÙHomG(·, A)

È�Ì3Ï�ÇÎÖY× Ð¢ÛYÐ÷Ñ.ÕNÙWÜoËNÍÎÏ�È�Ð.ÑYÈ {�ËNÔ"Ô�ÛY×�Õq×3ÇÎÉ�È6Æ/Ç ÕqÏ�Ì�ÕqÔ"ÔS~0ÑYÈvÙ@Ù@È�ÐËNÖ9È�Ì@Ù3×@È��!È�ÇÎͳÈ�È�ÇÎÐS�IË {�ËNÔ"ÓYͳÈ�� dzÙ3×^�−−−→ F1 ⊗Z[G] A −−−→ F0 ⊗Z[G] A −−−→ HomG(F0, A) −−−→ HomG(F1, A) −−−→

y

x

AGNG−−−→ AG

y

x

0 0.

áçÇÎÌ�ÑYÈ��.ÐYdzÈ�Ì@È�Ð�Ñ*dzÈ}¼ !GJ+��ZR��E #TE 4HE %KEQA �V�RA )�U^*[*P���Hn ÜoäYÌ n ∈ Z

ÕqͳÙ�Ñ*dzÈ2�IËNå.ËNÔ�ËNͳËNÏ�dzÈIÑYÈ�Ì�ËNÖ9È�Ì@Ù3×@È�Ð �!È�ÇÎͳÈ�éÆ/dzÈ7�?Õq×@Èdö+�IËNå.ËNÔ�ËNͳËNÏ�dzÈ

Hn(G, ·)ÖYÇÎͳÑYÈ�×IÈ�ÇÎÐ.È�ÐS{�ËNå.ËNÔ�ËNͳËNÏ�dzÙ�ï@å.È�ÐèøYÛYÐ,{±×@ËNÌ{é�z�ÙÏ�ÇÎÍÎ×IÑYÈ�Ì

Page 19: ¼ ½¿¾0À5¼ Áÿ½ÅÄ - Université du Luxembourgmath.uni.lu/~wiese/other/diplom.pdf · 2007-03-08 · ë ËNÖ!È Ç ˙ ÕqÛ*Ü A+ ×3Ì3ÇÎÉ Ç ÕqÍ ÛYÐ.ÑØÕqÛ*Ü

���9�=���5ø��KJ�â:�^å,�

Hn(G,A) =

Hn(G,A) n ≥ 1

AG/NGA n = 0

NGA/IGA n = −1

H−n−1(G,A) n ≤ −2A �CB�� Þ ä[� Æ=È�ÐèÈ�Ì@Ù3×@È�ÐàÛYÐ.ÑàͳÈ�×3òv×@È�Ð ø�ÕqÍÎÍ0ÍÎdzÈvÙ3×/Ô�ÕqÐÒÑ*ÇÎÌ@È/{�×=ÕqÛ.Ù=ÑYÈ�Ô Æ/Ç ÕqÏ�Ì�ÕqÔ"Ô ÕqÖQé9ø?äYÌn = 0,−1È�Ì3Ï�ÇÎÖY×WÙ3dzï@åôÑ.ÕNÙz�Ì3Ï�È�ÖYÐYdzÙWÑ.ÕqÌ�ÕqÛ.Ù^~-Ñ.Õ$�àÑ*dzÈ_�0ÜoÈ�ÇÎÍ³È Ð�ÕNï�åôÛYб×@È�ÐôÙ3ÛYÌoý3È/{±×3ÇÎÉàÛYÐ.ÑúÑ*Ç³È Ð�ÕNï@åôËNÖ9È�Ð ÇÎÐNý3È/{±×3ÇÎÉ

Ù@ÇÎÐ.Ñ-é2z�ÙdzÙ3×IäYÖYÍÎdzï@å�~!Ñ*dzÈ7£�È�ò{È�dzï@åYТÛYÐYÏ

H0(G,A) := H−1(G,A) = NGA/IGAòvÛÊë�ìqåYͳÈ�ÐQé0�QÌ3ÇÎÉ�Ç ÕqͳÈ�Ì3ëlÈ�dzÙ@È

È�Ì3å�ÕqÍÎ×@È�ÐàëyÇÎÌIÑ.ÕqÐYÐ Ñ*dzÈ=ÜgËNÍÎÏ�È�Ð.ÑYÈÝÈ��.Õ${±×@È���Èn ±Û.È�ÐYò�ÕqÖ9È�ͳÙ@ï@å.È�Ì=õ/Ì3ÛYÓYÓ!È�Ð��0→ H0(G,A)→ H0(G,A)

·NG−−−→ H0(G,A)→ H0(G,A)→ 0.ùÊÇÎ×3×@È�ͳÙ6ÑYÈ�Ì6È��*ÓYÍÎÇÎòvÇÎ×@È�Ð�£�ÈvÙ@ï�åYÌ@È�ÇÎÖYÛYÐYÏ ÑYÈ�ÌÚí�È�Ì3ÖYÇÎÐ.Ñ*ÛYÐYÏ�Ù3å.ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�È�Ð Ù3×@È�ÍÎÍÎ×ØÔ�ÕqÐçÜgÈvÙ3×^~�Ñ.Õ$�NG · im(δ0) = 0

Ï�ÇÎÍÎ×^~�ÛYÐ.ÑèÑ.Õ$�δ0äYÖ!È�Ì

NGAÜ�Õ${�×@ËNÌ3dzÙ3dzÈ�Ì3×{é�ø?äYÌ=Ñ*dzÈ��?Õq×@Èdö+�IËNå.ËNÔ�ËNͳËNÏ�dzÈØÈ�Ì3Ï�È�Ö!È�ÐèÙ3dzï@å

Ñ.ÕqÌ�ÕqÛ.ÙAÑ*dzÈ�í�È�Ì3ÖYÇÎÐ.Ñ*ÛYÐYÏ�Ù3å.ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�È�Ðδ−2 : H−2(G,A)→ H−1(G,A)

ÛYÐ.Ñδ0 : H0(G,A)→ H1(G,A)

ÕqÛ*Ü0Ð�Õq×3äYÌ3ÍÎdzï@å.ÈWáúÈ�dzÙ@È�é.Æ/dzÈ�ÞAÖYÖYÇÎͳÑ*ÛYÐYÏδ−1 : NGC/IGC → AG/NGAëyÇÎÌ@ÑÃûaɱÏ�Í�éQß}Þ�á âaÿyÑ*ÛYÌ@ï@åÊùàÛYÍÎ×3ÇÎÓYÍÎÇK{NÕq×3dzËNÐàÔ"ÇÎ×

NGÇÎÐ.Ñ*ÛYòvdzÈ�Ì3× û

C = B/Aÿ��

δ−1 : (b+A) + IGC 7→ NG · b+NGA.ùèÕqÐ äYÖ9È�Ì3ò{È�ÛYÏ�×IÙ3dzï�å ëydzÈvÑYÈ�Ì3ÛYÔ Í³È�dzï�å¢×IÉ�ËNÐ ÑYÈ�ÌIáúËNåYͳÑYÈ��.ÐYdzÈ�Ì3×3å.È�ÇÎ×{é��È�ÇÎÍ�ûaÖ�ÿ�ÑYÈvÙb�QÈ�Ô"Ô�ÕNÙIûl��éK��é©°�ÿ�Ö9ÈvÙ�ÕqÏ�×0ТÛYÐ�~�Ñ.Õ$�/ÜaäYÌ�È�ÇÎÐ.È�Ð ÇÎÐ.Ñ*ÛYòvdzÈ�Ì3×@È�Ð�ÖYòvëÝé${�ËNÇÎÐ.Ñ*ÛYòvdzÈ�Ì3×@È�ÐGö�ùÒ˱Ñ*ÛYÍ

AÏ�ÇÎÍÎ×^�

H−1(G,A) = H0(G,A) = 0é�ÞIÛ.Ù�ÛYÐ.Ù@È�Ì@È�ÔÅáçdzÙ@Ù@È�ÐçäYÖ!È�ÌÚæAËNÔ�ËNͳËNÏ�dzÈèÛYÐ.Ѥ�IËNå.ËNÔ�ËNͳËNÏ�dzÈ

Ù�ï@åYÍÎdzÈ/��È�ÐÊëyÇÎÌHn(G,A) = 0

ÜoäYÌIÕqÍÎͳÈn ∈ Z

éÞIÛ*ÜoÏ�Ì3ÛYÐ.ÑÃÑYÈ�ÌWÐ�Õq×3äYÌ3ÍÎdzï@å.È�Ðóæ/ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�È�Ð

HomG(Z[G], A) � AÛYÐ.Ñ

A ↪→ Z[G] ⊗Z AdzÙ@×

ý3ÈvÑYÈ�ÌGö�ùàË�Ñ*ÛYÍ�Ù@Ë&ëlËNåYÍ-ø�Õ${�×@ËNÌ,ö�ÕqͳÙ�ÕqÛ.ï@å\|AТ×@È�Ì3Ô�Ë�Ñ*ÛYÍ9È�ÇÎÐ.ÈvÙlÇÎÐ.Ñ*ÛYòvdzÈ�Ì3×@È�ÐÚÖYòvëÝé�{�ËNÇÎÐ.Ñ*ÛYòvdzÈ�Ì3×@È�Ð6ùÒ˱Ñ*ÛYͳÙ�~

ÛYÐ.Ñ ëyÇÎÌIÑ*äYÌ,ÜoÈ�Ð Ñ*dzÈ�ùàÈ�×3å.Ë�ÑYÈÝÑYÈ�ÌÆ/ÇÎÔ�È�Ð.Ù3dzËNÐ.Ù3É�È�Ì@Ù@ï�åYdzÈ�ÖYÛYÐYÏÚÇÎÐ6Ö!È�dzÑYÈ�ÐÊðIdzï@å±×3ÛYÐYÏ�È�ÐÊÖ9È�ТÛY×3ò{È�ÐQéÞIÛ*ÜlÑ*dzÈvÙ�È�áúÈ�dzÙ@È�{[�NÐYÐ.È�ÐèëyÇÎÌ=ÛQé!Õ*é!Ñ*dzÈWðAÈvÙ3×3Ì3ÇK{�×3dzËNÐàÛYÐ.ÑÒÑ*dzÈ��IËNÌ@ÈvÙ3×3Ì3ÇK{�×3dzËNÐúÕqÛ*Ü�ÕqÍÎÍ³È ÆAÇÎÔ�È�Ð.Ù@dzËNÐ.È�Ð

n ∈ ZÕqÛ.Ù�ÑYÈ�åYÐ.È�ÐQé

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0 : H0(G,A) = AG → H0(G/U,AU ) = (AU )G/UÏ�È�Ì�ÕNÑYÈ/Ñ*dzÈIùàÛYÍÎ×3ÇÎÓYÍÎÇK{�Õq×3dzËNÐ�Ô"ÇÎ×

NUdzÙ3×{é�áçÇÎÌlÙ@ï�åYÍÎdzÈ/��È�Ð�Ñ.Õqå.È�Ì�Ñ*dzÈ�z¯�*dzÙ3×@È�ÐYò=ÑYÈ�Ì�ÇÎÐ.Ñ*ÛYòvdzÈ�Ì3×@È�Ð6ÞAÖYÖYÇÎͳÑ*ÛYÐYÏ

ÑYÈdÜ −1 : H−1(G,A)→ H−1(G/U,AU ).áçÇÎÌ=Ù@È�×3ò{È�ÐÒÑYÈdÜ −n−1 :=

ÑYÈdÜnÜaäYÌ

n > 0é0z�Ù=È�Ì3Ï�ÇÎÖY×/Ù3dzï�åÒÑ*dzÈWÏ�È�ëyäYÐ.Ù@ï�å¢×@È í�È�Ì3×3Ì�ìqÏ�ÍÎdzï@å,{�È�ÇÎ×�Ô"ÇÎ×Aí�È�Ì3ÖYÇÎÐ*ö

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0 →A→ B → C → 0

Ñ.ÕNÙÆ/Ç ÕqÏ�Ì�ÕqÔ"ÔNGC/IGC

δ−1−−−→·NG

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.

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AUT�98

B GZV>XE^8GUB%;� o�l ���P� A &$E #TE 4HE %KEQA ��I/<�#SJ=)���C^�R! %�UT�98 B 8 �5C^��I.�VOQ��%���I�J5-tE^8���)

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i = 1, . . . , n− 1A �5%MJ5-x8[! �P�X�yI/Jx! U,<�#�8 �V�7���>=/UP���T6

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−−−→ Hp+q+1(G,A ⊗B)&$E 4�4�UTJV!GJ=�5C o

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−−−→ Hp+q+1(G,A⊗B)&$E 4�4�UTJV!GJ=�5C oáúÈ�ÇÎ×@È�ÌIÏ�ÇÎÍÎ×�ûaÉ�Ï�Í�éQßkÁª�±á�â+~,��Ì@ËNÓ9Ë�Ù3ÇÎ×3dzËNÐÂ��é ®.éjþ"òvÛ.Ù�ÕqÔ"Ô�È�Ð Ô"ÇÎ×IðAÈ�Ô�ÕqÌd{ÚÕqÛ*Üs�!é,®�¨�ÿ�ÑYÈ�ÌyÜgËNÍÎÏ�È�Ð.ÑYÈ

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−−−→ Hp+q(G,M)& E 4�4�U:JV!GJ=�5Cn-38 o # o �.I�A �5%MJ(δα) ∪ β + (−1)p(α ∪ δβ) = 0��NT)

α ∈ Hp(G,Hom(A,M))UB�98

β ∈ Hq−1(G,C) oÆ=ÕNÙÇÎÔ �*Õq×3ò�É�È�Ì3ëlÈ�Ð.ÑYÈ�×@È�ülÛYÓYÓYÌ@Ë�Ñ*Û,{�×IÈ�Ì3Ï�ÇÎÖY×IÙ3dzï�åÊÕqÛ.ÙÑYÈ�Ô Ð.ËNÌ3Ô�ÕqͳÈ�ÐàÑ*ÛYÌ@ï�å6ÜoËNÍÎÏ�È�Ð.ÑYÈ�æAÇÎТ×@È�Ì@È�ÇÎÐ*ö

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−−−→ Hn(G,Hom(A,M) ⊗A)ev−−−→ Hn(G,M).

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−−−→ Hq(G/U, (A ⊗B)U )& E 4�4�U:JV!GJ=�5Cn-38 o # o �.I�A �5%MJ �����(8��5�

(x) ∪ y) = x ∪�5�^�

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ûaÇÎÇ�ÿA =

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É�ËNÌ3Ð.È6Ö9È�Ô�È�Ìd{�×^~0ÖYÇÎͳÑYÈ�ÐãÑ*dzÈHi(G/U,AU )

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ÛYÐ.Ñ6ÜaäYÌÈ�ÇÎÐ.È�Ð2ö+�AËNÌ�ÕqÐ.Ñ�~�Ñ.Õ$�

c(1, τ) = 1.b(τ)− b(τ) + b(1) = b(1)Ï�ÇÎÍÎ×{é

ê�Ô ø.ËNÍÎÏ�È�Ð.ÑYÈ�ÐôÙ�È�Çc(σ, τ)

È�ÇÎÐÒí�È�Ì3×3Ì@È�×@È�Ì�É�ËNÐγé�áçÇÎÌ=ëlÈ�Ì@ÑYÈ�РбÛYÐúÑYÈ�Ð ����)��n� %5%'UT�BA$I�4HE^8 UT%

C(γ)Ù�Ë

ÑYÈ��.ÐYdzÈ�Ì@È�Ð�~�Ñ.Õ$��Ñ.ÕNÙY£�ÇÎͳÑ6É�ËNÐc(σ, τ)

ÇÎÐH2(G,C(γ))

ÁIÛYÍÎÍQdzÙ@×^�C(γ) := C ⊕B := C ⊕

16=σ∈G

BdzÙ@×�ÕqͳÙ@Ë�Ñ*dzÈÜoÌ@È�dzÈAÕqÖ9È�ͳÙ@ï�å.È=õ/Ì3ÛYÓYÓ9È=ÕqÛ*Ü��B�±ÔWÖ!ËNͳÈ�Ð

bσÜaäYÌ

σ 6= 1é±Æ/dzÈ

Gö�Þ�{�×3dzËNÐ6ÕqÛ*Ü

C(γ)ÑYÈ��.ÐYdzÈ�Ì@È�Ð

ëyÇÎÌIÛYÐ.Ù@È�Ì@È�Ô ��dzÈ�Í�È�б×@Ù3ÓYÌ@Èvï@å.È�Ð.ÑÒÑ*ÛYÌ@ï@åσbτ := bστ − bσ + c(σ, τ),

ë�ËNÖ!È�Çb1 = c(1, 1)

Ù@È�Ç�éz�Ù�dzÙ3×�Ð�Õq×3äYÌ3ÍÎdzï@å�òvÛ�äYÖ!È�Ì3ÓYÌ3ä*ÜgÈ�Ð�~¢Ñ.Õ$�WÈvÙ�Ù3dzï�å�ÛYÔ�È�ÇÎÐ.ÈGö�ÞD{�×3dzËNÐ�å�ÕqÐ.ÑYÈ�ÍÎ×{é±Æ�ÕNÙ�Ù@dzÈ�å¢×�Ô�ÕqÐ�ëydzÈyÜgËNÍÎÏ�×^�

1.bτ = bτ − b1 + c(1, τ) = bτ

ρ(σbτ ) = ρ(bστ − bσ + c(σ, τ))

= bρστ − bρ + bρσ + bρ + c(ρ, στ) − c(ρ, σ) + ρc(σ, τ)

= bρστ − bρσ + c(ρσ, τ) = (ρσ)bτ

Æ�ÕqÖ9È�ÇQëyÛYÌ@ÑYÈÝÑ*dzÈ�AËNòn�B{�È�ÍÎÌ@È�Í Õq×3dzËNÐÒÉ�ËNÐÊËNÖ9È�Ð Ö!È�бÛY×3òv×{éz�Ù�dzÙ@×}{�Í ÕqÌ^~�Ñ.Õ$�ÃÑYÈ�Ì�ùàË�Ñ*ÛYÍC(γ)

ëlËNåYͳÑYÈ��.ÐYdzÈ�Ì3×6dzÙ@×^~�Ñ.ÕúÈ�ÌØÖYdzÙ�ÕqÛ*ÜÝê�Ù@ËNÔ�ËNÌ3ÓYåYdzÈÊÐYdzï�å¢×�É�ËNÐ�ÑYÈ�ÌÞAÛ.Ù3ë�ÕqåYÍÑYÈvÙ í�È�Ì3×3Ì@È�×@È�Ì@Ù

c(σ, τ)É�ËNÐ

γÕqÖYå�ìqÐYÏ�×{é�Æ=È�ÐYÐ Ù@È�Ç

d(σ, τ) = σc(τ) − c(στ) + c(σ)È�ÇÎÐ

�AËNÌ�ÕqÐ.Ñ É�ËNÐCÛYÐ.Ñ Ù�È�Ç

C(γ)ÑYÈ�ÌyòvÛ

c(σ, τ) + d(σ, τ)Ï�È�ÖYÇÎͳÑYÈ�×@È��!È�Ì,ÜgìqÍÎÍÎÛYÐYÏ�Ù@Ô�˱Ñ*ÛYÍ�ÕqÛ*Üs�B��Ô�Ö9ËNͳÈ�Ð

bσ~

Ñ.ÕqÐYÐÊëyÇÎÌ@Ñ Ñ*dzÈÝê�Ù@ËNÔ�ËNÌ3ÓYåYdzÈÝÑ*ÛYÌ@ï@åbσ 7→ bσ − c(σ)

Ï�È�Ï�È�Ö9È�ÐQéz�ÙIÈ�Ì3Ï�ÇÎÖY×IÙ@dzï@åÊÙ@ËqÜgËNÌ3×ÑYÈ�ÌyÜgËNÍÎÏ�È�Ð.ÑYÈGö+z�ÓYÇÎÔ�ËNÌ3ÓYåYdzÙ3ÔWÛ.Ù

φ : C(γ) = C ⊕B → IG, c 7→ 0, bσ 7→ σ − 1.

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æAdzÈ�Ì�ÕqÛ.ÙIÈ�Ì3å�ÕqÍÎ×@È�ÐÊëyÇÎÌIÑ*dzÈÝÈ��.Õ${±×@È���Èn ±Û.È�ÐYò0 −−−→ C −−−→ C(γ)

φ−−−→ IG −−−→ 0,

ë�ËNÖ!È�Ç9Ñ*dzÈIÈ�Ì@Ù@×@ÈIÞAÖYÖYÇÎͳÑ*ÛYÐYÏ�Ñ*dzÈAÐ�Õq×3äYÌ3ÍÎdzï@å.Ȫz�ÇÎбÖ!È�×3×3ÛYÐYÏC ↪→ C⊕B

dzÙ3×{é�áúÈ�ÇÎ×@È�Ì�å�ÕqÖ9È�Ð�ëyÇÎÌ�Ñ*dzÈIÈ��.Õ${±×@È��Èn ±Û.È�ÐYò0 −−−→ IG −−−→ Z[G]

ε−−−→ Z −−−→ 0.

ÆAdzÈvÙIÏ�ÇÎÖY×ÑYÈ�Ð6ÜgËNÍÎÏ�È�Ð.ÑYÈ�ÐÊæAËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ@Ô�Û.ÙAÜaäYÌ0ý3ÈvÑYÈ|AТ×@È�Ì3Ï�Ì3ÛYÓYÓ9ÈH ≤ G

ÛYÐ.Ñ�ý,ÈvÑYÈvÙn ∈ Z

�δ2 : Hn(H,Z)

δ1−−−→ Hn+1(H, IG)δ2−−−→ Hn+2(H,C).z�ÌdzÙ3×ÑYÈ�ÌIæ/ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�Û.Ù�~�ÑYÈ�Ð ëyÇÎÌIÙ@Û.ï@å.È�Ð��

���9�=���y��ô��KJ�â:�^å,� ¿ ��)�EBO��MAB�®H7E 4HE 4HE )+*�#���I.4�UBIδ2 : Hn(H,Z)→ Hn+2(H,C)

��NT)����5�0���GZV>XE^8GUB%

CUT�98\���5�

γ ∈ H2(G,C)��I�J�A��RA��^Oc���r8 UT)d<�#

β 7→)Q�.I G

H(γ) ∪ β =: γH ∪ β.Ë"��)��0��)I����98g�z=/UT�5CF! %É����J+ÊÇ+� È

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A �CB�� Þ ä[� ��ÛYÐ�ìNï@å.Ù@×ØÌ@Èvï�åYÐ.È�ÐçëyÇÎÌØÛYТ×@È�ÌÚí�È�Ì3ëlÈ�Ð.Ñ*ÛYÐYÏóÑYÈ�Ì�È���ÓYÍÎÇÎòvÇÎ×@È�Ð�£�ÈvÙ@ï@åYÌ@È�ÇÎÖYÛYÐYÏóÑYÈ�Ì�í�È�Ì3ÖYÇÎÐ*öÑ*ÛYÐYÏ�Ù@å.ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�È�ÐèÐ�ÕNï@å�~�Ñ.Õ$�

δ2(1) = γHdzÙ@×yÜaäYÌ

1 = 1 + |H|Z ∈ H0(H,Z)��z�ÇÎÐf|AÌ3ÖYÇÎÍ³Ñ ÑYÈ�Ì

1ÇÎÐÚÑYÈ�ÐS²cöu�9�B{�È�ÍÎÐÚÖYòvÏ�Í�éZ[G]dzÙ3×lÏ�È�Ï�È�Ö9È�ÐÚÑ*ÛYÌ@ï@å Ñ*dzÈ

1Ñ�ÑYÈvÙ@Ù@È�Ð\£�ÇÎͳÑ�ÛYТ×@È�ÌyÑYÈ�Ô ð/ÕqÐ.Ñ*å.ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�Û.Ù

È�Ì3Ï�ÇÎÖY×lÑYÈ�Ð1ö+�AËNòn�B{�È�Í�ÖYòvÏ�Í�é

IGÕqͳÙ

δ1(1)(σ) = σ−1é z�ÇÎÐH|AÌ3ÖYÇÎͳÑ"Ñ.ÕvÉ�ËNÐ�ÇÎÐ�ÑYÈ�Ð

1öu�9�T{�È�ÍÎÐ"ÖYòvÏ�Í�é

C(γ)dzÙ3×

x(σ) = bσécÆ=ÈvÙ@Ù@È�Ð�£�ÇÎͳÑ�ÛYТ×@È�Ì0ÑYÈ�Ô5ð/ÕqÐ.Ñ*å.ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�Û.Ù?dzÙ3×

δ2◦δ1(1)(σ, τ) = σbτ−bσ,τ+bσ = c(σ, τ)é

ÆAdzÈ�ÞIÛ.Ù@Ù�ÕqÏ�Èδ2 = γH∪

ÜgËNÍÎÏ�×ТÛYÐÒÕqÛ.ÙD�*Õq×3òØûl��éK��éK�^¨�ÿdéz�Ù Ï�ÇÎÍÎ×H−1(H,Z) = H1(H,Z) = 0

~Ñ.ÕóÈvÙàÕqÛ*ÜaÏ�Ì3ÛYÐ.Ñ+ÑYÈ�Ìfz�Ð.Ñ*ÍÎdzï@å,{�È�ÇÎ×ÒÉ�ËNÐH{�È�ÇÎÐ.È ÐYdzï@å±×,ö

×3Ì3ÇÎÉ�Ç ÕqͳÈ�Ð>æ/ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�È�ÐH → Z

Ï�ÇÎÖY×^~�ÛYÐ.Ñ+ÕqÛ,��È�Ì@ÑYÈ�ÔNHZ = 0

dzÙ3×6ë�È�Ï�È�Ð>ÑYÈ�Ì êuÐNý,È/{�×3ÇÎɱÇÎ×�ìq×ÑYÈ�ÌlùÊÛYÍÎ×3ÇÎÓYÍÎÇK{�Õq×3dzËNÐ�Ô"ÇÎ×

|H|é�Æ�Õ

Z[G]{�ËNå.ËNÔ�ËNͳËNÏ�dzÙ@ï�åØ×3Ì3ÇÎÉ�Ç ÕqÍ.dzÙ3×^~qÜgËNÍÎÏ�×

Hn(H,Z) ∼= Hn+1(H, IG)ÛYÐ.Ñ

åYdzÈ�Ì�ÕqÛ.ÙH0(H, IG) = H2(H, IG) = 0

é9��ËNÔ"ÇÎ×AÈ�Ì3å�ÕqÍÎ×@È�ÐÒëyÇÎÌIÑYÈ�Ð ÜgËNÍÎÏ�È�Ð.ÑYÈ�Ðf��È�ÇÎÍ?È�ÇÎÐ.È�ÌIÍ ÕqÐYÏ�È�ÐÒÈ��.Õ${±×@È�Ð��Èn ±Û.È�ÐYò0→ H1(H,C)→ H1(H,C(γ))→ H1(H, IG)→ H2(H,C)→ H2(H,C(γ))→ 0, (∗)

ÑYÈ�ÐÊëyÇÎÌòvÛYÔ¬£�È�ëlÈ�dzÙAÑYÈ�Ì�²? ¢ÛYÇÎÉ�ÕqͳÈ�ÐYò{È�ÐÊå.È�Ì�ÕqÐYòvdzÈ�å.È�ÐQé(i) ⇒ (ii)

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H1(H,C) = 0dzÙ3×^~YÛYÐ.ÑàÑ.Õ$�

δ2ÛYÐ.Ñ�Ù@ËNÔ"ÇÎ×�ÕqÛ.ï@åδ2 : H0(H,Z)→ H2(H,C)

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Hi(G,A∗) ∼= H−i−1(G,A)∗.

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ÛYÐ.Ñ0→ Hom(A, IG)→ Hom(A,Z[G])→ Hom(A,Z)→ 0.æAdzÈ�Ì�ÕqÛ.ÙÚÙ@ï@åYÍÎdzÈ/��È�Ð ëyÇÎÌ�ТÛYÐ ÛYТ×@È�ÌÚí�È�Ì3ë�È�Ð.Ñ*ÛYÐYÏóÉ�ËNЧ�*Õq×3òóûl��éK��éK�G�&ÿØÑ*dzÈf�AËNÔ"Ô�ÛY×�Õq×3ÇÎÉ�ÇÎ×�ìq×6ÜgËNÍÎÏ�È�Ð.ÑYÈ�Ð

Æ/Ç ÕqÏ�Ì�ÕqÔ"Ô�Ù^�Hi−2(G,Hom(A,Z)) × H2−i(G,A)

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δ

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δ

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éB��ÛYÐ�ìNï@å.Ù@×�å�ÕqÖ!È�Ð"ëyÇÎÌ�Ñ.ÕNÙt{�ËNÔ"Ô�ÛY×�Õq×3ÇÎÉ�ÈIÆ/Ç ÕqÏ�Ì�ÕqÔ"Ô ÕqÛ.Ùs�*Õq×3òWûl��éK��éK�^­�ÿH2−i(G/U,A) × Hi(G/U,Hom(A,CU ))

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U-���NT) 8 �V�

U/V!�OQ��%'I/<�#Â�yI/J5-YUB�98X8���4ÃÀP��)^Od! �98@8��/)�?7E )�4}���BA )�U�*[*,���ÎOQ6cA % o

U o mDU�ab��)d8���4i��I�JT¡^�d8��ª�5�CU

����J5#:! %;J+�/�0��Y�OQ��)RA )�U^*[*P�_��������)�?�E )�4H�/�BA )�U^*[*P�³In�/%KO.I/Jx���5�0��?�E ).4}���TA ).U�*9Z*P� o��� �!�hÄ���ÅK6 �Æ#����8$'�z#���$S&��-�Ç��#!3 64$S¡%$z�S�G#�$'©��È�Ç

GkÈ�ÇÎÐ.ÈÊÓYÌ@˱È�Ð.Ñ*ÍÎdzï�å.È õ=Ì3ÛYÓYÓ!È�élá�ÇÎÌØÇÎÐ.Ñ*ÇÎòvdzÈ�Ì@È�ÐçÑ*dzÈèÕqÖYÏ�ÈvÙ@ï�åYͳË�Ù@Ù@È�Ð.È�Ð�ÁAËNÌ3Ô�ÕqÍÎ×@È�ÇÎͳÈ�ÌÚÉ�ËNÐ

GkÔ"ÇÎ×

ÜgËNÌ3Ô�ÕqͳÈ�ÐH���NÌ3Ó9È�Ì3ÐGK�Gk

ÛYÐ.Ñ Ù@ï@åYÌ@È�ÇÎÖ9È�Ð ÜoäYÌ?È�ÇÎÐ.È�òvëlÈ�Ç*ÜgËNÌ3Ô�ÕqͳÈ2���NÌ3Ó9È�ÌL|kÛYÐ.Ñ

K|kÔ"ÇÎ×

GL ⊆ GKÜoäYÌWÑYÈ�ÐÉÈAÛ.ËN×3dzÈ�Т×@È�ÐG(L|K) := GK/GL

é�á�ÇÎÌWÐ.È�ÐYÐ.È�Ð(Ñ*dzÈ�ÜoËNÌ3Ô�ÕqͳÈÒûaÏ¢ÕqͳËNdzÙ@Ù@ï@å.È&ÿ�z�Ì3ë�È�ÇÎ×@È�Ì3ÛYÐYÏL|KÕqÖ9È�ͳÙ@ï�å ÖYòvëÝé.È�ÇÎÐ.È

pö+z�Ì3ë�È�ÇÎ×@È�Ì3ÛYÐYÏP~.Ü�ÕqÍÎͳÙ

G(L|K)È�ÇÎÐ.È�ÕqÖ9È�ͳÙ@ï�å.È�ÖYòvëÝé.È�ÇÎÐ.È7��Ì@Ëqö

pö�õ/Ì3ÛYÓYÓ!ÈÝdzÙ3×{é

á�ÇÎÌÖ9È�×3Ì�ÕNï@å±×@È�ÐàÑ*dzÈ=ÜgËNÍÎÏ�È�Ð.ÑYÈ��±ÇÎ×3Û�Õq×3dzËNÐ��• (Gk, C)

Ù@È�ÇQÈ�ÇÎÐ.È��=Í ÕNÙ@Ù@È�Ð*ÜgËNÌ3Ô�Õq×3dzËNÐQé•z�Ù�È��*dzÙ3×3dzÈ�Ì@ÈÈ�ÇÎÐØͳÈ�É�È�Í öV{�ËNÔ"Ó�Õ${±×@È�Ì

Gkö�ùÒ˱Ñ*ÛYÍ

C0 ÛYÐ.Ñ�È�ÇÎÐ {�ËNå.ËNÔ�ËNͳËNÏ�dzÙ@ï@å�×3Ì3ÇÎÉ�Ç ÕqͳÈ�Ì Gk ö�ùàË�Ñ*ÛYÍ X ~Ù@Ë"Ñ.Õ$��Ñ*dzÈ���Èn ±Û.È�ÐYò1 −→C0 −→C −→X −→ 1

Ô"ÇÎ×Ù3×@È�×3ÇÎÏ�È�ÐÊæ/ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�È�ÐÒÈ��YÕ${�×dzÙ3×{é

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•ø?äYÌ0ý,ÈvÑYÈ�È�Ð.Ñ*ÍÎdzï�å.È�ÜgËNÌ3Ô�ÕqͳÈ7���NÌ3Ó9È�Ì@È�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ

K|kÙ@È�dzÈ�ÐÊÑ*dzÈ

G(K|k)ö�ùÒË�Ñ*ÛYÍÎÐ

CK := CGK , IK , EK , ClKÏ�È�Ï�È�Ö9È�Ð�~YëlÈ�ͳï�å.È�È�ÇÎÐ.ÈÝÈ��.Õ${±×@È���Èn ±Û.È�ÐYòÝÖYÇÎͳÑYÈ�Ð��

1→ EK → IK → CK → ClK → 1.ûl�&ÿ

•ø?äYÌ�È�Ð.Ñ*ÍÎdzï@å.È ÜoËNÌ3Ô�ÕqͳÈf���NÌ3Ó9È�Ì@È�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ�È�Ð

K ′|KÙ@È�dzÈ�Ð�ÞIÖYÖYÇÎͳÑ*ÛYÐYÏ�È�Ð

ιÏ�È�Ï�È�Ö9È�Ð�~�ëlËNÖ9È�Ç

ι :CK ↪→ CK′

Ñ*dzÈ7z�ÇÎТÖ9È�×3×3ÛYÐYÏØÙ@È�ÇV~YÙ@Ë"Ñ.Õ$�ØÑ.ÕNÙÆ/Ç ÕqÏ�Ì�ÕqÔ"Ô �����=�9�KÅ9Å��1 −−−→ EK′ −−−→ IK′ −−−→ CK′ −−−→ ClK′ −−−→ 1

x

ι

x

ι

x

ι

x

ι

1 −−−→ EK −−−→ IK −−−→ CK −−−→ ClK −−−→ 1x

NK′|K

x

NK′|K

x

NK′|K

x

NK′|K

1 −−−→ EK′ −−−→ IK′ −−−→ CK′ −−−→ ClK′ −−−→ 1{�ËNÔ"Ô�ÛY×�Õq×3ÇÎÉ6dzÙ@×{éYÞIÛ,��È�Ì@ÑYÈ�ÔîÖYÇÎͳÑYÈ�ÐàÑ*dzÈÝùàË�Ñ*ÛYÍÎÐ É�Ç ÕιÑ*ÇÎÌ@È/{�×@È��B��Ù@×@È�Ô�È�éYá�ÇÎÌÙ�È�×3ò{È�Ð��

I = lim−→K|k ´Vµ/¶/·WÊ IK, E = lim−→

K|k ´Vµ/¶/·WÊ EK , E(p) = lim−→K|k ´Vµ/¶�·WÊ p ËaÌ�«aÍ�Ê EK , C(p) = lim−→

K|k ´Vµ/¶�·WÊ p ËaÌ�«aÍ�Ê CK .

• IKå�Õq×yÏ�ÛY×@È�ÐÒõ�ÕqͳËNdzÙ,ö�ÞAÖ.Ù3×3dzÈ�ÏP~.Ñ-éYåQé

IG(K′|K)K′ = IK

éÞAÛ,��È�Ì@ÑYÈ�ÔîÏ�ÇÎÍÎ×

H1(G(K ′|K), IK′) = 1ÜoäYÌ0ý,ÈvÑYÈ�È�Ð.Ñ*ÍÎdzï@å.È�ÜgËNÌ3Ô�ÕqͳÈ7z�Ì3ë�È�ÇÎ×@È�Ì3ÛYÐYÏ

K ′|Ké

•ø?äYÌ�ý3ÈvÑYÈ=È�Ð.Ñ*ÍÎdzï@å.È=ÜgËNÌ3Ô�ÕqͳÈ���NÌ3Ó9È�Ì@È�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ

K|kdzÙ@×ClKÈ�Ð.Ñ*ÍÎdzï�å�~YÛYÐ.Ñ6ÑYÈ�ÌyðIÈ�òvÇÎÓYÌ@ËNòvÇÎ×�ìq×@Ù3å.Ëqö

Ô�ËNÔ�ËNÌ3ÓYåYdzÙ3ÔWÛ.ÙIÇÎÐ.Ñ*ÛYòvdzÈ�Ì3×IÈ�ÇÎÐ.È�ÐàêuÙ@ËNÔ�ËNÌ3ÓYåYdzÙ@Ô�Û.ÙClK ∼= CK/NL|KCL ∼= G(L|K) = GabK ,

ëlËNÖ9È�Ç�ÑYÈ�Ì�ÜgËNÌ3Ô�ÕqͳÈ7���NÌ3Ó9È�ÌLÑ*ÛYÌ@ï@å

GL = [GK , GK ]ÜgÈvÙ3×3Ï�È�ͳÈ�Ï�×dzÙ@×{é

•ûgÕ�ÿ

Hi(G(K|k), IK) = 1ÜoäYÌIÕqÍÎͳÈÝÈ�Ð.Ñ*ÍÎdzï@å.È�Ð

K|kÛYÐ.ÑàÕqÍÎͳÈ

i ∈ Zé

ûaÖ�ÿHi(G(K|k), IK) = 1

ÜoäYÌIÕqÍÎͳÈÝÈ�Ð.Ñ*ÍÎdzï@å.È�Ðpö+z�Ì3ë�È�ÇÎ×@È�Ì3ÛYÐYÏ�È�Ð

K|kÛYÐ.ÑÊÕqÍÎͳÈ

i ∈ Zé

•ûgÕ�ÿ�íAÇ Õ

ιÏ�ÇÎÍÎ×^�

lim−→K|k ´+µ�¶/·WÊ ClK = 1.

ûaÖ�ÿ�í/Ç ÕιÏ�ÇÎÍÎ×^�

lim−→K|k ´+µ�¶/·WÊ ClK = 1

ÛYÐ.Ñlim−→

K|k ´Vµ/¶�·WÊ p ËaÌ�«aÍ�Ê ClK(p) = 1é

���9�=���ÉÅPó������"����$�B�`��ËtNT)¡^�d8��.Ii ∈ Z

AG�RO/J��.Iv���`���5J=U,!GJ=�RE � Çu! È ���5�0���`& ! �9E �P��I/</#:����JVE.*:E %KEQA ��I/<�#,���p InE 4HE )+*�#���I.4�U�IHi(Gk, E) ∼= H2−i(Gk,Q/Z)∨UT�98 �5�S���=J=U:![J=�=E �jÇuO È �/���0�/�f& ! �9E �P��I/</#:���@JVE.*:E %KEQA ��I/</#:��� p I/EG4_EG)V*�#B�yI�4�UBI

Hi(Gk(p), E(p)) ∼= H2−i(Gk(p),Qp/Zp)∨.

ø�äYÌ�È�ÇÎÐ.È�ÓYÌ@Ë�È�Ð.Ñ*ÍÎdzï@å.È�õ/Ì3ÛYÓYÓ9ÈGÖ9È�ò{È�dzï@åYÐ.È�Ð ëyÇÎÌ�Ñ.ÕqÖ9È�Ç�Ô"ÇÎ×

G(p)ÇÎåYÌ@È�Ð Ô�ÕF�*ÇÎÔ�ÕqͳÈ�Ð���Ì@Ëqö

pöÎÈAÛ.Ëqö

×3dzÈ�ÐY×@È�ÐQé�ø?äYÌ�ÇÎåYÐ�Ï�ÇÎÍÎ×G(p) = G/R

~±ëlËNÖ9È�ÇRÜaäYÌ�ÑYÈ�ÐÚÆ/ÛYÌ@ï@å.Ù@ï�åYÐYÇÎ×3×ÕqÍÎͳÈ�ÌlË$¦-È�Ð.È�ÐgÁ/ËNÌ3Ô�ÕqÍÎ×@È�ÇÎͳÈ�Ì

H�GÙ@×@È�å¢×^~*ÜoäYÌIÑ*dzÈG/H

È�ÇÎÐ.Èpö�õ/Ì3ÛYÓYÓ9È=dzÙ3×{é

���9�=���ÉÅ9ô�� ö �B�Â�hâ��(G(p))ab

�yI/Js8���)�4_!�] ��4H! %É�ª13)dE ZpZ.q3U,EGJ=�V����J3CFE �

Gab-x8 o # o

(G(p))ab = (Gab)(p).

Page 33: ¼ ½¿¾0À5¼ Áÿ½ÅÄ - Université du Luxembourgmath.uni.lu/~wiese/other/diplom.pdf · 2007-03-08 · ë ËNÖ!È Ç ˙ ÕqÛ*Ü A+ ×3Ì3ÇÎÉ Ç ÕqÍ ÛYÐ.ÑØÕqÛ*Ü

A �CB�� Þ ä[� z�Ù�Ï�ÇÎÍÎ×^�(G/R)ab ∼= (G/R)/[G/R,G/R] ∼= G/[G,G]R

é?Æ�Õ:~QÜ�ÕqÍÎͳÙH � G

È�ÇÎÐ÷Ë$¦9È�Ð.È�ÌÁ/ËNÌ3Ô�ÕqÍÎ×@È�ÇÎͳÈ�Ì�dzÙ3×^~9ÕqÛ.ï@åH[G,G]/[G,G] �G/[G,G]

È�ÇÎÐÒË$¦-È�Ð.È�ÌÁAËNÌ3Ô�ÕqÍÎ×@È�ÇÎͳÈ�Ì=dzÙ3×^~�ÛYÐ.ÑèÈvÙIÛYÔ"Ï�È/{�È�åYÌ3×òvÛÚý3ÈvÑYÈ�Ô Ë$¦-È�Ð.È�Ð�Á/ËNÌ3Ô�ÕqÍÎ×@È�ÇÎͳÈ�Ì

N � G/[G,G]È�ÇÎÐ.È�ÐúË$¦-È�Ð.È�ÐhÁAËNÌ3Ô�ÕqÍÎ×@È�ÇÎͳÈ�Ì

H � GÔ"ÇÎ×

[G,G] ⊆ HÛYÐ.ÑN = H/[G,G]

Ï�ÇÎÖY×^~.È�Ì3Ï�ÇÎÖY×IÙ@dzï@åÊÑ*dzÈ�õ/ͳÈ�dzï�åYå.È�ÇÎ×IÉ�ËNÐ(G/R)ab

ÛYÐ.Ñ(Gab)(p)

é2£�È�É�ËNÌIëyÇÎÌòvÛYÔ¬£�È�ëlÈ�dzÙAÑYÈvÙY��å.ÈvËNÌ@È�Ô�ÙY{�ËNÔ"Ô�È�Ð�~�Ö!È�ë�È�dzÙ@È�Ð ëyÇÎÌòvÛYÐ�ìNï�å.Ù3×AÈ�ÇÎÐ

���9�=���ÉÅ�`�� ö �B�Â�hâ��E#T!GJ¯A U:J+����WY! %MEG�yI�Z=mO.I/J=�R�RA UT�98v�.IDA �5%;J��.NT)�! %5%K�

K|k

H1(GK , E) ∼= ClKUT�98 �54ÌË�! %5%tÇuO È 6nU�In�GJy6/%;�R<�#H1(GK(p), E(p)) ∼= ClK(p).

A �CB�� Þ ä[� áúÈ�Ï�È�Ðlim−→K|k

ClK = 1dzÙ@×?Ñ*dzÈ?��Èn ¢Û.È�ÐYò

1→ E → I → C → 1È��.Õ${±×{éG��Û�ÇÎåYÌ�Ö9È�×3Ì�ÕNï@å±×@È�Ð�ëyÇÎÌ

Ñ*dzÈ�Í ÕqÐYÏ�È"È��.Õ${±×@ÈH��Èn ±Û.È�ÐYò1 → EGK → IK → CK → H1(GK , E) → H1(GK , I) = 1

é9í�È�Ì3Ï�ͳÈ�dzï�å.È�ÐëyÇÎÌ�Ñ*dzÈvÙ@ÈWÔ"ÇÎ×=ÑYÈ�Ì/È��.Õ${±×@È�Ð¥��Èn ±Û.È�ÐYò

1 → EK → IK → CK → ClK → 1~!Ù@Ë�ÜoËNÍÎÏ�×=ÑYÈ�Ì/È�Ì@Ù@×@È���È�ÇÎÍ?ÑYÈ�Ì

ÞAÛ.Ù@Ù�ÕqÏ�È�é�±ÇÎÐ.Ñ6ëyÇÎÌÇÎÐ ÑYÈ�̳�±ÇÎ×3Û�Õq×3dzËNÐ÷ûaÖ�ÿ�~.Ù@Ë Ö9È�×3Ì�ÕNï@å¢×@È�ÐàëyÇÎÌGK(p) = GK/(

L|K ´Vµ/¶/·WÊ p ËPÌ�«aÍ/ÊGL) = GK/( lim←−

L|K ´Vµ/¶�·WÊ p ËaÌ�«aÍ�Ê GL).

Æ�Õqå.È�ÌdzÙ@×IÕqÛ*ÜaÏ�Ì3ÛYÐ.Ñ ÑYÈvÙAÈ�Ì@Ù3×@È�ÐS��È�ÇÎͳÙH1( lim←−

L|K ´Vµ/¶/·WÊ p ËaÌ�«aÍ�Ê GL, E)(p) = lim−→L|K ´Vµ/¶/·WÊ p ËaÌC«UÍ�Ê H

1(GL, E)(p) = lim−→L|K ´+µ�¶/·WÊ p ËaÌC«UÍ�Ê ClL(p) = 1.

ø�ËNÍÎÏ�ÍÎdzï@åàÙ�ï@åYÍÎdzÈ/��È�ÐÊëyÇÎÌ/ÕqÛ.ÙÑYÈ�ÌIêuÐ:Ï�Õq×3dzËNÐ*ö�ðIÈvÙ3×3Ì3ÇK{�×3dzËNÐ*öu��Èn ¢Û.È�ÐYò1→ H1(GK(p), E(p))

¸kµ�^−→ H1(GK , E)(p)

«�´ ¤−→ H1( lim←−

L|K ´Vµ/¶/·WÊ p ËPÌ�«aÍ/Ê GL, E)(p) = 1

Ñ*dzÈ�êuÙ@ËNÔ�ËNÌ3ÓYåYdzÈH1(GK(p), E(p)) ∼= ClK(p)

é2

A �CB�� Þ äSí9�9� �����=���ÉÅPó��^� ��ÛYÐ�ìNï�å.Ù3×�È�Ì3å�ÕqÍÎ×@È�Ð ëyÇÎÌ=ëlÈ�Ï�È�ÐúÑYÈ�Ì{�ËNå.ËNÔ�ËNͳËNÏ�dzÙ�ï@å.È�ÐÂ��Ì3ÇÎÉ±Ç ÕqÍÎÇÎ×�ìq×�É�ËNÐXÑ*dzÈ�æAËNÔH��ËNÔ�ËNÌ3ÓYåYdzÈ

Hi(Gk, C) ∼= Hi(Gk, C0)ÜoäYÌyÕqÍÎͳÈ

i ∈ Zé±Æ/dzÈvÙlå�Õq×�òvÛYÌ�ø.ËNÍÎÏ�ÈG~*Ñ.Õ$��Ù3dzï�å�Ñ*dzÈAÓYÌ@Ë�È�Ð.Ñ*ÍÎdzï@å.È/í�È�Ì@Ù3dzËNÐ6ÑYÈvÙ2�*Õq×3ò{ÈvÙ�É�ËNÐ\Á/Õ${�Õn�¢ÕqÔ�Õcö+�?Õq×@È

ûl��éK��驨G��ÿIÕqÛ.ï@åàÕqÛ*ÜCÕqТëlÈ�Ð.ÑYÈ�ÐàÍ ì$��×^~YÛYÐ.Ñ6ëyÇÎÌyÜoäYÌIÕqÍÎͳÈ

i ∈ ZÈ�ÇÎÐ.È�ÐÊæAËNÔH��ËNÔ�ËNÌ3ÓYåYdzÙ3ÔWÛ.Ù

Hi(Gk, C) ∼= H2−i(Gk,Z)∨

È�Ì3å�ÕqÍÎ×@È�ÐQé�Æ/dzÈ=Ï�ͳÈ�dzï@å.È�ÐfÐAÖ9È�Ì3ͳÈ�Ï�ÛYÐYÏ�È�ÐàÈ�Ì3Ï�È�Ö9È�Ð ÇÎÔîø�ÕqÍÎͳÈØûaÖ�ÿ�È�ÇÎÐ.È�æAËNÔH��ËNÔ�ËNÌ3ÓYåYdzÈHi(Gk(p), C(p)) ∼= H2−i(Gk(p),Z)∨.

z�Ö!È�Ð.Ù@Ë�È�Ì3å�ÕqÍÎ×@È�Ð ëyÇÎÌyÜoäYÌÈ�ÇÎÐ.ÈÝÈ�Ð.Ñ*ÍÎdzï@å.È�z�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏK|kê�Ù@ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�È�Ð

Hi(G(K|k), CK) ∼= H2−i(G(K|k),Z)∨,

ÕqÛ.ÙIÑYÈ�Ð.È�Ð6ëyÇÎÌIÕqÛ*Ü?Ñ*dzÈz�Ð.Ñ*ÍÎdzï�å,{�È�ÇÎ×AÑYÈ�ÌyÍÎÇÎÐ,{�È�ÐÒõ/Ì3ÛYÓYÓ9È�Ù@ï�åYÍÎdzÈ/��È�Ð�~.Ñ.Õ�Ñ*dzÈ=Ì@Èvï@å¢×@È�ÕqͳÙZö�ùàË�Ñ*ÛYÍ-È�Ð.Ñ*ÍÎdzï@å

È�Ì3ò{È�ÛYÏ�×AdzÙ3×yÛYÐ.Ñ6É�ËNÐ|G(K|k)|

ÕqÐYТÛYÍÎÍÎdzÈ�Ì3×/ëyÇÎÌ@Ñ-éá�ÇÎÌ×@È�ÇÎͳÈ�ÐàÑ*dzȪ®Nö+��È�Ì3Ô�Èdöu��Èn ±Û.È�ÐYò6ûl�&ÿ�ÕqÛ*Ü0ÇÎÐ

1→ EK → IK →MK → 1

Page 34: ¼ ½¿¾0À5¼ Áÿ½ÅÄ - Université du Luxembourgmath.uni.lu/~wiese/other/diplom.pdf · 2007-03-08 · ë ËNÖ!È Ç ˙ ÕqÛ*Ü A+ ×3Ì3ÇÎÉ Ç ÕqÍ ÛYÐ.ÑØÕqÛ*Ü

ÛYÐ.Ñ1→MK → CK → ClK → 1.áúÈ�Ï�È�ÐàÑYÈ�ÌY{�ËNå.ËNÔ�ËNͳËNÏ�dzÙ@ï@å.È�Ð��QÌ3ÇÎÉ�Ç ÕqÍÎÇÎ×�ìq×IÉ�ËNÐ

IKÈ�Ì3å�ÕqÍÎ×@È�Ð ëyÇÎÌIåYdzÈ�Ì�ÕqÛ.ÙyÜoäYÌ

i ∈ ZêuÙ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�È�Ð

Hi+1(G(K|k), EK )∼−→ Hi(G(K|k),MK )

ûR¨�ÿÜoäYÌIÕqÍÎͳÈ�È�Ð.Ñ*ÍÎdzï@å.È�Ðfz�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ�È�Ð

K|kÇÎÔîø�ÕqÍÎÍ�ûgÕ�ÿlÖYòvëÝé�ÜoäYÌÕqÍÎͳÈ�È�Ð.Ñ*ÍÎdzï@å.È�Ð

pö+z�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ�È�Ð

K|kÇÎÔ

ø�ÕqÍÎÍlûaÖ�ÿdé.ø.È�Ì3Ð.È�ÌAÖ!È/{�ËNÔ"Ô�È�Ð ëyÇÎÌAÑ*dzÈ�È��YÕ${�×@È���Èn ±Û.È�ÐYòHi(G(K|k),MK)→ Hi(G(K|k), CK )→ Hi(G(K|k), ClK)→ Hi+1(G(K|k),MK ).

ûR­�ÿá�ÇÎÌë�ËNÍÎͳÈ�Ð ÇÎÔ ÜgËNÍÎÏ�È�Ð.ÑYÈ�Ð6ÜoäYÌIÕqÍÎͳÈ

i ∈ ZÑ*dzÈ7z¯�*dzÙ3×@È�ÐYò�È�ÇÎÐ.ÈvÙIæAËNÔH��ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�Û.Ù

Hi(Gk, C) ∼= Hi+1(Gk, E)

ÇÎÔ ø�ÕqÍÎÍlûgÕ�ÿ�ÖYòvëÝé*ÇÎÔ ø�ÕqÍÎÍlûaÖ�ÿHi(Gk(p), C(p)) ∼= Hi+1(Gk(p), E(p))

ò{È�ÇÎÏ�È�Ð�~�ÑYÈ�ÐYÐÊÑ.ÕqÐYÐ6ÜgËNÍÎÏ�×ÑYÈ�̳�*Õq×3òÝÇÎÔîø�ÕqÍÎÍ�ûgÕ�ÿ�ëlÈ�Ï�È�ÐHi(Gk, E) ∼= Hi−1(Gk, C) ∼= H3−i(Gk,Z)∨ ∼= H2−i(Gk,Q/Z)∨

ÖYòvëÝé�ÑYÈ�ÌAÕqÐ�ÕqͳËNÏ�È�ÐàðAÈvï@åYТÛYÐYÏØÇÎÔîø�ÕqÍÎÍlûaÖ�ÿdéá�ÇÎÌÖ9È�å�ÕqÐ.ÑYÈ�ÍÎÐÊÑ*Ì@È�Ç�ø�ìqÍÎͳÈ�Ï�È�×3Ì@È�ÐYб×{éûoê3ÿ

i = 0� êuÐ Ñ*dzÈvÙ@È�Ô ø�ÕqÍÎÍQå�ÕqÖ9È�Ð6ëyÇÎÌyÜoäYÌ�ûgÕ�ÿlÜgËNÍÎÏ�È�Ð.ÑYÈÝêuÙ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�È�Ð��

Clk

Ï ´4ÐV¸§Ñ+«y½>ÐV¸ ¥ÓÒÎ¥−−−−−−−−→ Gabk

Ô ¸k¶+Õ+¬l·−−−−−→ H1(Gk,R/Z)∨

δ−−→ H2(Gk,Z)∨

Ñ�«�½c´Vµ/¶/·k¸k¹=º/´4«%Ö%¬�×�¬4Ø�¬�Ù�¬ÎËPÚz¬ ¥ ´−−−−−−−−−−−−−−−−−−→ H0(Gk, C).

ê�Ôîø�ÕqÍÎÍlûaÖ�ÿ�È�Ì3Ï�ÇÎÖY×AÙ3dzï�å��Clk(p)→ Gabk (p)→ H1(Gk(p),R/Z)∨ → H2(Gk(p),Z)∨ → H0(Gk(p), C(p)).

Æ�ÕqÖ!È�Ç-å�ÕqÖ9È�Ð ëyÇÎÌY�QÈ�Ô"Ô�Õàûl��éK��é©­�þ�ÿ�Ö!È�бÛY×3òv×{éùÊÇÎ×3×@È�ͳÙ��QÈ�Ô"Ô�Õ ûl��éK��é©­G°�ÿyÈ�Ì3å�ÕqÍÎ×@È�ÐàëyÇÎÌÑ*dzÈÝÞAÛ.Ù@Ù�ÕqÏ�È�é

ûoê@ê3ÿi > 0

� z�б×@Ù@ï@å.È�dzÑYÈ�Ð.ÑàdzÙ3×^~YÑ.Õ$�lim−→

K|k ´Vµ/¶/·WÊ Hi(

G(K|k), ClK)

= Hi(

Gk, lim−→K|k ´+µ�¶/·WÊ ClK

)

= Hi(

Gk, 1)

= 1

ÜaäYÌ�ûgÕ�ÿlÛYÐ.Ñ ÇÎÔ ø�ÕqÍÎÍlûaÖ�ÿlòvÛ.Ù�ìq×3òvÍÎdzï�ålim−→

K|k ´Vµ/¶/·WÊ p ËaÌC«UÍ�Ê Hi(

G(K|k), ClK)

= Hi(

Gk(p), lim−→K|k ´Vµ/¶/·WÊ p ËPÌ�«aÍ/Ê ClK(p)

)

= Hi(

Gk(p), 1)

= 1

Ï�ÇÎÍÎ×{é"£�ÇÎͳÑ*ÛYÐYÏ ÑYÈvÙ Ñ*ÇÎÌ@È/{±×@È�Ðj�-ÇÎÔ�ÈvÙ É�ËNÐ ûR¨�ÿ�ÛYÐ.ÑçûR­�ÿÝäYÖ!È�Ì�ÕqÍÎͳÈ�È�Ð.Ñ*ÍÎdzï@å.È�Ð`z�Ì3ë�È�ÇÎ×@È�Ì3ÛYÐYÏ�È�ÐK|kÇÎÔ ø�ÕqÍÎÍ�ûgÕ�ÿ=ÛYÐ.ÑôÇÎÔ ø�ÕqÍÎÍ=ûaÖ�ÿ=äYÖ9È�ÌWÑ*dzÈ�È�Ð.Ñ*ÍÎdzï�å.È�Ð

pö+z�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ�È�ÐóÈ�Ì3Ï�ÇÎÖY× Ñ.ÕqÐYÐô×@ËNÓ9ËNͳËNÏ�dzÙ@ï@å.È

êuÙ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�È�ÐHi+1

(

Gk, E)

∼= Hi(

Gk, lim−→MK

) ÖYòvëÝéHi+1

(

Gk(p), E(p))

∼= Hi(

Gk(p), lim−→MK

)

ÛYÐ.ÑHi

(

Gk, lim−→MK

)

∼= Hi(

Gk, C) ÖYòvëÝé

Hi(

Gk(p), lim−→MK

)

∼= Hi(

Gk(p), C(p))

.

Page 35: ¼ ½¿¾0À5¼ Áÿ½ÅÄ - Université du Luxembourgmath.uni.lu/~wiese/other/diplom.pdf · 2007-03-08 · ë ËNÖ!È Ç ˙ ÕqÛ*Ü A+ ×3Ì3ÇÎÉ Ç ÕqÍ ÛYÐ.ÑØÕqÛ*Ü

ûoê@ê@ê3ÿi < 0

� z�б×@Ù@ï@å.È�dzÑYÈ�Ð.ÑàdzÙ3×^~YÑ.Õ$��ÇÎÔîø�ÕqÍÎÍ�ûgÕ�ÿlim←−

K|k ´Vµ/¶/·WÊ Hi(

G(K|k), ClK)

= 1

ÖYòvëÝé*ÇÎÔ ø�ÕqÍÎÍlûaÖ�ÿlim←−

K|k ´+µ�¶/·WÊ p ËaÌ�«aÍ�Ê Hi(

G(K|k), ClK)

= 1

Ï�ÇÎÍÎ×{é9Æ=ÕNÙ�Ù@dzÈ�å¢×�Ô�ÕqÐ ëydzÈWÜoËNÍÎÏ�×^�-ÞAÛ.Ù7�*Õq×3òÊûl��éK��é©­:�&ÿ/Ù@ï@åYÍÎdzÈ/��È�ÐôëyÇÎÌ�ÕqÛ*Ü�Ñ*dzÈ��IËNÔ"ÔWÛY×�Õq×3ÇÎɱÇÎ×�ìq×�ÑYÈ�ÌÆ/Ç ÕqÏ�Ì�ÕqÔ"Ô�È

ClK′

NK′|K−−−−→ ClK ClK′(p)

NK′|K−−−−→ ClK(p)

Ï ´ ÐV¸§Ñ+« y

Ï ´ ÐV¸§Ñ+« y

ÛYÐ.Ñ Ï ´4ÐV¸§Ñ+« y

Ï ´4ÐV¸§Ñ+« y

GabK′×�¬lµ�½dµ/¸ ¤ ¹=º−−−−−−−→ GabK GabK′(p)

×�¬lµ�½dµ/¸ ¤ ¹=º−−−−−−−→ GabK (p)

Ô"ÇÎ×�Ù@È�Ð,{�Ì@Èvï@å±×@È�ÐØê�Ù@ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�È�ÐQé�á ìqåYͳÈ�Ð�ëyÇÎÌK ′|K

Ù�Ë,~�Ñ.Õ$�G(K ′|K) = GabK

Ï�ÇÎÍÎ×^~�Ñ-é åQéGK′ =

[GK , GK ]~YÑ.ÕqÐYРdzÙ3×Ñ*dzÈ{�ÕqÐ.ËNÐYdzÙ@ï@å.ÈWÞIÖYÖYÇÎͳÑ*ÛYÐYÏØËNÖ!È�Ð�~.Ñ*dzÈ=ÇÎÐ.Ñ*ÛYòvdzÈ�Ì3×AëyÇÎÌ@Ñ6É�ËNÐ

GK′ = [GK , GK ]→ GabK = GK/[GK , GK ],

Ñ*dzȥÁAÛYÍÎÍ ÕqÖYÖYÇÎͳÑ*ÛYÐYÏ.éAÆ/dzÈvÙàå�Õq×ÊòvÛYÌÒø.ËNÍÎÏ�ÈG~AÑ.Õ$��ÇÎÐHÑ*dzÈvÙ@È�Ô ø�ÕqÍÎÍ�ÕqÛ.ï�åClK′

NK′|K−−−−→ ClK

ÛYÐ.ÑClK′(p)

NK′|K−−−−→ ClK(p)

Ñ*dzÈ7ÁAÛYÍÎÍ ÕqÖYÖYÇÎͳÑ*ÛYÐYÏ�È�ÐàÙ3ÇÎÐ.Ñ�~Yë�ËNÌ�ÕqÛ.ÙIÙ3dzï�åÊÑ*dzȣ�È�å�ÕqÛYÓY×3ÛYÐYÏØÈ�Ì3Ï�ÇÎÖY×{éá�ÇÎÌyë�ËNÍÎͳÈ�ÐÊбÛYÐ ëydzÈ�ÇÎÔ ø�ÕqÍÎÍ�ûoê@ê,ÿlÑ*ÛYÌ@ï�åfÐAÖ!È�Ì3Ï¢ÕqÐYÏ�ÇÎÐôûR¨�ÿ�ÛYÐ.ÑôûR­�ÿ�òvÛYÔîÓYÌ@Ëqý3È/{±×3ÇÎÉ�È�Ðf�-ÇÎÔ�ÈvÙÉ�Ç ÕÑYÈ�Ð Æ=È�Ï�Õq×3dzËNÐ.È�Ð äYÖ!È�ÌÑ*dzÈ�È�Ð.Ñ*ÍÎdzï�å.È�Ðfz�Ì3ë�È�ÇÎ×@È�Ì3ÛYÐYÏ�È�Ð

K|kÇÎÔîø�ÕqÍÎÍlûgÕ�ÿlÖYòvëÝé*äYÖ9È�ÌÑ*dzÈ�È�Ð.Ñ*ÍÎdzï@å.È�Ð

pö+z�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ�È�ÐÒÇÎÔîø�ÕqÍÎÍlûaÖ�ÿ�ÑYÈ�ÐS£�È�ë�È�dzÙAÕqÖ.Ù@ï�åYÍÎdzÈ/��È�ÐQé�Æ�ÕqòvÛ Ö9È{ÕNï@å¢×@È�ÐàëyÇÎÌ^~YÑ.Õ$�ØÕqÛ.ÙÑYÈ�ÌYz�Ð.Ñ�öÍÎdzï@å,{�È�ÇÎ×�É�ËNÐ

ClKÑ*dzÈvz�Ð.Ñ*ÍÎdzï@å,{�È�ÇÎ×ØÉ�ËNÐ

Hi(G(K|k), ClK)ÜoËNÍÎÏ�×^~0ÛYÐ.Ñ(ëydzÈ6ËNÖ9È�ÐóÏ�ÈvÙ@È�å.È�ÐãÕqÛ.ï@å

Ñ*dzÈÊõ/Ì3ÛYÓYÓ9ÈHi(G(K|k), CK)

È�Ð.Ñ*ÍÎdzï�å dzÙ3×{é�Æ�Õqå.È�Ì�Ù@ï@åYÍÎdzÈ/��È�Ð�ëyÇÎÌ�ÕqÛ.Ù ûR­�ÿ"ÕqÛ*Ü/Ñ*dzÈvz�Ð.Ñ*ÍÎdzï@å,{�È�ÇÎ×É�ËNÐ

Hi(G(K|k),MK )é�ÞAͳÙ@ËèÖYͳÈ�ÇÎÖY×ØÑ*dzÈSzb�YÕ${�×3å.È�ÇÎ×�Ö9È�ÇÎÔ ÐIÖ9È�Ì3Ï¢ÕqÐYÏúòvÛYÔ ÓYÌ@Ëqý,È/{�×3ÇÎÉ�È�Т�-ÇÎÔ�ÈvÙ

È�Ì3å�ÕqÍÎ×@È�ÐQé��ËNÔ"ÇÎ×IÈ�Ì3Ï�ÇÎÖY×IÙ3dzï�å

Hi+1(

Gk, E)

∼= lim←−K|k ´Vµ/¶/·WÊ H

i(

G(K|k),MK

) ÖYòvëÝé

Hi+1(

Gk(p), E(p))

∼= lim←−K|k ´Vµ/¶/·WÊ p ËPÌ�«aÍ/Ê H

i(

Gk(p),MK

) ÛYÐ.Ñ

lim←−K|k ´Vµ/¶/·WÊ H

i(

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)

∼= Hi(

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lim←−K|k ´+µ�¶/·WÊ p ËaÌC«UÍ�Ê H

i(

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)

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2

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p

(K×p ,U

(0)p ).

Á=ÕNï@å��*Õq×3òØûl��驨*驨�ÿ�dzÙ3×Ñ*dzÈÝê�ÑYÈ�ÍÎÏ�Ì3ÛYÓYÓ9È�Í³Ë {NÕqÍ�{�ËNÔ"Ó�Õ${±×ÛYÐ.ÑÊæ/ÕqÛ.Ù@ÑYËNÌl¦-Ù@ï�åQé|Iб×@È�ÌIÈ�ÇÎÐ.È�Ô >XE�8 UB% ë�ËNÍÎͳÈ�Ð ëyÇÎÌIÈ�ÇÎÐ6ÜgËNÌ3Ô�ÕqͳÈvÙY��Ì@Ë�Ñ*Û,{±×m =

p

pnp

É�È�Ì@Ù@×@È�å.È�Ð�~.ëlËNÖ9È�Çnp ≥ 0

ÜoäYÌIÕqÍÎͳÈ�ÛYÐ.Ñnp = 0

ÜoäYÌ�ÜgÕNÙ3×ÕqÍÎͳÈpÉ�ËNÌ�ÕqÛ.Ù@Ï�ÈvÙ@È�×3òv×IÙ@È�Ç�é

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ø�äYÌAÈ�ÇÎÐ.È�ÐÊùàË�Ñ*ÛYÍmÑYÈ��.ÐYdzÈ�Ì@È�ÐÊëyÇÎÌIÑ*dzÈ�p 8B��% AG).U�*[*,��4HE^8GUB%KE

mÕqͳÙ

ImK :=

p|m

U(np)p ×

p-m

U(0)p ,

Ñ*dzÈWÕqͳÙIÕqÖYÏ�ÈvÙ@ï@åYͳË�Ù�Ù@È�Ð.È�|AТ×@È�Ì3Ï�Ì3ÛYÓYÓ9ÈÝÉ�ËNÐIKÕqÛ.ï@åàÍ³Ë {NÕqÍ�{�ËNÔ"Ó�Õ${�×dzÙ3×{é��È�Ç

SÈ�ÇÎÐ.È7��Ì3ÇÎÔ�Ù3×@È�ÍÎͳÈ�ÐYÔ�È�ÐYÏ�ÈWÉ�ËNÐ

KéYø?äYÌÙ3dzÈ�ÑYÈ��.ÐYdzÈ�Ì@È�ÐÊëyÇÎÌAÑ*dzÈ

SZ p 8���%kA )�U^*�*,�

IS(K) :=∏

p∈S

(K×p ,U

(0)p )×

p6∈S

U(0)p ,

Ñ*dzÈ�È�Ö9È�Ð*ÜgÕqÍÎͳÙ�ÕqÖYÏ�ÈvÙ@ï�åYͳË�Ù@Ù@È�Ð ÇÎÐIKÛYÐ.Ñ�Í³Ë {�ÕqÍT{�ËNÔ"Ó�Õ${±×0dzÙ3×{écáúÈ�ÇÎ×@È�Ì0å�ÕqÍÎ×@È�Ð�ëyÇÎÌ?ÜgÈvÙ3×^~&Ñ.Õ$�ÝÕqÛ*ÜoÏ�Ì3ÛYÐ.Ñ�ÛYÐ.Ù@Èdö

Ì@È�Ì�Æ=È��.ÐYÇÎ×3dzËNÐØÉ�ËNÐU

(0)p

ÕqÐØÌ@ÈvÈ�ÍÎͳÈ�Ðv�±×@È�ÍÎͳÈ�ÐIS(K)

Ï�ͳÈ�dzï@åIT (K)

dzÙ3×^~�ÜgÕqÍÎͳÙlÙ3dzï@åÚÑ*dzÈ���Ì3ÇÎÔ�Ù3×@È�ÍÎͳÈ�ÐYÔ�È�ÐYÏ�È�ÐSÛYÐ.Ñ

TТÛYÌAÕqÐÊÛYÐ.È�Ð.Ñ*ÍÎdzï@å.È�Ðh�±×@È�ÍÎͳÈ�Ð ÛYТ×@È�Ì@Ù@ï�å.È�dzÑYÈ�ÐQé

ùÊÇÎ×OKëlËNÍÎͳÈ�ÐãëyÇÎÌ�ëydzÈÚÏ�È�ë3�NåYÐYÍÎdzï@åçÑYÈ�Ð÷ðIÇÎÐYÏ ÑYÈ�Ì"Ï¢ÕqÐYò{È�ÐÎ�9ÕqåYͳÈ�ÐóÉ�ËNÐ

KÖ9È�ò{È�dzï@åYÐ.È�ÐQé�ø.È�Ì3Ð.È�Ì

Ù�È�×3ò{È�Ð ëyÇÎÌyÜoäYÌÈ�ÇÎÐ.È7��Ì3ÇÎÔ�Ù3×@È�ÍÎͳÈ�ÐYÔ�È�ÐYÏ�ÈS

OK,S := {a ∈ K | vp(a) ≥ 0ÜoäYÌIÕqÍÎͳÈ

p 6∈ S, p -∞},

Ñ-é.åQéYÑ.Õ$��Ï�ÇÎÍÎ×O×K,S = {a ∈ K | vp(a) = 0

ÜaäYÌIÕqÍÎͳÈp 6∈ S, p -∞} = IS(K) ∩K×.

áçÇÎÌIÑYÈ��.ÐYdzÈ�Ì@È�Ð6ÜoäYÌIÈ�ÇÎÐ.È�ÐÊùÒË�Ñ*ÛYÍm =

p pnp

O×K,m := {a ∈ OK | a ∈ U

(np)p

ÜoäYÌIÕqÍÎͳÈp} = Im

K ∩K×.

áúÈ�ÇÎ×@È�ÌyÖ!È�ò{È�dzï�åYÐ.ÈJKÑ*dzÈ7p 8��Q! % AG).U�*[*,� É�ËNÐ

OKé:�±Ç³È/dzÙ3×yÑ*dzÈAÜaÌ@È�dzÈ=ÕqÖ9È�ͳÙ@ï@å.È�õ/Ì3ÛYÓYÓ9È/Ô"ÇÎ×yÑYÈ�Ð\��Ì3ÇÎÔ ö

dzÑYÈ{ÕqͳÈ�ÐóÕqͳÙ�z�Ì3ò{È�ÛYÏ�È�Ð.ÑYÈ�ÐQé�|AТ×@È�ÌJ SKëlËNÍÎͳÈ�Ð(ëyÇÎÌ�Ñ*dzÈØêuÑYÈ{ÕqÍÎÏ�Ì3ÛYÓYÓ9È�É�ËNÐ

OK,SÉ�È�Ì@Ù@×@È�å.È�Ð�~�ë�È�ͳï@å.ÈØdzÙ@Ëqö

Ô�ËNÌ3ÓYåÊdzÙ3×�òvÛYÌ�ÜaÌ@È�dzÈ�ÐÊÕqÖ9È�ͳÙ@ï@å.È�Ðèõ/Ì3ÛYÓYÓ!È�È�Ì3ò{È�ÛYÏ�×IÉ�ËNÐ ÑYÈ�ÐS��Ì3ÇÎÔ"dzÑYÈ{ÕqͳÈ�ÐÒÕqÛ,��È�Ì3å�ÕqÍÎÖ É�ËNÐSé,z�Ö!È�Ð.Ù@Ë"Ù�È�Ç

J mK

Ñ*dzÈ\|Iб×@È�Ì3Ï�Ì3ÛYÓYÓ!È�É�ËNÐJKÑYÈ�Ì�òvÛ

m×@È�ÇÎͳÈ�Ì,ÜoÌ@È�Ô�ÑYÈ�Ð÷ê�ÑYÈ{ÕqͳÈG~�ëlÈ�ͳï@å.È6Ù�ËNÔ"ÇÎ×WdzÙ@ËNÔ�ËNÌ3ÓYå÷òvÛ

J SKdzÙ3×^~

ë�ËNÖ!È�ÇSåYdzÈ�ÌIÑ*dzÈÝùÒÈ�ÐYÏ�È�ÕqÍÎͳÈ�Ì

m×@È�ÇÎͳÈ�Ð.ÑYÈ�Ðf��Ì3ÇÎÔ"dzÑYÈ{ÕqͳÈ�Ö!È�ò{È�dzï�åYÐ.È�×{é

ê\âPß5� Þ ä/ë_æ:F�� Þ �0�êuÔ ÜgËNÍÎÏ�È�Ð.ÑYÈ�ÐÊëlËNÍÎͳÈ�ÐÊëyÇÎÌIÈ�ÇÎÐ.ÈWõ�ÕqͳËNdzÙ,ö�ÞD{�×3dzËNÐàÕqÛ*Ü0ÑYÈ�ÐàêuÑYÈ�ͳÈ�Ð ÑYÈ��.ÐYdzÈ�Ì@È�ÐQé���È�Ç�Ñ.ÕqòvÛ

K|kÈ�ÇÎÐ.È�È�Ð.Ñ*ÍÎdzï@å.È

Ï¢ÕqͳËNdzÙ�Ù@ï@å.È�z�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ6Ô"ÇÎ×�õ�ÕqͳËNdzÙ,ö�õ/Ì3ÛYÓYÓ9ÈG(K|k)

é9z�ÇÎÐσ ∈ G(K|k)

ËNÓ9È�Ì3dzÈ�Ì@È�бÛYÐèÕqÛ*Ü�È�ÇÎÐ.È�Ô êuÑYÈ�Ía = (aP)P

Ñ*ÛYÌ@ï@å(σa)P := σaσ−1P.ø?äYÌ�È�ÇÎÐ.ÈÊùÒÈ�ÐYÏ�È

SÉ�ËNФ��Ì3ÇÎÔ�Ù3×@È�ÍÎͳÈ�ÐçÉ�ËNÐ

kë�È�Ì@ÑYÈàÑ*dzÈàùàÈ�ÐYÏ�ÈàÕqÍÎͳÈ�Ìg��Ì3ÇÎÔ�Ù@×@È�ÍÎͳÈ�Ð�É�ËNÐ

K~�Ñ*dzÈÊäYÖ!È�Ì

ÑYÈ�Ð.È�ÐôÉ�ËNÐSÍÎdzÈ�Ï�È�Ð�~-È�Ö9È�Ð*Ü�ÕqÍÎͳÙ�Ô"ÇÎ×

SÖ!È�ò{È�dzï�åYÐ.È�×{é�ê�Ù3×

m =∏

i pnii

È�ÇÎÐúùÒË�Ñ*ÛYÍ?É�ËNÐk~-Ù@Ë�Ö9È�ò{È�dzï@åYÐ.È�Ð

ëyÇÎÌ=È�Ö9È�Ð*ÜgÕqÍÎͳÙAÔ"ÇÎ×mÑYÈ�ÐÒùÒ˱Ñ*ÛYÍ ∏

i(Pi,1 · · ·Pi,ri)einiÉ�ËNÐ

K~�ëlËNÖ9È�Ç?Ñ*dzÈ

Pi,jÑ*dzÈ�äYÖ9È�Ì

piÍÎdzÈ�Ï�È�Ð.ÑYÈ�Ð��Ì3ÇÎÔ�Ù3×@È�ÍÎͳÈ�ÐÊÉ�ËNÐ

KÙ3ÇÎÐ.Ñ�~*ÛYÐ.Ñ

eiÑYÈ�Ìyí�È�Ì3òvëlÈ�ÇÎÏ�ÛYÐYÏ�Ù3ÇÎÐ.ÑYÈ��àÉ�ËNÐ

Pi,jäYÖ!È�Ì

pidzÙ@×{éz�Ù�dzÙ3× {�Í ÕqÌ^~�Ñ.Õ$�÷Ñ*dzÈX|AТ×@È�Ì3Ï�Ì3ÛYÓYÓ9È�Ð

IS(K)ÛYÐ.Ñ

ImK

ÛYТ×@È�Ì6Ñ*dzÈvÙ@È�Ì6ÞD{±×3dzËNÐ ÕqÖYÏ�ÈvÙ@ï@åYͳË�Ù�Ù@È�Ð Ù3ÇÎÐ.Ñ-éÆ/dzÈ�êuÑYÈ{ÕqÍÎÏ�Ì3ÛYÓYÓ9È

JKÉ�È�Ì@Ù�È�å.È�ÐÒëyÇÎÌ/Ô"ÇÎ×�ÑYÈ�ÌAÏ�È�ë3�NåYÐYÍÎdzï@å.È�Ð(õÝÕqͳËNdzÙ,ö�Þ�{�×3dzËNÐ�~-ÕqͳÙ@Ë�ÑYÈ�Ì/ÞAТë�È�Ð.Ñ*ÛYÐYÏ6É�ËNÐ

σ ∈ G(K|k)ÕqÛ*Ü�Ñ.ÕNÙê�ÑYÈ{ÕqÍ�é.Æ=È�ÌIæAËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ@Ô�Û.Ù

χ : IK → JK , a 7→∏

p-∞

pvp(ap)

dzÙ@×È�ÇÎÐGö�æAËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ@Ô�Û.Ù^�

σ.χ(a) = σ∏

pvp(ap) =∏

(σp)vp(ap) =∏

pvp(σaσ−1p) = χ(σa).ø.È�Ì3Ð.È�ÌIÏ�ÇÎÍÎ×Ð�ÕNï�åX�*Õq×3ò�ê@ê�ê�驨*éjþ ÕqÛ.ÙÝßkÁIö+����â+~YÑ.Õ$�

I(K)Ï�ÛY×@È�ÐÒõ�ÕqͳËNdzÙ3ö�ÞIÖ.Ù3×3dzÈ�Ï"å�Õq×^~�Ñ.Õ$�ØÕqͳÙ@Ë

Ik = IG(K|k)K

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dzÙ@×^~�ë�ÕNÙ�Ù3dzï�åóÐ�Õq×3äYÌ3ÍÎdzï@å�ÕqÛ.ï�å�ÕqÛ*ÜImK

ÛYÐ.ÑIS(K)

äYÖ9È�Ì3×3Ì�ìqÏ�×{é�Æ=ÕqÔ"ÇÎ×�È�Ì3å�ÕqÍÎ×@È�ÐãëyÇÎÌ ÜaäYÌ"È�ÇÎÐ.ÈÊõ�ÕqͳËNdzÙ,öz�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏM |K

Ñ*dzÙd{±Ì@È�×@ÈG(M |K)

ö�ùàË�Ñ*ÛYÍÎÐÊëydzÈ/ÜgËNÍÎÏ�×^�IM = lim−→

L|K,L⊆M

IL =⋃

L|K,L⊆M

IG(M |L)M

ûaÖYòvëÝé!õ/ͳÈ�dzï@å.ÈvÙIÜaäYÌIS(K)

ÛYÐ.ÑImK

ÿdéá�ÇÎÌ_{[�NÐYÐ.È�ÐóбÛYÐãÕqÛ.ï@å Ñ*dzÈv��Õq×@Èdö+�AËNå.ËNÔ�ËNͳËNÏ�dzÈàÑYÈ�Ì�ê�ÑYÈ�ÍÎÏ�Ì3ÛYÓYÓ9ÈÚÖ!È�×3Ì�ÕNï�å¢×@È�ÐQé�Æ/dzÈvÙ@È�å�Õq× ÜgËNÍÎÏ�È�Ð.ÑYÈ

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Hi(G(K|k), IK) ∼=⊕

p

Hi(Gp(K|k),K×p ).

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i ∈ ZÊHi(G(K|k), IS(K)) = 1 o

A �CB�� Þ ä[� ê�Ù3×p ∈ S

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Hi(G(Kp|kp),Up) = 1dzÙ3×{é

2

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×

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G(K|k)K ,

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G(K|k)K → H1(G(K|k),K×)

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a ∈ IkÑ*ÛYÌ@ï@å

N (a) :=∏

p

|ap|p.

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C0k

Ö!È�ò{È�dzï�åYÐ.È�Ð ëyÇÎÌyÑYÈ�ÐS��È�Ì3Ð ÑYÈ�Ì�ÞAÖ.Ù@ËNÍÎÛY×3Ð.ËNÌ3Ô ÕqÛ*ÜCké

ÞAͳÙ�ÜoÛYÐ.Ñ.ÕqÔ�È�Т×�ÕqÍ�ëyÇÎÌ@Ñ Ù3dzï@åàÑ*dzÈ7�IËNÔ"Ó�Õ${�×3å.È�ÇÎ×AÑYÈ�Ì/õ=Ì3ÛYÓYÓ!ÈC0k

È�Ì3ëlÈ�dzÙ@È�Ð÷û,ßkÁöu�0��â+~���å.ÈvËNÌ@È�Ôîí/êdéK��é©°�ÿdé

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Æ/È�ÐÊÍ³Ë {NÕqÍ�{�ËNÔ"Ó�Õ${�×@È�Ð�~�æ=ÕqÛ.Ù@ÑYËNÌl¦-Ù@ï@å.È�ÐG(K|k)

ö�ùàË�Ñ*ÛYÍCS(K) = IK/K

×UK,S,

ë�ËNÖ!È�ÇÑ*dzÈg{�ËNÔ"Ó�Õ${�×@ÈÊõ/Ì3ÛYÓYÓ9ÈUK,S

Ñ*ÛYÌ@ï@å ∏

p∈S{1} ×∏

p6∈S U(0)p

Ï�È�Ï�È�Ö9È�ÐóÙ@È�ÇV~?Ð.È�ÐYÐ.È�ÐóëyÇÎÌ Ñ*dzÈSZp 8B��%Ã&F%M!$I.In���TA ).U�*[*,� é�ø�ÕqÍÎͳÙ

K|kÕqÛ,��È�Ì3å�ÕqÍÎÖ6É�ËNÐ

SÛYТÉ�È�Ì3òvë�È�ÇÎÏ�×dzÙ3×^~±Ç³Ù3×�ÑYÈ�Ì�ùÒË�Ñ*ÛYÍ

UK,S{�ËNå.ËNÔ�ËNͳËNÏ�dzÙ@ï@å

×3Ì3ÇÎÉ�Ç ÕqÍV~9ë�ÕNÙÝÕqÛ.Ù�ÑYÈ�Ô Í³Ë {�ÕqͳÈ�ÐúðIÈvÙ@ÛYÍÎ×�Õq×AÜoäYÌ=ÛYТÉ�È�Ì3òvë�È�ÇÎÏ�×@È}��Ì3ÇÎÔ�Ù3×@È�ÍÎͳÈ�ÐpÉ�ËNÐ

kÜgËNÍÎÏ�×"ûaÉ�Ï�Í�é0ßkÁö+���yâ+~�*Õq×3ò ê@êdé ®.é©­�ÿ�~.ÛYÐ.ÑàÑ.ÕqÌ�ÕqÛ.Ù^~�Ñ.Õ$��ëyÇÎÌAÑ*dzÈ�z�Ì3ë�È�ÇÎ×@È�Ì3ÛYÐYÏ

C|RÕqͳÙIÉ�È�Ì3òvë�È�ÇÎÏ�×ÝÕqÐ.Ù@È�å.È�ÐQé�êuÐàÑ*dzÈvÙ�È�̪�±ÇÎ×3Û�Õq×3dzËNÐ

å�Õq×�ÑYÈ�Ì�ùÒ˱Ñ*ÛYÍCS(K)

Ï�ÛY×@È�ÐÚõÝÕqͳËNdzÙ,ö�ÞIÖ.Ù@×3dzÈ�ÏP~�Ñ.ÕÝÑ*dzÈD��Èn ±Û.È�ÐYò0→ Uk,S → Ik → (CS(K))G(K|k) → 0È��.Õ${�×ydzÙ3×{é|Iб×@È�Ì ÑYÈ�ÌHp 8���%'&�%K!$IcI^���BA )�U�*[*,�S4_E�8 UB%KE

më�ËNÍÎͳÈ�Ð(ëyÇÎÌ ÑYÈ�Ð÷È�Ö!È�Ð*Ü�ÕqÍÎͳÙ�Í³Ë {NÕqÍ3{�ËNÔ"Ó�Õ${±×@È�Ð÷ÛYÐ.Ñ(æ/ÕqÛ.Ù3ö

ÑYËNÌl¦-Ù@ï�å.È�ÐG(K|k)

ö�ùÒË�Ñ*ÛYÍCmK = Im

KK×/K× = Im

K/O×K,mÉ�È�Ì@Ù@×@È�å.È�ÐQé

z�ÙIÙ@È�ÇPKÑ*dzÈ�õ/Ì3ÛYÓYÓ9ÈÝÑYÈ�Ìæ/ÕqÛYÓY×3dzÑYÈ{ÕqͳÈ�É�ËNÐ

OKÛYÐ.Ñ6ë�È�ÇÎ×@È�Ì

PmK := {(a) ∈ J m

K | a ∈ O×K

ÛYÐ.Ña ∈ U

(np)p

ÜoäYÌp | m}

ÛYÐ.ÑPSK := {(a) ∈ J SK | a ∈ K

×}.í�ËNÐÊê�б×@È�Ì@ÈvÙ@Ù@È=ÜoäYÌyÛYÐ.ÙÙ3ÇÎÐ.Ñ Ð.È�Ö!È�Ð ÑYÈ�ÌÏ�È�ë3�NåYÐYÍÎdzï@å.È�Ðhp 8��d! %'&�%K!$IcI^���BA )�U�*[*,�ClK = JK/PK

ÕqÛ.ï�åàÑ*dzÈ�Ñ*ÛYÌ@ï�åClmK = J m

K /PmK

ÖYòvëÝéClS(K) = J SK/P

SKÑYÈ��.ÐYdzÈ�Ì3×@È�p 8��d! %'&�%K!$IcI^���BA )�U�*[*,��4HE^8GUB%KE

mÖYòvëÝé.Ñ*dzÈ

SZ p 8��d! %'&�%K!$IcI^���BA )�U�*[*,� é

á�ÇÎÌØÖ!È�×3Ì�ÕNï�å¢×@È�Ð ÑYÈ�ÐÎz�ÓYÇÎÔ�ËNÌ3ÓYåYdzÙ@Ô�Û.ÙJK � J SK , a 7→ OK,Sa

~�ëlÈ�ͳï�å.È�Ì6ÑYÈ�Фz�ÓYÇÎÔ�ËNÌ3ÓYåYdzÙ@Ô�Û.ÙJK � J SK/P

SK = ClS(K)

ÇÎÐ.Ñ*ÛYòvdzÈ�Ì3×{éPKÍÎdzÈ�Ï�×WÇÎÔõ��È�Ì3ÐÃÑ*dzÈvÙ�ÈvÙ�æ/ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�Û.Ù^~?Ù@Ë Ñ.Õ$�

ClS(K)È�ÇÎÐàø�Õ${±×@ËNÌIÉ�ËNÐClKÛYÐ.ÑÊÙ@ËNÔ"ÇÎ×IÈ�Ð.Ñ*ÍÎdzï@åàdzÙ3×{é

ÆAdzÈIõ=Ì3ÛYÓYÓ!ÈCK/C

mK

å.È�ÇK��× �9J=)d!$#�%'&�%M! IcIn�/�BA )�U^*[*P�³4HE^8 UT%MEméNêuÔ ÜoËNÍÎÏ�È�Ð.ÑYÈ�Ð"ë�ËNÍÎͳÈ�Ð�ëyÇÎÌ�Ù3dzÈ�dzÑYÈ{ÕqÍÎ×3å.ÈvËqö

Ì@È�×3dzÙ�ï@åÊÖ9ÈvÙ@ï�åYÌ@È�ÇÎÖ!È�Ð ÛYÐ.Ñ ÇÎåYÌ@Èz�Ð.Ñ*ÍÎdzï@å,{�È�ÇÎ×�ÕqÖYͳÈ�ÇÎ×@È�ÐQé�Æ�ÕqòvÛ Ö9È�×3Ì�ÕNï@å±×@È�ÐÊëyÇÎÌIÑ*dzÈ�õ/Ì3ÛYÓYÓ9ÈI

(m)K = {a ∈ IK | ap ∈ U

(np)p

ÜoäYÌp | m}.

ÞAÛ.ÙÑYÈ�Ô ÞAÓYÓYÌ@Ën�*ÇÎÔ�Õq×3dzËNÐ.Ù@Ù�Õq×3ò�ÜoËNÍÎÏ�×^~YÑ.Õ$�IK = I

(m)K K× Ç³Ù3×{é.Æ=È�ÌæAËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ@Ô�Û.Ù

χ : I(m)K → J m

K , a 7→∏

p-∞,p-m

pvp(ap)

dzÙ@×�Ô"ÇÎ×�ÑYÈ�Ô É�ËNÌ3å.È�Ì�ÑYÈ��.ÐYdzÈ�Ì3×@È�ÐχÉ�È�Ì3×3Ì�ìqÏ�ÍÎdzï�å�ÇÎÐÚÑYÈ�Ô �±ÇÎÐYÐ.ÈG~*Ñ.Õ$��Ñ*dzÈvÙ@È�Ì�Ù@dzï@å6ÕqÛ.Ù

a 7→ (a)ÜaäYÌ

a ∈ K×ÛYÐ.ÑÊËNÖYÇÎÏ�È�ÔχòvÛ.Ù�ÕqÔ"Ô�È�Ð.Ù@È�×3òv×{é

á�ÇÎÌÖ9È�×3Ì�ÕNï@å±×@È�ÐàÑ*dzÈ��±ÛYÌoý3È/{�×3dzËNÐχ : I

(m)K � J m

K/PmK .��È�Ç

a ∈ I(m)K

ÇÎÔ¬��È�Ì3Ð É�ËNÐχéYÆ=ÕqÐYÐÊÏ�ÇÎÖY×AÈvÙÈ�ÇÎÐ

(x) ∈ PmK

~YÑ-é åQéx ∈ U

(np)p

ÜoäYÌp | m

~*Ô"ÇÎ×(x) = χ(a)

éÆ�ÕqÐYРdzÙ3×

apx−1 È�ÇÎÐ@z�ͳÈ�Ô�È�Т×�É�ËNÐ U (0)

p

ÜoäYÌp - m

ÛYÐ.ÑèÉ�ËNÐU

(np)p

~�ÜgÕqÍÎͳÙpÑYÈ�ÐúùÒ˱Ñ*ÛYÍ

m×@È�ÇÎÍÎ×{é-Æ=Õqå.È�Ì

Ï�ÇÎÍÎ×ker(χ) = I

(m)K ∩ Im

KK×.ÞAͳÙ@Ë"È�Ì3å�ÕqÍÎ×@È�ÐàëyÇÎÌ

CK/CmK∼= IK/I

mKK

× ∼= I(m)K K×/Im

KK× ∼= I

(m)K /(I

(m)K ∩ Im

KK×) ∼= J m

K/PmK .

Page 40: ¼ ½¿¾0À5¼ Áÿ½ÅÄ - Université du Luxembourgmath.uni.lu/~wiese/other/diplom.pdf · 2007-03-08 · ë ËNÖ!È Ç ˙ ÕqÛ*Ü A+ ×3Ì3ÇÎÉ Ç ÕqÍ ÛYÐ.ÑØÕqÛ*Ü

z�Ù�Ù@È�ÇmfÑYÈ�Ì�È�Ð.Ñ*ÍÎdzï@å.ÈÞAТ×@È�ÇÎÍ.ÑYÈvÙ�ùàË�Ñ*ÛYͳÙ

mÕqÛ*ÜoÏ�ÈdÜgÕ$��×�ÕqͳÙ�ê�ÑYÈ{ÕqÍYÉ�ËNÐ

OKé Á=ÕNï@å�ÑYÈ�Ô_ï�åYÇÎÐ.ÈvÙ3dzÙ@ï@å.È�Ð

ðAÈvÙ3×@Ù�Õq×3ò�Ï�ÇÎÍÎ×IÑ*dzÈ�ê�Ù@ËNÔ�ËNÌ3ÓYåYdzÈOK/mf

∼=∏

p|mf

OK/pvp(np).

ÞAÛ.ÙÊÑYÈ�Ô êuÙ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�Û.Ù(OK/p

s)× ∼= Up/U(s)p , a + ps 7→ aU

(s)p

È�Ì3å�ÕqÍÎ×@È�Ð5ëyÇÎÌ(OK/mf )

× ∼=∏

p|mfUp/U

(np)p∼= IS∞(K)/I

mf

K .æAdzÈ�Ì�ÕqÛ.ÙͳÈ�ÇÎ×@È�Ð ëyÇÎÌÑ*dzÈÝê�Ù@ËNÔ�ËNÌ3ÓYåYdzÈ

IS∞(K)/ImK∼=

p|mf

Up/U(np)p ×

p|m, p «�´V´V·k· , np=1

R×/R×+

ÕqÖ�~*ÕqÛ.Ù�ÑYÈ�ÌlëyÇÎÌ�Ñ*dzȳz�Ð.Ñ*ÍÎdzï@å,{�È�ÇÎ×yÉ�ËNÐIS∞(K)/Im

K

ÛYб×@È�Ì2£�È�бÛY×3òvÛYÐYÏ ÑYÈ�Ì?z�Ð.Ñ*ÍÎdzï�å,{�È�ÇÎ×yÉ�ËNÐOK/p

vp(np)È�Ì3å�ÕqÍÎ×@È�ÐQé9£�È�×3Ì�ÕNï@å¢×@È�ÐÒëyÇÎÌyТÛYÐÒÑ*dzÈ�È��YÕ${�×@È���Èn ±Û.È�ÐYòIS∞(K)K×/Im

KK× → IK/I

mKK

× → IK/IS∞(K)K× → 0,

Ù�Ë�È�Ì3å�ÕqÍÎ×@È�ÐÒëyÇÎÌAÑ*dzÈ�z�Ð.Ñ*ÍÎdzï@å,{�È�ÇÎ×=É�ËNÐIK/I

mKK

× ∼= CK/CmK

~�ÑYÈ�ÐYÐIK/IS∞(K)K× Ç³Ù@×IdzÙ@ËNÔ�ËNÌ3ÓYåàòvÛYÌê�ÑYÈ{ÕqÍK{�Í ÕNÙ@Ù@È�ÐYÏ�Ì3ÛYÓYÓ9È

ClK~NÛYÐ.Ñ

IS∞(K)K×/ImKK

× = IS∞(K)/ImK(K×∩IS∞(K))

dzÙ@×0È�ÇÎÐ�ø�Õ${±×@ËNÌ�É�ËNÐIS∞(K)/Im

K

é*áçÇÎÌyå�ÕqÍÎ×@È�Ð6ÜgÈvÙ3×^����9�y���5ø��KJ�â:�^å,� ¿ �R�D�0J=)l!$#B%Ã&�%K!$I.In���BAG).U�*[*,��4HE^8 UT%ME

m�yI/Js���98:%;�=</#[-"UT�98 L �5)Y#:!�OQ���XC^�R! χ 8 �V� p InE 4HE )V*9#��R�

CK/CmK∼= J m

K /PmK .

���h}��h}��Xê�ëì&�6X&�í�#���$'&°î�&�ï���&���©&��êuÐ�Ñ*dzÈvÙ@È�ÔÍ�ÝÕqÓYÇÎ×@È�Í�ë�È�Ì@ÑYÈ�ÐØëyÇÎÌlòvëlÈ�Ç!È��YÕ${�×@ȳ��Èn ±Û.È�ÐYò{È�Ð�å.È�Ì3ͳÈ�ÇÎ×@È�Ð�~*ÕqÛ*ÜQÑYÈ�Ð.È�Ð�Ñ*dzÈÜoËNÍÎÏ�È�Ð.ÑYÈ�Ð�ÆAÛ�ÕqÍÎÇÎ×�ìq×@Ù3öÙ�ìq×3ò{È�Ö!È�Ì3ÛYå.È�ÐQéá�ÇÎÌIÙ@×�ÕqÌ3×@È�Ð òvÛYÐ�ìNï@å.Ù3×IÔ"ÇÎ×IÈ�ÇÎÐ.È�Ô

���9�y���aj�� ö �B�¥�¥â��������0 −→ A −→ B

φ−→ C −→ 0

�/���0�v�Q][!$&FJ+� ���>=/UP���T6SCFE �`!�OQ��%'I/</#:���ÎJVE�*:E %KEQA ��I/<�#:�/�W2)�U�*[*,��� ÇuOc6 L o G ZV>�E^8 UT%'� È -Y����8���)}!G%�%É�àH�E 4HE 4HE )+*�#���I.4}���ÂI�J+�/J=�KAvI����08 o exI�In���V���g�n��)��0��)�6 L ����!�ORAB��ZIn<�#�%KE$IcI^���0� �"��J+��)RA ).U�*[*,�/�¥ÇuOQ6 L o Z+4HE^8 UT%;� È M ⊆ B UT�98 D ⊆ C AB�RA��^OQ�/�,-¯I/E�8[!da φ(M) ⊆ D��I�J o ¿ ! �P�AG��%MJ+Ê

ÇÆ! Èh¿ �V�ª���>=/UP���T60 −→φ−1(D)/M −→B/M

φ−→ C/D −→ 0�yI/J3�d][! &FJ34��=J¯I/J+�/J=�MAB���IH7E 4HE 4HE )+*�#���I.4}��� oÇlO È¥p I�J�6nU�In�GJy6/%;�R<�#

φ(M) = D-¯InE}��I�J38 �V�ª���½=/U,�/�T6

0 −→A/A ∩M −→B/Mφ−→ C/D −→ 0�d][!$&FJ34��=J¯I�J+�/J=�KA����IH�E 4HE 4HE )V*9#��yI�4}��� o

A �CB�� Þ ä[�φ−1(D) ≤ B

dzÙ@×IÈ�ÇÎÐÒÕqÖYÏ�ÈvÙ@ï�åYͳË�Ù@Ù@È�Ð.È�̪|Iб×@È�Ì3Ô�˱Ñ*ÛYÍ�é!Æ/dzÈ��±×@È�×3ÇÎÏG{�È�ÇÎ×=ÑYÈ�ÌAÞAÖYÖYÇÎͳÑ*ÛYÐYÏ�È�ÐàÇÎÐûgÕ�ÿ�ÛYÐ.ÑôûaÖ�ÿldzÙ3×?{�Í ÕqÌ{éYáúÈ�Ï�È�Ð

C ∼= B/AÏ�ÇÎÍÎ×

D ∼= φ−1(D)/Aé�Æ=Õqå.È�ÌÜoËNÍÎÏ�×

C/D ∼= (B/A)/(φ−1(D)/A) ∼= B/φ−1(D),

ë�ÈvÙ3å�ÕqÍÎÖàÑ*dzÈ���Èn ±Û.È�ÐYòÝÇÎÐÃûgÕ�ÿyÈ��YÕ${�×ydzÙ3×{é��ÛÒûaÖ�ÿ?Ï�È�бäYÏ�×lÈvÙ�òvÛ�ò{È�ÇÎÏ�È�Ð�~¢Ñ.Õ$�φ−1(D) = AM

ûaÔ�ÛYÍÎ×3ÇÎÓYÍÎÇK{�Õq×3ÇÎÉ�Ï�ÈvÙ@ï@åYÌ3dzÈ�Ö9È�Ð�ÿ0Ï�ÇÎÍÎ×{é±Æ/dzÈvÙ�äYÖ!È�Ì3ÓYÌ3ä*Üo×Ô�ÕqÐàÙ@ËqÜoËNÌ3×{é

2

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���9�y���ak�� ö �B�¥�¥â�� ¿ �V�x���½=/U,�/�T60 −→IS(K) −→IK

χ−→ J SK −→ 0

��I�J��d][! &FJ5-�UB�98�8 �R�0H7E 4HE 4HE )+*�#���I.4}���I����08_I/J+�/J=�MA - L E�OQ��� J SK 4��5J38���)_q2U,EGJ=�R�/��J+����JVE.*,E %MEcA �R��C$��).In�.#:�/��In��� oá�ÇÎÌIÑYÈ��.ÐYdzÈ�Ì@È�ÐIK,S :=

p∈S

(K×p ,U

(0)p ).

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Ñ+«y½�ð−−→ IK,S → 0

û5®±ÿòvÛYÌAø.ËNÍÎÏ�È�éYáúÈ�ÇÎ×@È�ÌAÑYÈ��.ÐYdzÈ�Ì@È�ÐÊëyÇÎÌ

CK,SÑ*ÛYÌ@ï@åÊÑ*dzÈÝÈ��.Õ${�×@È���Èn ¢Û.È�ÐYò

0→ O×K,S → IK,S → CK,S → 0.

ûgþ�ÿÞAͳÙÞAТë�È�Ð.Ñ*ÛYÐYÏØÉ�ËNÐS�QÈ�Ô"Ô�Õàûl��驨*é©��ÿ�È�Ì3å�ÕqÍÎ×@È�ÐÊëyÇÎÌyòvÛYÐ�ìNï�å.Ù3×AÑ.ÕNÙ���9�y���=�nm�� ö �B�Â�hâ���Ë�E % AB���98������½=/U,�/�T6�����I����08g�d]�!$&FJ5-xUT�98g8 �V��H�E 4HE 4HE )+*�#���I.4}���fI.�5�98_I/J+�/J=�KA��ªmO^O��5%'Z8GUB�BAB���0ÊÇÆ! È

0→ K×IS(K)→ IK → ClS(K)→ 0ÇlO È0→ IS(K)/O×

K,S → CK → ClS(K)→ 0Çu< È0→ CK,S → CS(K)→ ClS(K)→ 0

A �CB�� Þ ä[� á�ÇÎÌAÏ�È�å.È�ÐÒÕqÛ.ÙIÉ�ËNÐàÑYÈ�ÌAÈ��.Õ${�×@È�ÐX��Èn ¢Û.È�ÐYò0 −→IS(K) −→IK

χ−→ J SK −→ 0

ÕqÛ.ÙIÉ�ËNÌ3ÇÎÏ�È�Ô��È�Ô"Ô�ÕÊûl��驨*驱�ÿdéø�äYÌ�ûgÕ�ÿ�ë�ìqåYͳÈ�Ð6ëyÇÎÌÑ*dzȳ|AТ×@È�Ì3Ï�Ì3ÛYÓYÓ9È�Ð

PSK ≤ JSK

ÛYÐ.ÑK×IS(K) ≤ IK

~¢ÜoäYÌyÑ*dzÈχ(K×IS(K)) =

PSKÏ�ÇÎÍÎ×{éP�QÈ�Ô"Ô�Õ ûl��驨*é©��ÿyÍÎdzÈdÜoÈ�Ì3×yбÛYÐàÑ*dzÈ�ÞAÛ.Ù@Ù�ÕqÏ�È�éø�äYÌÝûaÖ�ÿ�ë�ìqåYͳÈ�Ð6ëyÇÎÌ

PSK ≤ JSK

ÛYÐ.ÑK× ≤ IK

éTz�Ù�Ï�ÇÎÍÎ×^�χ(K×) = PSK

~¢ë�ËNÌ�ÕqÛ.ÙÑ*dzÈAÞAÛ.Ù@Ù�ÕqÏ�ÈAÜgËNÍÎÏ�×{éø�äYÌ ûoï&ÿ�ë�ìqåYͳÈ�ÐÒëyÇÎÌAëydzÈvÑYÈ�Ì3ÛYÔ

PSK ≤ JSK

~�ÕqÖ9È�ÌбÛYÐK×UK,S ≤ IK

éPz�ÙIÏ�ÇÎÍÎ×χ(K×UK,S) = PSK

éÞAÛ.ÙY�QÈ�Ô"Ô�Õàûl��驨*é©��ÿ�È�Ì3å�ÕqÍÎ×@È�Ð ëyÇÎÌIÑ*dzÈ�È��YÕ${�×@È���Èn ¢Û.È�ÐYò

0→ IS(K)/(IS(K) ∩K×UK,S)→ CS(K)→ ClS(K)→ 0.

ùàÇÎ×3×@È�ͳÙØû5®±ÿÝÙ@È�å.È�ÐúëyÇÎÌ^~�Ñ.Õ$�ÊÑYÈ�Ì���È�Ì3ÐúÉ�ËNÐproj : IS(K) → CK,S ∼= IK,S/O

×K,S

Ï�È�Ï�È�Ö!È�ÐÃdzÙ3×�Ñ*ÛYÌ@ï�åUK,SO

×K,S∼= UK,S(K×∩IS(K)) ∼= K×UK,S ∩IS(K)

é¢ÞAͳÙ@Ë�Ï�ÇÎÍÎ×CK,S ∼= IS(K)/(IS(K)∩K×UK,S)

~ë�ËNÔ"ÇÎ×/ÕqÍÎͳÈvÙyÖ9È�ëydzÈvÙ@È�РdzÙ@×{é

2

ÞIÛ.Ù�ûaÖ�ÿ�ÛYÐ.Ñ Ñ*ÛYÌ@ï@åX��Û.Ù�ÕqÔ"Ô�È�Ð.Ù@È�×3ò{È�ÐÊÉ�ËNÐÃûgþ�ÿ�Ô"ÇÎ×Y��È�ÇÎÍ�ûoï&ÿ�È�Ì3å�ÕqÍÎ×@È�ÐàëyÇÎÌIÑYÈ�Ð���9�y���=�����KJ�â:�^å,�������

S�/���0�>����TA���CFE ��13)���4�I�J+�/%�%É��� o ¿ ! �P�fI.�5�98}8G�R�7���>=nU,���:6����

0→ O×K,S → IS(K)→ CK → ClS(K)→ 0

UT�980→ O×

K,S → IK,S → CS(K)→ ClS(K)→ 0�Q][!$&FJ34��5JtI�J+�/J=�KA����IH�E 4HE 4HE )V*9#��yI�4}��� oÁIÛYÐàëlËNÍÎͳÈ�ÐÒëyÇÎÌIÈ�ÇÎÐÒÕqÐ�ÕqͳËNÏ�ÈvÙ³z�Ì3Ï�È�ÖYÐYdzÙAÜaäYÌAÑ*dzÈ

mZ ¼ #:�QE ).�V� È�Т×�ëydzïc{�È�ÍÎÐQé���È�Ç�Ñ.ÕqòvÛ

m =∏

p pnpÈ�ÇÎÐ

ùÒË�Ñ*ÛYÍ�éqêuåYÔ ËNÌ@Ñ*Ð.È�Ð"ëyÇÎÌ�Ñ*dzÈ�È�Ð.Ñ*ÍÎdzï�å.ÈyùàÈ�ÐYÏ�ÈSÑYÈ�Ìb��Ì3ÇÎÔ�Ù3×@È�ÍÎͳÈ�Ð"òvÛ�~�Ñ*dzÈ�ÇÎÐ

mÕqÛ*ÜoÏ�È�å.È�ÐQéNáçÇÎÌ?ÑYÈ��.ÐYdzÈ�Ì@È�Ð

Ñ*dzÈÝÍ³Ë {NÕqÍ�{�ËNÔ"Ó�Õ${�×@ÈG~�æ=ÕqÛ.Ù@ÑYËNÌl¦-Ù@ï@å.ÈG~�ÕqÖ9È�ͳÙ@ï�å.ÈWõ/Ì3ÛYÓYÓ9ÈIK,m :=

p|m

U(np)p .

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z�ÙIÈ�Ì3Ï�ÇÎÖY×IÙ3dzï�å ëydzÈvÑYÈ�ÌIÑ*dzÈ�È��.Õ${±×@È���Èn ¢Û.È�ÐYò0→ UK,S → I

mK

proj−−→ IK,m→ 0.

ûR°�ÿáúÈ�ÇÎ×@È�Ì3åYÇÎÐ ÑYÈ��.ÐYdzÈ�Ì@È�Ð6ëyÇÎÌyÕqÐ�ÕqͳËNÏ òvÛØí�ËNÌ3å.È�Ì3ÇÎÏ�È�ÔîÑ*dzÈÝõ/Ì3ÛYÓYÓ9È

CK,mÑ*ÛYÌ@ï�åÚÑ*dzȪz¯�.Õ${�×3å.È�ÇÎ×�ÑYÈ�ÌY��Èn ±Û.È�ÐYò

0→ O×K,m → IK,m → CK,m → 0.

ûV»�ÿ���9�y���=�:��� ö �B�Â�hâ�� ¿ �R����>=nU,���:6

0→ ImK/O

×K,m → CK → CK/C

mK → 0��I�J2�d][!$&$Jx4��=JbI/J+�/J=�MAB���@H�E 4HE 4HE )+*�#���I.4}��� o

A �CB�� Þ ä[� ��È�Ì/Æ/È��.ÐYÇÎ×3dzËNÐÊå�ÕqÖ!È�Ð ëyÇÎÌAÑ*dzÈ�È��YÕ${�×@È���Èn ¢Û.È�ÐYò0→ K×Im

K → IK → CK/CmK → 0.

ø�Õ${±×@ËNÌ3dzÙ3dzÈ�Ì@È�ÐèëyÇÎÌ�ÕqÛ.ÙAÑYÈ�ÐèÈ�Ì@Ù3×@È�ÐàÖ!È�dzÑYÈ�Ðúõ/ÍÎdzÈvÑYÈ�Ì3ÐK× å.È�Ì�ÕqÛ.Ù^~�Ù�ËØÈ�Ì3å�ÕqÍÎ×@È�ÐÒëyÇÎÌ/Ñ.ÕNÙ³z�Ì3Ï�È�ÖYÐYdzÙ=ÛYб×@È�Ì£�È{ÕNï�å¢×3ÛYÐYÏ�É�ËNÐ

K×ImK/K

× ∼= ImK/I

mK ∩K

× ∼= ImK/O

×K,m

é2

ÆAdzÈÝòvÛf�*Õq×3òØûl��驨*éK�G�&ÿÕqÐ�ÕqͳËNÏ�È�ÐÊÈ��.Õ${�×@È�ÐX��Èn ¢Û.È�ÐYò{È�ÐàÙ3ÇÎÐ.Ñ Ñ*dzÈ/ÜgËNÍÎÏ�È�Ð.ÑYÈ�Ð�����9�y���=�TÅ��KJ�â:�^å,�������

m =∏

p pnp�/����>XE�8 UB%ªUT�98

S8 �V�@���98,%'�R<�#:��>����BA��¥8���)X���

m! U��dA��.#:���08����12).�54�I�J+��%5%K�/� o ¿ !G�P��I����08 8 �V����>=/UP���T6F���

0→ O×K,m → I

mK → CK → CK/C

mK → 0UT�98

0→ O×K,m → IK,m → CS(K)→ CK/C

mK → 0�Q][!$&FJ34��5JtI�J+�/J=�KA����IH�E 4HE 4HE )V*9#��yI�4}��� o

A �CB�� Þ ä[� ÆAdzÈ�z¯�YÕ${�×3å.È�ÇÎ×IÑYÈ�ÌÈ�Ì@Ù3×@È�Ð���Èn ¢Û.È�ÐYò�ÜgËNÍÎÏ�×ÛYÐYÔ"ÇÎ×3×@È�ÍÎÖ�ÕqÌ=ÕqÛ.ÙY�QÈ�Ô"Ô�ÕÊûl��驨*éK�^¨�ÿdéÆAdzÈ�ê�Ð,{�ÍÎÛ.Ù3dzËNÐ

K×UK,S ⊆ K×Im

K

ÍÎdzÈdÜgÈ�Ì3×IÑ*dzÈÝÈ��.Õ${�×@È���Èn ¢Û.È�ÐYò0→ K×Im

K/K×UK,S → IK/K

×UK,S = CS(K)→ IK/K×Im

K = CK/CmK → 0.

ÞAÐ�ÕqͳËNÏ�ëydzÈÝÇÎÐf�QÈ�Ô"Ô�Õàûl��驨*éK�^²�ÿ�Ö9È�×3Ì�ÕNï@å¢×@È�ÐèëyÇÎÌIÑ*dzÈ�Ð�Õq×3äYÌ3ÍÎdzï@å.È�ÞIÖYÖYÇÎͳÑ*ÛYÐYÏImK

Ñ�«�½�ð−−→ IK,m/O

×K,m

~�Ñ*dzÈÕqÛ*ÜoÏ�Ì3ÛYÐ.ÑØÉ�ËNÐèûR°�ÿ�ÑYÈ�Ð ��È�Ì3Ð

UK,SO×K,m = UK,S(Im

K ∩K×) = Im

K ∩UK,SK× å�Õq×{é�Æ=ÕqÔ"ÇÎ×�È�Ì3å�ÕqÍÎ×@È�Ð�ëyÇÎÌÑ*dzÈ�ê�Ù@ËNÔ�ËNÌ3ÓYåYdzÈ

K×ImK/K

×UK,S ∼= ImK/(I

mK ∩ UK,SK

×) ∼= IK,m/O×K,m∼= CK,m

éYÆAdzÈ=òvë�È�ÇÎ×@È���Èn ±Û.È�ÐYòÜgËNÍÎÏ�×бÛYÐ Ô"ÇÎ×3×@È�ͳٳ��Û.Ù�ÕqÔ"Ô�È�Ð.Ù�È�×3ò{È�ÐÊÉ�ËNÐÃûV»�ÿlÔ"ÇÎ×IÑYÈ�ÌIÏ�È�Ì�ÕNÑYÈÝÈ�Ì3å�ÕqÍÎ×@È�Ð.È�Ð���Èn ±Û.È�ÐYò�é

2

���h}��h����¶å3s345�&�12&�6s��&å5/3 ,��0#!3 &¹*¹34#!�S�G&������/�z��&��'$z.�&�,��C64&��È�Ç

kÈ�ÇÎÐ6ÕqÍÎÏ�È�ÖYÌ�ÕqdzÙ�ï@å.È�ÌD�9ÕqåYÍK{[�NÌ3Ó9È�ÌyÛYÐ.Ñ

K|kÈ�ÇÎÐ.È/È�Ð.Ñ*ÍÎdzï�å.È=Ï¢ÕqͳËNdzÙ@Ù@ï�å.ȳz�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ.é.Æ/dzÈ/Í³Ë {NÕqͳȪ�=Í ÕNÙ,ö

Ù�È�Ð,{[�NÌ3Ó!È�Ì3×3å.ÈvËNÌ3Ç³È ûaÉ�Ï�Í�é�È�×uë�ÕèßkÁIö+����â+~0�ÝÕqÓYÇÎ×@È�Í0ê�ê,ÿyÍÎdzÈdÜoÈ�Ì3×AÜaäYÌ=È�Ð.Ñ*ÍÎdzï@å.È_��Ì3ÇÎÔ�Ù3×@È�ÍÎͳÈ�ÐpÉ�ËNÐ

kp ��CF! )���Z!G��J+���9!�O�O���%K8 UT�BA����

ÇÎТÉKp|kp

: H2(G(Kp|kp),K×p )

∼−→

1

[Kp : kp]Z/Z.

ø?äYÌÛYÐ.È�Ð.Ñ*ÍÎdzï�å.È���Ì3ÇÎÔ�Ù@×@È�ÍÎͳÈ�РdzÙ3×yТÛYÌAÑYÈ�ÌIø�ÕqÍÎÍKp = C

ÛYÐ.Ñkp = R

É�ËNÐ ðAÈ�ͳÈ�ÉNÕqÐYò�é.ø?äYÌyÇÎåYÐ ÑYÈ��.ÐYdzÈ�Ì@È�ÐëyÇÎÌ

ÇÎТÉC|R : H2(G(C|R),C×) ∼= H0(G(C|R),C×) ∼= R/R+

∼−→

1

2Z/Z

Ñ*ÛYÌ@ï�åàÑ.ÕNÙyí�ËNÌ3ò{È�dzï�å.È�ÐQé�ÞAÛ*ÜoÏ�Ì3ÛYÐ.ÑÚÉ�ËNÐX�*Õq×3òØûl��驨*é©­�ÿ3{[�NÐYÐ.È�ÐàëyÇÎÌyТÛYÐàÈ�ÇÎÐ.È�ÐàæAËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3ÔWÛ.ÙÇÎТÉ

K|k : H2(G(K|k), IK)→1

[K : k]Z/Z

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Ñ*ÛYÌ@ï�å6Ñ*dzÈ�ûoÈ�Ð.Ñ*ÍÎdzï�å.È&ÿ?�±ÛYÔ"Ô�ÈAÑYÈ�Ì�ê�бÉNÕqÌ3Ç Õqб×@È�Ð�ÕqÖYÖYÇÎͳÑ*ÛYÐYÏ�È�ÐàÙ@È�ÇÎÐ.È�Ì2�IËNÔ"Ó9ËNÐ.È�Т×@È�Ð ÑYÈ��.ÐYdzÈ�Ì@È�ÐQé,z�ÇÎÐ�ÜoÛYÐ*öÑ.ÕqÔ�È�б×�ÕqͳÈvÙDz�Ì3Ï�È�ÖYÐYdzÙ�ûaÉ�Ï�Í�é-ßkÁö+���yâ+~��*Õq×3ò�ê@ê@êdéjþ*éjþ�ÿ�Ö!ÈvÙ�ÕqÏ�×

c ∈ H2(G(K|k),K×) ⇒ÇÎбÉ

K|kc = 0,

ë�ËNÔ"ÇÎ×AÑYÈ�ÌIæ/ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3ÔWÛ.ÙÇÎТÉ

K|k : H2(G(K|k), CK)→1

[K : k]Z/Z

ë�ËNåYͳÑYÈ��.ÐYdzÈ�Ì3×AdzÙ@×{éYÞIÛ.ï�åÊÑ*dzÈvÙ@È�ÐÊæ/ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�Û.ÙAÐ.È�ÐYÐ.È�ÐÊëyÇÎÌDp ��CF! )��=! �9J+���9!�O^O/��%K8 UT�BA éÁIÛYÐ_{[�NÐYÐ.È�Ð"ëyÇÎÌ�Ñ.ÕNÙ?ÜgËNÍÎÏ�È�Ð.ÑYÈ?��å.ÈvËNÌ@È�Ô ÜoËNÌ3ÔWÛYÍÎdzÈ�Ì@È�Ð�~NÜoäYÌ0ÑYÈvÙ�Ù@È�ÐH£�È�ëlÈ�dzÙ�ëyÇÎÌ�ÕqÛ*Ü?ßkÁ�±á�â+~G��å.ÈvËNÌ@È�Ô�*éK��驨G¨:~�ÖYòvëÝé.ÕqÛ*ÜßkÁö+����â+~9�*Õq×3òWê@ê@êdé©°*驱:~±É�È�Ì3ë�È�dzÙ@È�ÐQé���9�y���=�Bó������"����$�B�`�������

k���5�¢! %kA��^O�)d! ��I/<�#,��)\�"!$#�%'&$( )+*,��)vUT�98

k���5�¢! %kA��^O�)d! ��I/<�#:�/) mO.I/<�#B%'U/a o C =

lim−→K|k

CKIn�/�¯8 �V� p 8��/%Ã&�%K!$I.In���BAG).U�*[*,��UT�98

G = G(k|k)8G�R��WY!G%ME ��IlAG).U�*[*,� o ¿ ! �P�S��I�J (G, C) ���5�0�Y��%M! IcIn�/�PZ

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u ∈ UKÛYÐ.Ñ

π : G(L|K)ab → G(T |K)abÑ*dzÈèÐ�Õq×3äYÌ3ÍÎdzï�å.ÈôÞAÖYÖYÇÎͳÑ*ÛYÐYÏ.éÆ�ÕqÐYРdzÙ3×

π((u,L|K)) = (u,L|K)G0(L|K) = (u, T |K) = ϕvK(u)T |K = 1

~YÑ.ÕqÔ"ÇÎ×Ï�ÇÎÍÎ×AÕqͳÙ@Ë(u,L|K) ∈ G0(L|K)

é£�È�×3Ì�ÕNï�å¢×@È�ÐàëyÇÎÌyТÛYÐτ ∈ G0(L|K)

Ô"ÇÎ×(a, L|K) = τ

ÜaäYÌyÈ�ÇÎÐa ∈ K× é*Æ�ÕNÙ�ÇÎÔ"ÓYÍÎÇÎòvdzÈ�Ì3× (a, T |K) =

ϕvK (a)T |K = 1

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b ∈ L× Ô"ÇÎ× vK(a) = fvL(b)ë�ìqåYͳÈ�ÐQéz�ÙAdzÙ@×

e · vK(NL|Kb) = vL(NL|Kb) = n · vL(b) = e · vK(a)ÛYÐ.ÑÒÙ@ËNÔ"ÇÎ×

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kp → CkAG��%MJ+Ê

NK|kCK ∩ k×p = NKp|kp

K×p

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a = (. . . , 1, xp, 1, . . . ) ∈ CK

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dzÙ3×yëlÈ�Ï�È�Ðh�*Õq×3òØûl��驨*éû�ÿyË$¦-È�Ð�~�ÕqͳÙ@Ë�È�ÇÎÐ.È7ÁAËNÌ3Ô�È�ÐYÏ�Ì3ÛYÓYÓ9È�é�Æ�ÕqÔ"ÇÎ×/Ù3ÇÎÐ.Ñ ÕqÛ.ï@åèÕqÍÎͳÈÒAÖ9È�Ì3Ï�Ì3ÛYÓYÓ9È�ÐfÁ/ËNÌ3Ô�È�ÐYÏ�Ì3ÛYÓYÓ!È�ÐQé|IÔ"Ï�È/{�È�åYÌ3×/Ù@È�ÇIÈ�ÇÎÐ.È7Á/ËNÌ3Ô�È�ÐYÏ�Ì3ÛYÓYÓ9ÈG~�ÕqͳÙ@Ë�È�ÇÎÐ.È�Ë$¦9È�Ð.È7|AТ×@È�Ì3Ï�Ì3ÛYÓYÓ9ÈÝÉ�ËNÐ

Cké.Æ/dzÈ�åP�Nå.È�Ì@È�Ð�z�ÇÎÐ.Ù3ö

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kp

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IéIdzÙ3×

ÕqͳÙ�Ë ÒAÖ!È�Ì3Ï�Ì3ÛYÓYÓ9È�É�ËNÐCmK

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2

Æ/È�Ì �*Õq×3ò6Ö9ÈvÙ�ÕqÏ�×�ÕqͳÙ@Ë,~�Ñ.Õ$�Úý3ÈvÑYÈ6È�Ð.Ñ*ÍÎdzï@å.È Ï¢ÕqͳËNdzÙ@Ù@ï@å.È ÛYÐ.Ñ ÕqÖ!È�ͳÙ@ï�å.Èvz�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏK|kÇÎÐ È�ÇÎÐ.È�Ô�0J=)l!$#B%Ã&�%K!$I.In���,& ( )V*P��)

km È�б×3å�ÕqÍÎ×@È�Ð"dzÙ3×{éqÆAdzÈvÙ0Ù@È�dzÈ�Ð Ñ*dzÈlòvÛ Cmk

Ï�È�åP�NÌ3ÇÎÏ�È�Ð_���NÌ3Ó9È�Ì{éqÆ�ÕqÔ"ÇÎ×�Ù3ÇÎÐ.ÑWÑ*dzÈyõÝÕqͳËNdzÙ,öõ=Ì3ÛYÓYÓ!È�ÐÊÑ*dzÈvÙ�È�Ì�z�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ�È�ÐÒÏ�È�Ì�ÕNÑYÈÝÑ*dzÈ��±×3Ì�ÕqåYÍK{�Í ÕNÙ@Ù@È�ÐYÏ�Ì3ÛYÓYÓ9È�Ð��

G(km|k) ∼= Ck/Cmk

ÞIͳÙY�AËNÐ.Ù@Èn ±Û.È�ÐYò�ÕqÛ.ÙD�*Õq×3òØûl��驨*éK�^±�ÿÝûoï&ÿ�È�Ì3å�ÕqÍÎ×@È�ÐÊëyÇÎÌÑYÈ�Ð���9�y���y�,ó��KJ�â:�^å,�������

k���5�r�"!$#�%'&$( )+*,��)g4��5J�!G% A���O�)l!G�yIn<�#:��4 mO.In<�#�%;U�a

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k o ���/Jy6F� m =∏

p∈S p o ¿ ! �P�¤��I�J�8��/)v�9J=)d!$#�%'&�%M! IcIn�/�,&$( )+*,��) km8 �R��4H!^] �54H! %É�!BOQ��%'I/<�#:��-Y!GU�ab��).#T! %ÉO

SUB�PC ��)d6 L ���KAGJ+��UT�98g�����0��).#T! %ÉO S 6�!$#�4 C$��)Q6 L ���KAGJ+�A�! %MEG�yI.I/<�#:��e3) L �/�5J+��)�UT�BA\CFE �

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G(km|k) ∼= Ck/Cmk .>f�5J

km(p)OQ�Q6F���R<�#��0��� L �5)78 �R�4H!^] �54_!G%K�7��� km

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G(km(p)|k) ∼= (Ck/Cmk )(p).

á�ÇÎÌ ë�ËNÍÎͳÈ�ÐHÐ.Ë�ï@å {±ÛYÌ3ò(ÕqÛ*Ü"È�ÇÎÐ.ÈÃÕqÍÎ×@È�Ì3Ð�Õq×3ÇÎÉ�ÈÃø.ËNÌ3ÔWÛYÍÎdzÈ�Ì3ÛYÐYÏ ÑYÈvÙÊðIÈ�òvÇÎÓYÌ@ËNòvÇÎ×�ìq×@Ù@Ï�ÈvÙ@È�×3ò{ÈvÙ ÜoäYÌ Ö9ÈdöÙ�ï@åYÌ�ìqÐ,{�×@ÈIí�È�Ì3òvë�È�ÇÎÏ�ÛYÐYÏ�È�ÇÎÐYÏ�È�å.È�ÐQé:��È�Ç

SÈ�ÇÎÐ.È/È�Ð.Ñ*ÍÎdzï@å.È/ùàÈ�ÐYÏ�ÈAÉ�ËNÐ ��Ì3ÇÎÔ�Ù@×@È�ÍÎͳÈ�Ð�É�ËNÐ

ké¢Æ/dzÈ/Æ/È��.ÐYÇÎ×3dzËNÐ

É�ËNÐCS(K) = IK/K

×UK,SdzÙ3×/Ñ.ÕNÑ*ÛYÌ@ï�å Ô�ËN×3ÇÎɱdzÈ�Ì3×^~-Ñ.Õ$��ÇÎÐ ÑYÈ�Ð¥�±×@È�ÍÎͳÈ�Ð

p 6∈ SÇÎÔ æAÇÎТÖYÍÎdzïQ{ ÕqÛ*Ü2�*Õq×3ò

ûl��驨*éK�^±�ÿ�ûaÖ�ÿÑ*Ç³È õ/Ì3ÛYÓYÓ9È�ÐUp

Ù3×@È�×@ÙAÇÎÐÒÑYÈ�ÐXÁAËNÌ3Ô�È�ÐYÏ�Ì3ÛYÓYÓ9È�Ð È�Т×3å�ÕqÍÎ×@È�Ð Ù3ÇÎÐ.Ñ-é�áúÈ�Ï�È�ÐK× ∩ UK,S = 1{[�NÐYÐ.È�ÐóëyÇÎÌ

UK,SÕqͳÙ_|AТ×@È�Ì3Ï�Ì3ÛYÓYÓ9ÈÚÉ�ËNÐ

CKÕqÛ:¦-ÕNÙ�Ù@È�ÐQé0ÞIÛ*ÜoÏ�Ì3ÛYÐ.ÑóÑYÈ�Ì�{�ËNå.ËNÔ�ËNͳËNÏ�dzÙ@ï@å.È�Т��Ì3ÇÎÉ±Ç ÕqÍÎÇÎ×�ìq×

É�ËNÐUK,S

ûoÙ3dzÈ�å.È��!éP­G­�ÿ�ÜgËNÍÎÏ�×IÕqÛ.ÙÑYÈ�ÌÈ��YÕ${�×@È�Ð���Èn ¢Û.È�ÐYò0→ UK,S → CK → CS(K)→ 0,

Ñ.Õ$� ÜaäYÌÕqÍÎͳÈi ∈ Z

Ï�ÇÎÍÎ×^�Hi(G(K|k), CK ) ∼= Hi(G(K|k), CS(K)).

ðIÛ*Üa×IÔ�ÕqÐÊÙ3dzï@åÊÜoÈ�Ì3Ð.È�ÌAÑYÈ�ÐÊÏ�ÛY×@È�ÐèõÝÕqͳËNdzÙ,ö�ÞIÖ.Ù@×3dzÈ�Ï�É�ËNÐCS(K)

ÇÎÐXz�Ì3ÇÎÐYÐ.È�Ì3ÛYÐYÏP~!È�Ì3å�ìqÍÎ×IÔ�ÕqÐÊÙ@ËqÜgËNÌ3×^~.Ñ.Õ$�(GS , CS)

È�ÇÎÐ.Ȫ�=Í ÕNÙ@Ù�È�Ð*ÜoËNÌ3Ô�Õq×3dzËNÐ6ÖYÇÎͳÑYÈ�×{é*Æ�ÕqÖ!È�Ç-Ù@È�ÇGSÑ*dzÈÝõ�ÕqͳËNdzÙ3ö�õ/Ì3ÛYÓYÓ!È=ÑYÈ�Ì�Ô�ÕF�*ÇÎÔ�ÕqͳÈ�Ð6ÕqÛ,��È�Ì3å�ÕqÍÎÖ

SÛYбÉ�È�Ì3òvëlÈ�ÇÎÏ�×@È�Ð�z�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ

kSÉ�ËNÐ

k~�ÛYÐ.Ñ

CS = lim−→CS(K)~�ëlËNÖ9È�ÇQÑYÈ�ÌY�-ÇÎÔ�ÈvÙ�äYÖ9È�ÌyÑ*dzÈ�È�Ð.Ñ*ÍÎdzï@å.È�Ð

Ï¢ÕqͳËNdzÙ�Ù@ï@å.È�Ð�z�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ�È�ÐKÉ�ËNÐ

kÍ ÕqÛ*ÜgÈG~�Ñ*dzÈ�ÇÎÐ

kSÈ�б×3å�ÕqÍÎ×@È�ÐèÙ@ÇÎÐ.Ñ-é�Æ/dzÈ�Á/ËNÌ3Ô�È�ÐYÏ�Ì3ÛYÓYÓ!È�ÐèÈ�Ì3Ï�È�Ö9È�Ð

Ù@dzï@å"ÕqÛ.Ù?ÑYÈ�Ð�Ï�È�ëx�NåYÐYÍÎdzï�å.È�Ð�Ñ*ÛYÌ@ï@å"ø�Õ${�×@ËNÌ3ÖYÇÎͳÑ*ÛYÐYÏ�Ð�ÕNï@åUK,SécÆ�ÕqÔ"ÇÎ×0Ù3ÇÎÐ.ÑWÈvÙ�Ï�È�Ì�ÕNÑYÈ�ëydzÈvÑYÈ�Ì?Ñ*dzÈ�Ë$¦-È�Ð.È�Ð|AТ×@È�Ì3Ï�Ì3ÛYÓYÓ9È�ÐàÈ�Ð.Ñ*ÍÎdzï�å.È�ÐÒêuÐ.ÑYÈ��YÈvÙ�É�ËNÐ

CS(k)é

���h}��hµ����å&���ø�#!����$z6s��&�#!3 �G#%$'©æ:7�än�^FâF��^�-÷�â,ä^äF�"�"���ÛYÐ�ìNï�å.Ù3×/ÜoËNÌ3ÔWÛYÍÎdzÈ�Ì@È�Ð ëyÇÎÌ/È�ÇÎÐ.È ÕqÖ.Ù3×3Ì�Õ${�×@ÈWø�ÕNÙ@Ù3ÛYÐYÏÚÑYÈvÙAæ=ÕqÛYÓY×3dzÑYÈ{ÕqͳÙ�Õq×3ò{ÈvÙ=ÇÎÐèÑYÈ�ÌAÇÎÔ ÞIÖ.Ù�ï@åYÐYÇÎ×3×���éK��é©°Ö9È�ТÛY×3òv×@È�ÐXÁAËN×�Õq×3dzËNÐQé��È�Ç

G = GkÈ�ÇÎÐ.È�ÓYÌ@Ë�È�Ð.Ñ*ÍÎdzï@å.ÈØõ=Ì3ÛYÓYÓ!È ÛYÐ.Ñ

CÈ�ÇÎÐ.È��=Í ÕNÙ@Ù@È�Ð*ÜgËNÌ3Ô�Õq×3dzËNÐúòvÛ

Gé9áçÇÎÌ/ÇÎÐ.Ñ*ÇÎòvdzÈ�Ì@È�ÐúÑ*dzÈ

ÕqÖYÏ�ÈvÙ�ï@åYͳË�Ù@Ù@È�Ð.È�ÐSÁ/ËNÌ3Ô�ÕqÍÎ×@È�ÇÎͳÈ�Ì�É�ËNÐGkÔ"ÇÎ×lÜgËNÌ3Ô�ÕqͳÈ�Ð\���NÌ3Ó9È�Ì3Ð

K|kÛYÐ.Ñ�Ù@ï�åYÌ@È�ÇÎÖ!È�Ð�ëydzÈvÑYÈ�Ì

G(L|K) =GK/GL

ÜoäYÌyòvëlÈ�Ç�ÜoËNÌ3Ô�ÕqͳÈ7�D�NÌ3Ó!È�ÌL|kÛYÐ.Ñ

K|kÔ"ÇÎ×

GL ⊆ GKéYÞAÛ,��È�Ì@ÑYÈ�Ô Ù@È�×3ò{È�Ð6ëyÇÎÌ

CK = CGKé

á�ÇÎÌ/Ù@È�×3ò{È�ÐàÇÎÐÒÑ*dzÈvÙ@È�Ô ÞIÖ.Ù@ï�åYÐYÇÎ×3×AÉ�ËNÌ�ÕqÛ.Ù�~�Ñ.Õ$��ÜaäYÌ=ÕqÍÎͳÈWÈ�Ð.Ñ*ÍÎdzï�å.È�ÐàÜgËNÌ3Ô�ÕqͳÈ�Ð����NÌ3Ó9È�Ì@È�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ�È�ÐK|kÑ*dzÈ�ÞAÖ!È�ÍÎdzÙ3dzÈ�Ì3ÛYÐYÏ

GabKÈ�Ð.Ñ*ÍÎdzï@åÊdzÙ@×{é

á�ÇÎÌx�,��dzÈ�Ì@È�Ð6È�ÇÎÐ.È/È�Ð.Ñ*ÍÎdzï@å.È/ÜgËNÌ3Ô�ÕqͳȪ���NÌ3Ó9È�Ì@È�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏK|kÛYÐ.Ñ�Ö9È�ò{È�dzï@åYÐ.È�Ð6Ô"ÇÎ×

K1ÑYÈ�Ðv���NÌ3Ó9È�Ì

Ô"ÇÎ×GabK = G(K1|K)

ÛYÐ.Ñ Ô"ÇÎ×K2ÑYÈ�ÐS�D�NÌ3Ó!È�ÌAÔ"ÇÎ×

GabK1= G(K2|K1).

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Á=ÕNï@åàÆ/È��.ÐYÇÎ×3dzËNÐàÈ�Ì3å�ÕqÍÎ×@È�Ð6ëyÇÎÌ^~.Ñ.Õ$�GK2 = G′

K1

È�ÇÎÐ.È|Iб×@È�Ì3Ï�Ì3ÛYÓYÓ!ÈÝÉ�ËNÐG′K

dzÙ@×^~*ÛYÐ.Ñ6Ñ.Õ$�GK1 = G′

KÏ�ÇÎÍÎ×ÚûoÑ.ÕqÖ!È�Ç�Ö!È�ò{È�dzï�å.È�ý3È�ëlÈ�ÇÎͳÙG′ Ñ*dzÈH�IËNÔ"ÔWÛY×�Õq×@ËNÌ3Ï�Ì3ÛYÓYÓ!È [G,G]

ÿdé�Æ=ÕqÌ�ÕqÛ.ÙWÙ@ï@åYÍÎdzÈ/��È�ÐÃëyÇÎÌ�Ñ*dzÈ�õ=ͳÈ�dzï@å*öå.È�ÇÎ×@È�Ð

G(K2|K)ab = G(K1|K)ÛYÐ.Ñ

G(K2|K)′ = G(K2|K1).Á/ÕNï�åX�*Õq×3òØûl��éK��é©­:�&ÿ�dzÙ@×Ñ.ÕNÙÆ/Ç ÕqÏ�Ì�ÕqÔ"ÔCK/NK1|KCK1

(·,K2|K)−−−−−→

∼G(K2|K)ab

i

y ù ´4« yCK1/NK2|K1

CK2

(·,K2|K1)−−−−−−→

∼G(K2|K1)

{�ËNÔ"ÔWÛY×�Õq×3ÇÎÉ0~�ëlËNÖ9È�ÇiÑYÈ�ÌÉ�ËNÐàÑYÈ�Ìê�Ð,{�ÍÎÛ.Ù3dzËNÐ

CK ↪→ CK1

ÇÎÐ.Ñ*ÛYòvdzÈ�Ì3×@È�æ/ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3ÔWÛ.ÙIdzÙ3×{éÆAdzÈ�Ö9È�Ì3äYåYÔ�×@ÈyÏ�Ì3ÛYÓYÓ9È�Т×3å.ÈvËNÌ@È�×3dzÙ�ï@å.Èí�È�Ì@Ù@dzËNÐ ÑYÈvÙ�æ=ÕqÛYÓY×3dzÑYÈ{ÕqͳÙ�Õq×3ò{ÈvÙ�É�ËNÐ"øYÛYÌ3×uë�ìqÐYÏ�ͳÈ�Ì=ûaɱÏ�Í�é*ßkÁIöu�0��â+~��å.ÈvËNÌ@È�Ô í/êdéû�é©°�ÿldzÙ3×Ñ.ÕNÙÜoËNÍÎÏ�È�Ð.ÑYÈ

���9�y���y��ô������"����$�B�`�������G���������/�98:%;�=<�#���)Q6���UGAGJ+�\W2)�U�*[*,�.-tInEH��I�J38 �V�ºÀP�/).%K!^A���)�UT�BA

ÀP�/): Gab → (G′)ab8G�R��J=)���C^�R! %É�ªmO^O��5%M8 UT�BA o

Æ=ÕqÌ�ÕqÛ.ÙIòvdzÈ�å.È�Ð ëyÇÎÌIÑ*dzÈ=ÜoäYÌyÛYÐ.ÙyÌ@È�ͳÈ�É�ÕqТ×@È��IËNÐ.Ù@Èn ±Û.È�ÐYò[����9�y���y��`���ñ}�9F�0ß5ß5â,B� ¿ ��)�CFE �@8���) p �,&F%'UBI.�RE �X���98GUG6n�R��)/J+�­H7E 4HE 4HE )+*�#���I.4�UBI

i : CK/NK1|KCK1 → CK1/NK2|K1CK2��I�JxJ=).�5C^�R! % o

áv�B:9vâP�"ì"� Þ ;���âPß5ä^â:�^å Þ �ú;��� �9ß5��7�âPß5�B�¤ñSß5â,ä^ä^�B�<F�ò9Fìs��F�����B�� Þ ���È�Ç

kÈ�ÇÎÐ��9ÕqåYÍK{[�NÌ3Ó9È�Ì/Ô"ÇÎ×Y�,�*dzÈ�Ì3×@È�Ô ÕqÍÎÏ�È�ÖYÌ�ÕqdzÙ@ï�å.È�Ð ÞIÖ.Ù@ï�åYÍÎÛ,�

k|kÛYÐ.Ñ

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kéLSÙ@È�Ç�Ñ*dzÈIÔ�ÕF��ÇÎÔ�ÕqͳÈAÛYТÉ�È�Ì3òvë�È�ÇÎÏ�×@Èz�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ"É�ËNÐ

k~¢Ñ*dzÈIÇÎÐ

SÉ�ËNÍÎÍ�ò{È�Ì3ͳÈ�Ï�×�dzÙ3×{é�z�ÇÎÐ.ÈIí�È�Ì3ÖYÇÎÐ.Ñ*ÛYÐYÏ

òvÛYÔ ÕqÖ.Ù3×3Ì�Õ${±×@È�Ðàø�ÕqÍÎÍ�È�Ì3å�ÕqÍÎ×@È�ÐÊëyÇÎÌÑ*ÛYÌ@ï�åàÑ*dzÈÝÆ=È��.ÐYÇÎ×3dzËNÐGk := G(LS |k)

ÛYÐ.ÑC := CLS

.

ùÊÇÎ×k1Ö!È�ò{È�dzï�åYÐ.È�Ð6ëyÇÎÌyÑ*dzÈIÔ�ÕF�*ÇÎÔ�ÕqͳÈÝÕqÖ!È�ͳÙ�ï@å.È/ÛYТÉ�È�Ì3òvë�È�ÇÎÏ�×@È�ÛYÐ.Ñ�ÇÎÐ

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pö+z�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ�È�ÐQéQÆ�ÕqÐYÐf{�ËNÔ"Ô�ÛY×3dzÈ�Ì@È�Ð

Ñ*dzÈ�ÆAÇ ÕqÏ�Ì�ÕqÔ"Ô�È

ClS(k)χ

←−−−∼

Ck/Nk1|kCk1 ClS(k)(p)χ

←−−−∼

(Ck/Nk1(p)|kCk1(p))(p)

µ�¬ ¥ y

i

y

ÛYÐ.Ñ µ�¬ ¥ y

i

y

ClS(k1)χ

←−−−∼

Ck1/Nk2|k1Ck2 ClS(k1(p))(p)χ

←−−−∼

(Ck1(p)/Nk2(p)|k1(p)Ck2(p))(p)

.

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ClS(k)→ ClS(k1)UT�98

ClS(k)(p)→ ClS(k1(p))(p)I����08}J=)���C��=! % o

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ÁIÛYÐ(Ù@È�ÇmÈ�ÇÎÐÃùÒË�Ñ*ÛYÍ�ÛYÐ.Ñ

Lm Ñ*dzÈ"Ô�ÕF��ÇÎÔ�ÕqͳÈgz�Ì3ë�È�ÇÎ×@È�Ì3ÛYÐYÏÒÉ�ËNÐ k ÇÎÐYÐ.È�Ì3å�ÕqÍÎÖÃÉ�ËNÐ k ~�Ù@ËÊÑ.Õ$� ÜoäYÌLm ⊃ K ⊃ k

Ù3×@È�×@ÙKab ⊂ km Ï�ÇÎÍÎ×{é.Æ/dzÈ=í�È�Ì3ÖYÇÎÐ.Ñ*ÛYÐYÏ�òvÛYÔ ÕqÖ.Ù@×3Ì�Õ${±×@È�ÐÊø�ÕqÍÎÍQëyÇÎÌ@Ñ6Ï�È�Ï�È�Ö9È�ÐÊÑ*ÛYÌ@ï@å

Gk := G(Lm|k)ÛYÐ.Ñ

C := CLm .

á�ÇÎÌ�Ù�ï@åYÌ@È�ÇÎÖ9È�Ðk1 := km ÛYÐ.Ñ k2 := km

1

ÜaäYÌ�Ñ*dzȳ�±×3Ì�ÕqåYÍK{±Í ÕNÙ@Ù�È�Ð,{[�NÌ3Ó!È�ÌlÔ�˱Ñ*ÛYͳËmÉ�ËNÐ

kÖYòvëÝé

k1ÛYÐ.Ñ

ëydzÈvÑYÈ�Ìk1(p)

ÖYòvëÝék2(p)

ÜoäYÌ�Ñ*dzÈyÇÎÐk1ÖYòvëÝé

k2È�б×3å�ÕqÍÎ×@È�Ð.È�Ð�Ô�ÕF�*ÇÎÔ�ÕqͳÈ�Ð�Ï¢ÕqͳËNdzÙ@Ù�ï@å.È�Ð

pö+z�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ�È�ÐQé

k2dzÙ3×yÏ¢ÕqͳËNdzÙ@Ù@ï�å äYÖ9È�Ì

ké,z�ÙY{�ËNÔ"ÔWÛY×3dzÈ�Ì@È�ÐàÑ*dzÈÝÆ/Ç ÕqÏ�Ì�ÕqÔ"Ô�È

J mk /P

mk

χ←−−−

∼Ck/C

mk (J m

k /Pmk )(p)

χ←−−−

∼(Ck/C

mk )(p)

µ�¬ ¥ y

i

y

ÛYÐ.Ñ µ+¬ ¥ y

i

y

J mk1/Pm

k1

χ←−−−

∼Ck1/C

mk1

(J mk1(p)/P

mk1(p))(p)

χ←−−−

∼(Ck1(p)/C

mk1(p))(p)

.

��ËNÔ"ÇÎ×AÈ�Ì3Ï�ÇÎÖY×IÙ3dzï�åÊÑ.ÕNÙ���9�y���y��j���ñ}�9F�0ß5ß5â,B� ¿ �V��0!GJ=NB)�%;�=<�#,����mO^O��5%M8 UT�BA��/�

J mk /P

mk → J

mkm/Pm

km

UT�98(J m

k /Pmk )(p)→ (J m

km(p)/Pmkm(p))(p)UT�98\8 �V�C$E �h8���) p �,&F%'UBI.�RE �X���98GUG6n�R��)/J+���SmO^O��5%M8GUB�BAB���

i : Ck/Cmk → Ckm/Cm

km

UT�98i : (Ck/C

mk )(p)→ (Ckm(p)/C

mkm(p))(p)I����08}J=)���C��=! % o�±Ó!È�òvdzÈ�ÍÎÍAÜaäYÌ

m = 1dzÙ3×ØÑYÈ�Ìv�±×3Ì�ÕqåYÍK{±Í ÕNÙ�Ù@È�Ð,{[�NÌ3Ó!È�Ì�Ï�È�Ì�ÕNÑYÈÒÑ*Ç³È Ô�ÕF�*ÇÎÔ�ÕqͳÈÒÕqÖ9È�ͳÙ@ï@å.ÈàÛYТÉ�È�Ì3òvë�È�ÇÎÏ�×@Èz�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏP~ ÇV&�%K�/���0�/) È Hª�5%KOQ�/)�J5In<�#:��) ��%K!$IcI^���,&$( )+*,��) Ï�È�Ð�ÕqÐYТ×{é�Æ/dzÈ�ÞIÛ.Ù�Ù�ÕqÏ�ÈØÑYÈvÙ &�%K!$I.I.��I/<�#:�/�¹H7! U�*0J=Z�R8��d!G%ÃIn!GJy6��.I dzÙ3×yбÛYÐ�~�Ñ.Õ$��ý3ÈvÑYÈvÙê�ÑYÈ{ÕqÍ-É�ËNÐ

kÇÎÔ æIÇÎÍÎÖ9È�Ì3×@Ù@ï�å.È�Ð��=Í ÕNÙ@Ù@È�Ð,{[�NÌ3Ó9È�Ì

k1È�ÇÎÐÊæ/ÕqÛYÓY×3dzÑYÈ{ÕqÍ�ëyÇÎÌ@Ñ-é

���h}��hÄ���ÅK6 �ô� �0#�3s6i$'¡%$z�G�G#%$'© û>ü��å6 �S

ó<,/3 3X©&��C3 &�5!$'&ÇÅ��'ë�&�6i$'&����0��5/&��áçÇÎÌÐ.È�åYÔ�È�РбÛYÐÊÑ*dzÈ�í�ËNÌ�ÕqÛ.Ù@Ù�È�×3òvÛYÐYÏ�È�ÐàÑYÈ�ÌIÞAÌ3Ö!È�ÇÎ×Ýß á�â�é•��È�Ç

kÈ�ÇÎÐÊÕqÍÎÏ�È�ÖYÌ�ÕqdzÙ@ï�å.È�Ì�9ÕqåYÍK{[�NÌ3Ó9È�ÌÛYÐ.Ñ

SÈ�ÇÎÐ.ÈÝùÒÈ�ÐYÏ�È�É�ËNÐS��Ì3ÇÎÔ�Ù@×@È�ÍÎͳÈ�ÐÊÉ�ËNÐ

• LSÙ@È�Ç�Ñ*dzÈ�Ô�ÕF�*ÇÎÔ�ÕqͳÈÝÛYбÉ�È�Ì3òvëlÈ�ÇÎÏ�×@È�ÛYÐ.Ñ ÇÎÐ

SÉ�ËNÍÎÍQò{È�Ì3ͳÈ�Ï�×@È�z�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ�É�ËNÐ

k~YÛYÐ.Ñ

LS(p)Ñ*dzÈ

Ô�ÕF�*ÇÎÔ�ÕqͳÈ�ÇÎÐLSÈ�б×3å�ÕqÍÎ×@È�Ð.È�Ï¢ÕqͳËNdzÙ�Ù@ï@å.È

pö+z�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ.é

•ø?äYÌ�È�ÇÎÐ.ÈÈ�Ð.Ñ*ÍÎdzï@å.È=õ�ÕqͳËNdzÙ,ö+z�Ì3ë�È�ÇÎ×@È�Ì3ÛYÐYÏ

K|kÙ@È�Ç

ES(K)Ñ*dzÈAõ/Ì3ÛYÓYÓ9ÈyÑYÈ�Ì

Sö+z�ÇÎÐYå.È�ÇÎ×@È�Ð

O×K,S

~NÛYÐ.ÑëyÇÎÌÙ@È�×3ò{È�Ð

ES(LS) := lim−→K|k,K⊆LS

ES(K)ÖYòvëÝé

ES(LS(p)) := lim−→K|k,K⊆LS(p)

ES(K).

ÞIÛ.ÙÑYÈ�Ô ÕqÖ.Ù3×3Ì�Õ${±×@È�ÐàÆAÛ�ÕqÍÎÇÎ×�ìq×@Ù�Ù�Õq×3òØû=��å.ÈvËNÌ@È�Ô2ûl��éK��é©­$®±ÿ3ÿÙ@ï@åYÍÎdzÈ/��È�ÐÊëyÇÎÌ���å.ÈvËNÌ@È�Ô ��éK�=É�ËNÐôß á�â�é���9�y���y��k������"����$�B�`�HÇÆ! È ËsNB)�!G%�%É�

i ∈ ZA �VO/J?�.I&$! �9EG�P�yIn<�#:��JVE.*:EG%MEQAG�yIn<�#:� p InE 4HE )V*9#��yI�4}���

Hi(

G(LS |k), ES(LS))

∼= H2−i(

G(LS |k),Q/Z)∨.

ÇlO È ËtNT)�! %5%É�i ∈ Z

A �VO/J2��I&$! �9E ���yIn<�#:��JVE.*:E %KEQA ��I/</#:� p I/EG4_EG)V*�#B�yI�4H�/�Hi

(

G(LS(p)|k), ES(LS(p)))

∼= H2−i(

G(LS(p)|k),Qp/Zp)∨.

Page 49: ¼ ½¿¾0À5¼ Áÿ½ÅÄ - Université du Luxembourgmath.uni.lu/~wiese/other/diplom.pdf · 2007-03-08 · ë ËNÖ!È Ç ˙ ÕqÛ*Ü A+ ×3Ì3ÇÎÉ Ç ÕqÍ ÛYÐ.ÑØÕqÛ*Ü

A �CB�� Þ ä[� áçÇÎÌWÔWä.Ù@Ù@È�ÐóÑ*dzÈ\£�ÈvÑYÈ�ÛY×3ÛYÐYÏúÑYÈ�Ì�ÇÎÔ ÕqÖ.Ù3×3Ì�Õ${�×@È�ÐóÆAÛ�ÕqÍÎÇÎ×�ìq×@Ù�Ù�Õq×3ò6É�È�Ì3ëlÈ�Ð.ÑYÈ�×@È�Ð`£�È�ò{È�dzï@å*öбÛYÐYÏ�È�Ð ÜgÈvÙ3×3ͳÈ�Ï�È�ÐQéø�äYÌ�È�ÇÎÐ.È"È�Ð.Ñ*ÍÎdzï�å.È Ï¢ÕqͳËNdzÙ@Ù@ï@å.È_z�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ

K ⊆ LSÉ�ËNÐ

kÖ9È�ò{È�dzï@åYÐ.È

CKÑ*dzÈ�ê�ÑYÈ�ÍK{�Í ÕNÙ@Ù@È�ÐYÏ�Ì3ÛYÓYÓ9È

ÛYÐ.ÑC0K

ÑYÈ�ÐS{�ËNÔ"Ó�Õ${�×@È�Ðf��È�Ì3ÐÊÑYÈ�ÌÞAÖ.Ù@ËNÍÎÛY×3Ð.ËNÌ3Ô é0��È�×3ò{È�ÐÊëyÇÎÌC = lim−→

LS⊃K|k

CKÛYÐ.Ñ

C0 = lim−→LS⊃K|k

C0K ,

Ù�Ë Ç³Ù3×C0 ͳÈ�É�È�Í öV{�ËNÔ"Ó�Õ${�×^~.ÛYÐ.Ñ ëyÇÎÌå�ÕqÖ!È�Ð Ñ*dzÈ�È��YÕ${�×@È���Èn ¢Û.È�ÐYò

0→ C0 → C → R+ → 0.

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Sö+z�ÇÎÐYå.È�ÇÎ×@È�Ð

ES(K)äYÖ9È�Ì3Ð.ËNÔ"Ô�È�ÐÃÛYÐ.ÑúÑ*dzÈ�É�ËNÐ

IKÑ*ÛYÌ@ï�å

IS(K)~*ëlÈ�ͳï�å.È�ÜoäYÌ

K ⊂ LSÐ�ÕNï@åX�IËNÌ@ËNÍÎÍ ÕqÌ�ûl��驨*éjþ�ÿ2{�ËNå.ËNÔ�ËNͳËNÏ�dzÙ@ï@åÒ×3Ì3ÇÎÉ±Ç ÕqÍ-dzÙ@×{é

ClKë�È�Ì@ÑYÈ�Ì@È�ÓYÌ�ìNÙ@È�Т×3dzÈ�Ì3×�Ñ*ÛYÌ@ï�å

ClS(K)~cë�È�ͳï@å.ÈvÙ�Ð�ÕNï@å �*Õq×3ò�ûl��驨*驨:�&ÿ�dzÙ@ËNÔ�ËNÌ3ÓYå"òvÛYÌ�õ�ÕqͳËNdzÙ,ö�õ=Ì3ÛYÓYÓ!È

ÑYÈ�ÌIÔ�ÕF��ÇÎÔ�ÕqͳÈ�Ð ÛYбÉ�È�Ì3òvëlÈ�ÇÎÏ�×@È�ÐúÕqÖ9È�ͳÙ@ï@å.È�Ð ÛYÐ.Ñ ÇÎÐSÉ�ËNÍÎÍ-ò{È�Ì3ͳÈ�Ï�×@È�Ðfz�Ì3ë�È�ÇÎ×@È�Ì3ÛYÐYÏØÉ�ËNÐ

KdzÙ@×{é

ÆAdzÈ�z�Ð.Ñ*ÍÎdzï@å,{�È�ÇÎ×/É�ËNÐClS(K)

å�ÕqÖ!È�Ð ëyÇÎÌIÕqÛ*Üx��È�ÇÎ×@È�­G­�Ö9È�ëydzÈvÙ@È�ÐQéÆAdzÈX�QÌ3ÇÎÉ�Ç ÕqÍÎÇÎ×�ìq×6É�ËNÐ

lim−→LS⊃K|k

ClS(K)ÛYÐ.Ñ

lim−→LS(p)⊃K|k

ClS(K)(p)ÜoËNÍÎÏ�×6ÕqÛ.ÙØÑYÈ�ÔÅæ=ÕqÛYÓY×3dzÑYÈ{ÕqͳÙ�Õq×3ò

û=�AËNÌ@ËNÍÎÍ ÕqÌ�ûl��驨*驨[»�ÿ3ÿdéÆAdzÈ�z¯�YÕ${�×3å.È�ÇÎ×AÑYÈ�̪��Èn ¢Û.È�ÐYò{È�ÐàÇÎÐàÑYÈ�ÔîÛYÐ.Ñ Ñ*dzÈ��IËNÔ"ÔWÛY×�Õq×3ÇÎɱÇÎ×�ìq×�ÑYÈvÙÆ/Ç ÕqÏ�Ì�ÕqÔ"Ô�Ù ûl��éK��é©­G­�ÿ�ÜgËNÍÎÏ�È�Ð

ÕqÛ.Ù³�*Õq×3òØûl��驨*éK�G�&ÿ�ÖYòvëÝéYÍ ÕNÙ�Ù@È�Ð Ù3dzï@åàÙ@ËqÜoËNÌ3×yÐ�ÕNï�åYÌ@Èvï@åYÐ.È�ÐQéÆ=ÕNÙ���å.ÈvËNÌ@È�Ô È�Ì3Ï�ÇÎÖY×IÙ3dzï�å ТÛYÐÒÕqÛ.Ù���å.ÈvËNÌ@È�Ô ûl��éK��é©­$®±ÿdé

2

���h}��Óý<��ÅK6 �ô� �0#�3s6i$'¡%$z�G�G#%$'© û>ü��-��&��S�C.0�z¡!����$'&¹òZ&��z©)ë�&�645<�0��5ÁAÛYÐ Ö9È�×3Ì�ÕNï@å±×@È�ÐÊëyÇÎÌIÑ*dzÈ/ÜgËNÍÎÏ�È�Ð.ÑYÈ��±ÇÎ×3Û�Õq×3dzËNÐ��•��È�Ç

pÈ�ÇÎÐ.ȳ��Ì3ÇÎÔ"ò&ÕqåYÍ-ÛYÐ.Ñ

kÈ�ÇÎÐ ÕqÍÎÏ�È�ÖYÌ�ÕqdzÙ@ï�å.È�ÌD�9ÕqåYÍK{[�NÌ3Ó!È�Ì^~�ëlÈ�ͳï�å.È�Ð ëyÇÎÌ�ÇÎÔ ø�ÕqÍÎͳÈ

p = 2ÕqͳÙ�×@ËN×�ÕqÍ

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•��È�Ç

SÈ�ÇÎÐ.ÈÝÈ�Ð.Ñ*ÍÎdzï�å.È�ùÒÈ�ÐYÏ�È�É�ËNÐ���Ì3ÇÎÔ�Ù3×@È�ÍÎͳÈ�ÐàÉ�ËNÐ

kÔ"ÇÎ×

S ∩ Sp = ∅,

ëlËNÖ9È�ÇSpÑ*dzÈÝùÒÈ�ÐYÏ�ÈÝÑYÈ�ÌyäYÖ9È�Ì

pÍÎdzÈ�Ï�È�Ð.ÑYÈ�Ð���Ì3ÇÎÔ�Ù@×@È�ÍÎͳÈ�ÐÊÉ�ËNÐ

kÙ@È�Ç�é

•ùÊÇÎ×

kSÖ9È�ò{È�dzï@åYÐ.È�ÐèëyÇÎÌ�Ñ*dzÈWÔ�ÕF��ÇÎÔ�ÕqͳÈ�Ï¢ÕqͳËNdzÙ@Ù@ï�å.È�z�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ É�ËNÐ

kÇÎÐk~!Ñ*dzÈ"ÕqÛ,��È�Ì3å�ÕqÍÎÖÒÉ�ËNÐ

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p~*ÛYбÉ�È�Ì3òvëlÈ�ÇÎÏ�×=dzÙ3×{é

áúÈ�ÇÎ×@È�ÌIÙ@È�ÇkS(p)

Ñ*dzÈ�Ô�ÕF��ÇÎÔ�ÕqͳÈG~�Ñ.ÕqÌ3ÇÎÐÊÈ�Т×3å�ÕqÍÎ×@È�Ð.ÈWÏ¢ÕqͳËNdzÙ@Ù@ï�å.Èpö+z�Ì3ë�È�ÇÎ×@È�Ì3ÛYÐYÏ.é�Á/ÕNï�å í�ËNÌ�ÕqÛ.Ù@Ù@È�×,ö

òvÛYÐYÏ"dzÙ3×kS(p)

äYÖ!È�ÌkÇÎÐÊÕqÍÎͳÈ�Ð���Ì3ÇÎÔ�Ù3×@È�ÍÎͳÈ�Ð

p ∈ Sò&ÕqåYÔîÉ�È�Ì3òvëlÈ�ÇÎÏ�×{é

•ø?äYÌÝÈ�ÇÎÐ.È"È�Ð.Ñ*ÍÎdzï@å.È�Ï¢ÕqͳËNdzÙ�Ù@ï@å.ÈHz�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ

kS ⊃ K|kÙ@ï�åYÌ@È�ÇÎÖ!È�ÐúëyÇÎÌ

ES(K) := O×k,m

ÜoäYÌ�ÑYÈ�ÐùÒ˱Ñ*ÛYÍ

m :=∏

p∈S pé

áçÇÎÌÙ@È�×3ò{È�ÐES(kS) := lim−→

kS⊃K|k

ES(K)ÛYÐ.Ñ

ES(kS(p)) := lim−→kS(p)⊃K|k

ES(K).

���9�y���ÉÅ�m������"����$�B�`��ËtNT)�! %5%É�i ∈ Z

A �VO/J?�.Iª&$! �9E ���yIn<�#:��JVE.*:E %KEQA ��I/</#:� p I/EG4_EG)V*�#B�yI�4H�/�Hi

(

G(kS(p)|k), ES(kS(p)))

∼= H2−i(

G(kS(p)|k),Qp/Zp)∨.

A �CB�� Þ ä[� á�ÇÎÌyÔWä.Ù@Ù@È�ÐàÕqÛ.ï�åÊåYdzÈ�ÌÑ*dzÈ7ÐAÖ!È�Ì@Ù�È�×3òvÛYÐYÏ�È�Ð òvÛYÔ ÕqÖ.Ù3×3Ì�Õ${�×@È�ÐÊø�ÕqÍÎÍ�ÕqÐYÏ�È�Ö9È�ÐQéø�äYÌ�È�ÇÎÐ.È"È�Ð.Ñ*ÍÎdzï�å.È Ï¢ÕqͳËNdzÙ@Ù@ï@å.È_z�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ

K ⊆ kSÉ�ËNÐ

kÙ@È�Ç�ëydzÈvÑYÈ�Ì

CKÑ*dzÈ�ê�ÑYÈ�ÍK{�Í ÕNÙ@Ù@È�ÐYÏ�Ì3ÛYÓYÓ9È

ÛYÐ.ÑC0K

ÑYÈ�ÌY{�ËNÔ"Ó�Õ${�×@È7��È�Ì3ÐÊÑYÈ�ÌIÞAÖ.Ù@ËNÍÎÛY×3Ð.ËNÌ3Ô é0��È�×3ò{È�ÐÊëyÇÎÌIÕqÛ.ï@åÊåYdzÈ�ÌC = lim−→

kS⊃K|k

CKÛYÐ.Ñ

C0 = lim−→kS⊃K|k

C0K ,

Page 50: ¼ ½¿¾0À5¼ Áÿ½ÅÄ - Université du Luxembourgmath.uni.lu/~wiese/other/diplom.pdf · 2007-03-08 · ë ËNÖ!È Ç ˙ ÕqÛ*Ü A+ ×3Ì3ÇÎÉ Ç ÕqÍ ÛYÐ.ÑØÕqÛ*Ü

Ù�Ë Ç³Ù3×C0 ͳÈ�É�È�Í öV{�ËNÔ"Ó�Õ${�×^~.ÛYÐ.Ñ ëyÇÎÌå�ÕqÖ!È�Ð Ñ*dzÈ�È��YÕ${�×@È���Èn ¢Û.È�ÐYò

0→ C0 → C → R+ → 0.

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ëyÛYÌ@ÑYÈ�éClKë�È�Ì@ÑYÈÚÌ@È�ÓYÌ�ìNÙ@È�Т×3dzÈ�Ì3×�Ñ*ÛYÌ@ï@å Ñ*dzÈÊõ/Ì3ÛYÓYÓ9È

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~?ëlÈ�ͳï�å.È6Ð�ÕNï@åÎ�*Õq×3òôûl��驨*驨$®±ÿWdzÙ�ËNÔ�ËNÌ3ÓYåóòvÛYÌõÝÕqͳËNdzÙ,ö�õ/Ì3ÛYÓYÓ9ÈØÑYÈ�ÌWÔ�ÕF��ÇÎÔ�ÕqͳÈ�ÐóÕqÛ,��È�Ì3å�ÕqÍÎÖ

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lim−→kS⊃K|k

CK/CmK

ÛYÐ.Ñlim−→

kS(p)⊃K|k

CK/CmK(p)ÜoËNÍÎÏ�×ÕqÛ.ÙyÑYÈ�Ô æ=ÕqÛYÓY×3dzÑYÈ{ÕqͳÙ�Õq×3ò�û=�IËqö

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ÕqÛ.Ù³�*Õq×3òØûl��驨*éK�^­�ÿ�ÖYòvëÝéYÍ ÕNÙ�Ù@È�Ð Ù3dzï@åàÙ@ËqÜoËNÌ3×yÐ�ÕNï�åYÌ@Èvï@åYÐ.È�ÐQéÆ=ÕNÙ���å.ÈvËNÌ@È�Ô È�Ì3Ï�ÇÎÖY×IÙ3dzï�å ТÛYÐÒÕqÛ.ÙIÑYÈ�Ôîø�ÕqÍÎÍlûaÖ�ÿ�É�ËNÐ���å.ÈvËNÌ@È�Ô ûl��éK��é©­$®±ÿdé

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2

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H1(G,Z/prZ) = Hom(G/Gpr [G,G],Z/prZ) = (G/Gpr [G,G])∨

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S4�������4H! %'-"InE�AG��%MJ

|S| = dimFpH1(N)G.

ê�Ô ÜgËNÍÎÏ�È�Ð.ÑYÈ�Ð6ëlËNÍÎͳÈ�Ð ëyÇÎÌÑYÈ�ÐS£�È�Ï�Ì3ÇM¦ ÑYÈ�Ì�ÜaÌ@È�dzÈ�ÐS��Ì@Ëqöpö�õ/Ì3ÛYÓYÓ!È�È�ÇÎÐ*ÜaäYåYÌ@È�Ð6ÛYÐ.ÑÚÇÎåYÐ6Ö9È�ТÛY×3ò{È�Ð�~YÛYÔ

ÑYÈ�Ðf£�È�Ï�Ì3ÇM¦úÑYÈ�ÌðAÈ�Í Õq×3dzËNÐ òvÛÊÑYÈ��.ÐYdzÈ�Ì@È�Ð ÛYÐ.Ñ6òvÛàï@å�ÕqÌ�Õ${�×@È�Ì3dzÙ3dzÈ�Ì@È�ÐQé�Q���=���yô0��áv�Cy¯� Þ � Þ �0�t�����/�

X���5�0� >@���BA�� o e3�5�0� ÜoÌ@È�dzÈg��Ì@Ëqö p ö�õ/Ì3ÛYÓYÓ9È ! U.� X ��I�J7���5�0�H13)dE Z

pZ.W2)�U^*�*,�

FX6/UBI/!G4�4}����4��=Jx���5�0��)³mO^O/��%K8 UT�BA

i : X → FX-bI/Eg8[!daÂA���%MJ+���0Ê

Ç+� È�� �d8���Ed�D���0�f���9J+��)=AG).U�*[*,�U ≤ FX

����J5#T� %MJ9�/!$I/Jx! %5%É�De3%É��4}����J+��C$E �i(X) oÇV�5� È ËtNT)2¡n�Q8���13)dE Z

pZ.W2)�U^*[*P�

G4��=JsmO^O��5%K8 UB�TA

j : X → GA �ROnJD�.I7A����9! U¥���5�0���-H7E 4HE 4HE )+*�#���I.4�UBI

f : FX → G-bInE 8[!Qa

f ◦ i = jA �5%;J o

ùÒÕqÐ�È�Ì3å�ìqÍÎ×lÈ�ÇÎÐ.È�ÜoÌ@È�dzÈ���Ì@Ëqöpö�õ=Ì3ÛYÓYÓ!ÈAÑ*ÛYÌ@ï@å\£lÇÎͳÑ*ÛYÐYÏ�ÑYÈvÙ�ÓYÌ@Ëqý3È/{±×3ÇÎÉ�È�Ðg�-ÇÎÔ�ÈvÙläYÖ!È�ÌlÕqÍÎͳȭÈAÛ.ËN×3dzÈ�Т×@È�Ð

F/U~�ëlËNÖ9È�Ç

FÑ*dzÈ�Ï�È�ëx�NåYÐYÍÎdzï�å.ÈWÜaÌ@È�dzÈWõ=Ì3ÛYÓYÓ!È�Ö9È�ò{È�dzï@åYÐ.È�×^~�ÛYÐ.Ñ

UÑ*dzÈ�ÁAËNÌ3Ô�ÕqÍÎ×@È�ÇÎͳÈ�ÌAÉ�ËNÐ

FÉ�ËNÐÊÈ�Ð.Ñ�ö

ÍÎdzï�å.È�Ô êuÐ.ÑYÈ��ÚÑ*ÛYÌ@ï�åYÍ ìqÛ*Üa×^~�Ñ*dzÈ=ÜgÕNÙ@×yÕqÍÎͳÈ�z�ͳÈ�Ô�È�Т×@Èx ∈ X

È�б×3å�ÕqÍÎ×@È�Ð�~�ÛYÐ.Ñ�ÜoäYÌÑ*dzÈF/U

È�ÇÎÐ.Èpö�õ/Ì3ÛYÓYÓ9È

dzÙ@×{éqÆ/dzÈ?z¯�*dzÙ3×@È�ÐYòyÛYÐ.Ñ�z�ÇÎÐ.ÑYÈ�ÛY×3ÇÎÏG{�È�ÇÎ×�ÑYÈvÙ�æAËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3ÔWÛ.Ùf : FX → G

ÜoËNÍÎÏ�×�Ñ.ÕqÐYÐ"ÇÎÔ áúÈvÙ@È�б×3ÍÎdzï@å.È�ÐÕqÛ.ÙWÑYÈ�ÌWÙ3×@È�×3ÇÎÏ�È�Ð(ø.ËNÌ3×@Ù@È�×3òvÛYÐYÏ ÕqÛ*Ü

FXÑYÈvÙ�È�ÇÎÐ.ÑYÈ�ÛY×3ÇÎÏ�È�Ð÷æ/ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�Û.Ù ÑYÈ�Ì�Ï�È�ë3�NåYÐYÍÎdzï@å.È�Ð÷ÜoÌ@È�dzÈ�Ð

õ=Ì3ÛYÓYÓ!È�ÕqÛ*ÜXÐ�ÕNï�å

ÞIÛ.Ù�ÑYÈ�Ì�ÛYÐYÇÎÉ�È�Ì@Ù@È�ÍÎͳÈ�Ðvz�ÇÎÏ�È�Ð.Ù�ï@å�ÕcÜo×�ÑYÈ�ÌlÜaÌ@È�dzÈ�Ðg��Ì@Ëqöpö�õ/Ì3ÛYÓYÓ9È�Ð�Ù�ï@åYÍÎdzÈ/��È�ÐÚëyÇÎÌ^~¢Ñ.Õ$� ÈvÙ�òvÛ�É�ËNÌ3Ï�È�Ï�È�Ö!Èdö

Ð.È�ÌD��Ì@Ëqöpö�õ/Ì3ÛYÓYÓ9È

GÔ"ÇÎ×�z�Ì3ò{È�ÛYÏ�È�Ð.ÑYÈ�Ð.Ùd��Ù@×@È�Ô

SÈ�ÇÎÐ.ÈÝÈ��.Õ${±×@È���Èn ±Û.È�ÐYò

1→ R→ F → G→ 1

Ï�ÇÎÖY×^~.ëlËNÖ9È�ÇFÑ*dzÈ/ÜoÌ@È�dzÈ7��Ì@Ëqö

pö�õ/Ì3ÛYÓYÓ9È�ÕqÛ*Ü

SdzÙ3×{éz�ÇÎÐ�« ��%K!GJ=�=EG�0���,I�wFI/J+��4

RÉ�ËNÐ

GÖYòvÏ�Í�é

SdzÙ3×�È�ÇÎÐvz�Ì3ò{È�ÛYÏ�È�Ð.ÑYÈ�Ð.ÙQ��Ù3×@È�Ô É�ËNÐ

RÕqͳÙ2ÁAËNÌ3Ô�ÕqÍÎ×@È�ÇÎͳÈ�ÌyÉ�ËNÐ

Fé�ÞIÛ.ï�å6ÜaäYÌÑ*dzÈvÙ�ÈvÙyå�ÕqÖ!È�ÐÊëyÇÎÌÈ�ÇÎÐ.È{�ËNå.ËNÔ�ËNͳËNÏ�dzÙ@ï@å.È"ülå�ÕqÌ�Õ${�×@È�Ì3dzÙ3dzÈ�Ì3ÛYÐYÏ.é

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p

�Q���=���a`��KJ�â:�^å,�������G�/���0� ���98:%;�R<�#@��)Q6��/U AGJ+��13)dE Z

pZ.W2)�U^*�*,�_4��=J24����P�54H! %É��4¬e3)d6F��UGA����98��/�,I.w$I�J+�/4

S o¿ ! �P�vA �5%;J9��NB)����5�X4��5�P�54H! %K��I « �/%M!GJ=�RE �0�/�,I.w$I�J+�/4ROQ6QA % o S

|R| = dimFpH2(G).

¿ �V�.In�7�"!$#�%¯OQ�Q6��/�=<�#B�0��� L �5)4��=J r(G)UB�98}�0�����0���fI.�V��8B��� ðAÈ�Í Õq×3dzËNÐ.È�ÐYÌ�ÕqÐYÏ CFE �

G oA �CB�� Þ ä[� á�ÇÎÌyÖ9È�×3Ì�ÕNï@å±×@È�ÐÒÑ*dzÈÝø?äYÐ*Üaö+��È�Ì3Ô�Èdöu��Èn ±Û.È�ÐYò�ÕqÛ.Ùª�*Õq×3òØûR¨*驨*éû�ÿ

0→ H1(G)→ H1(F )→ H1(R)G → H2(G)→ H2(F ).

Æ�Õ Ð�ÕNï�åôßkÁ�±áãâ+~,��Ì@ËNÓ9Ë�Ù3ÇÎ×3dzËNÐ�­*éjþ*é©�:~H2(F ) = 0

Ï�ÇÎÍÎ×^~�È�Ì3å�ÕqÍÎ×@È�Ð ëyÇÎÌòvÛ.Ù�ÕqÔ"Ô�È�Ð Ô"ÇÎת�*Õq×3òØûR¨*éK��é ®±ÿ0 = dimFpH

1(G)− dimFpH1(F ) + dimFpH

1(R)G − dimFpH2(G)

= d(G) − |S|+ |R| − dimFpH2(G).

2á�ÇÎÌë�È�Ì@ÑYÈ�ÐàÙ3Ó�ìq×@È�Ì�ÜgËNÍÎÏ�È�Ð.ÑYÈ��*ìq×3ò{È�Ö9È�ÐP�N×3ÇÎÏ�È�ÐQé�Q���=���5ø��KJ�â:�^å,�������

G�/���0��/�98:%;�=<�# ��)Q6��/U AGJ+�D13)dE Z

pZ�W2).U�*[*,�4��5J¯��������) « �/%M!GJ=�RE �,-¯8 o # o r(G) = 1 o ¿ !G�P���I�J

d(G) = 1E^8���)

G#:!GJ

Zp! %ÃI_q2U,EGJ=�R�/��J+��� o

A �CB�� Þ ä[� êuÙ3×r(G) = 1

~YÙ@Ë��.Ð.ÑYÈ�ÐàëyÇÎÌÈ�ÇÎÐ.ÈÝÈ��.Õ${±×@È���Èn ±Û.È�ÐYò1→ R→ F → G→ 1,

ë�ËNÖ!È�ÇRÉ�ËNÐàÈ�ÇÎÐ.È�Ô z�ͳÈ�Ô�È�Т×�ÕqͳٳÁAËNÌ3Ô�ÕqÍÎ×@È�ÇÎͳÈ�Ì=ÑYÈ�ÌÜoÌ@È�dzÈ�Ð���Ì@Ëqö

pö�õ=Ì3ÛYÓYÓ!È

FûaÖYòvÏ�Í�é�È�ÇÎÐ.È�Ô Ô"ÇÎÐYÇÎÔ�ÕqͳÈ�Ðz�Ì3ò{È�ÛYÏ�È�Ð.ÑYÈ�Ð.Ùd��Ù@×@È�Ô

SÉ�ËNÐ

Gÿ�È�Ì3ò{È�ÛYÏ�×�ëyÇÎÌ@Ñ-é�Á=ÕNï@å\�QÈ�Ô"Ô�ÕØûR¨*éK��é©­�ÿ�dzÙ3×

RF ′/F ′ È�Т×uëlÈvÑYÈ�Ì�×3Ì3ÇÎÉ�Ç ÕqÍ!Ë�ÑYÈ�ÌëyÇÎÌ@Ñ É�ËNÐÊÈ�ÇÎÐ.È�Ô¬z�ͳÈ�Ô�È�б×/ÕqͳÙY|AТ×@È�Ì3Ï�Ì3ÛYÓYÓ9ÈÝÉ�ËNÐF/F ′ = F ab

È�Ì3ò{È�ÛYÏ�×{é�Æ/dzÈ���Èn ¢Û.È�ÐYò1→ RF ′/F ′ → F ab → Gab → 1

dzÙ@×�È��.Õ${�×^~*ÛYÐ.ÑÚÕqÍÎͳÈ=É�ËNÌd{�ËNÔ"Ô�È�Ð.ÑYÈ�Ð õ/Ì3ÛYÓYÓ9È�Ð Ù3ÇÎÐ.ÑZpö�ùàË�Ñ*ÛYÍÎÐ�~*ëlËNÖ9È�Ç

rkZp Fab = |S|

dzÙ@×{éYÆ=Õqå.È�Ì�Ï�ÇÎÍÎ×rkZpG

ab ≥ |S| − 1~.ÛYÐ.Ñ6Ñ*dzÈ7£�È�å�ÕqÛYÓY×3ÛYÐYÏ�ÜgËNÍÎÏ�×{é

2

�Q���=���aj��KJ�â:�^å,�������G���5�0�����98:%;�R<�#S�/)d6F��U A[J+�713)dE Z

pZ�W2).U�*[*,� o ¿ ! �P�X�yI/Jx8 �V�ª���>=/UP���T6

0→ (pGab)∨ → H2(G,Z/pZ)→ pH

2(G,Qp/Zp)→ 0

�Q][!$&FJ o p �,InOQ��I/E �98B��)d�_Oc��)d�Q<�#��0�/J?I.�R<�#v8B��) « ��%K!GJ=�RE �0����)l! �TA}C$E � G 6nUr(G) = dimFp pG

ab + dimFp pH2(G,Qp/Zp).

A �CB�� Þ ä[� á�ÇÎÌyÏ�È�å.È�ÐàÕqÛ.ÙIÉ�ËNÐ ÑYÈ�ÌIÈ��.Õ${±×@È�ÐX��Èn ±Û.È�ÐYò0→

1

pZ/Z→ Qp/Zp

p−→Qp/Zp → 0.

Æ/dzÈvÙ@È�ÌAËNÌ@Ñ*Ð.È�Ð ëyÇÎÌIÑ*dzÈ=Í ÕqÐYÏ�ÈÝÈ��.Õ${±×@È���Èn ¢Û.È�ÐYòH1(G,Qp/Zp)

p−→H1(G,Qp/Zp)→ H2(G,Z/pZ)→ H2(G,Qp/Zp)

p−→H2(G,Qp/Zp)

òvÛQé�ÆAdzÈ���Èn ±Û.È�ÐYò0→ pG

ab → Gabp−→GabdzÙ@×È��YÕ${�×^~YëlËNÌ�ÕqÛ.ÙIëyÇÎÌIÑ*ÛYÌ@ï@åàÆAÛ�ÕqÍÎdzÙ@dzÈ�Ì3ÛYÐYÏØÑ*dzÈ�È�Ö9È�Ð*ÜgÕqÍÎͳÙAÈ��YÕ${�×@È���Èn ¢Û.È�ÐYò

H1(G,Qp/Zp)p−→H1(G,Qp/Zp)→ H1(pG

ab,Qp/Zp)→ 0

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È�Ì3å�ÕqÍÎ×@È�ÐQé�Æ=ÈvÙ3å�ÕqÍÎÖS{[�NÐYÐ.È�Ð ëyÇÎÌIÙ�ï@åYÍÎdzÈ/��È�Ð�~�Ñ.Õ$�0→ H1(pG

ab,Qp/Zp)→ H2(G,Z/pZ)→ pH2(G,Qp/Zp)→ 0

È��.Õ${�×ydzÙ3×{éYáúÈ�Ï�È�ÐH1(pG

ab,Qp/Zp) = Hom(pGab,Qp/Zp) = (pG

ab)∨

È�Ì3Ï�ÇÎÖY×AÙ3dzï@åÊÑYÈ�̪�*Õq×3ò�é2

Æ/È�ÌÜoËNÍÎÏ�È�Ð.ÑYÈ��*Õq×3ò�dzÙ3×È�ÇÎÐ ÞIÐ�ÕqͳËNÏ�ËNÐÊòvÛY̳�!È�Ì3ͳÈ�Ï�ÛYÐYÏØÈ�ÇÎÐ.ÈvÙùÒË�Ñ*ÛYͳÙyäYÖ!È�ÌÈ�ÇÎÐ.È�Ô æ/ÕqÛYÓY×3dzÑYÈ{ÕqÍÎÌ3ÇÎÐYÏØÇÎÐÈ�ÇÎÐ.È�Ð ÜaÌ@È�dzÈ�Ð ÛYÐ.ÑÊÈ�ÇÎÐ.È�Ð���ËNÌ@Ù3dzËNÐ.Ù@×@È�ÇÎÍ�é�Q���=���ak��KJ�â:�^å,�������

G���5�0�����98:%;�R<�#S�/)d6F��U A[J+�713)dE Z

pZ�W2).U�*[*,� o ¿ ! �P�vA ��%MJ+Ê

d(G) = dimFp(G/Φ(G)) = rkZp(Gab) + dimFp(pG

ab).

A �CB�� Þ ä[� áúÈ�Ï�È�Ðd(Gab) = d(G)

Ñ*äYÌ,ÜgÈ�ÐôëyÇÎÌGÕqͳÙWÕqÖ9È�ͳÙ@ï�åôÉ�ËNÌ�ÕqÛ.Ù@Ù@È�×3ò{È�ÐQé?Æ�ÕqÐYÐôdzÙ3×

GÑ*ÇÎÌ@È/{�×@ÈvÙ��Ì@˱Ñ*Û,{�×�É�ËNÐ

r�IËNÓYdzÈ�Ð6É�ËNÐ

ZpÛYÐ.Ñ

s = d(G)− rõ/Ì3ÛYÓYÓ9È�Ð6ÑYÈ�Ì�ø�ËNÌ3Ô

Z/peiZé*Æ�ÕqÖ!È�Ç!dzÙ3×

rÑYÈ�Ìyð/ÕqÐYÏ

É�ËNÐGÕqͳÙ

Zpö�ùÒ˱Ñ*ÛYÍ�é9ÞAÛ*ÜaÏ�Ì3ÛYÐ.ÑàÉ�ËNÐ

pZp = 0Ï�ÇÎÍÎ×

pG = p(∏

Z/peiZ)é!ø?äYÌ/Ñ*dzÈ

Fpö�Æ/ÇÎÔ�È�Ð.Ù3dzËNÐèÜgËNÍÎÏ�×

бÛYÐdimFp(pG) = s

é2

}d� �!�h}����®6s3i$'�C64&��C�0�05�&��éó<,/���Z�z,!(p(¸äå�C�0�0��&��

ægß=ß5���B�¥� Þ �"��äz�ÇÎÐ.È bt%MJ=).�V��)/J+�}W2)�U�*[*,� dzÙ3×È�ÇÎÐ.È�õ/Ì3ÛYÓYÓ9È

GòvÛ.Ù�ÕqÔ"Ô�È�Ð Ô"ÇÎ×AÈ�ÇÎÐ.È�ÌIÞIÖYÖYÇÎͳÑ*ÛYÐYÏ

ω : G→ N ∪ {∞},

Ù�Ë"Ñ.Õ$�"ÜaäYÌIÕqÍÎͳÈx, y ∈ G

ω(xy−1) ≥ min(ω(x), ω(y))

Ï�ÇÎÍÎ×{é�Æ/È��.ÐYdzÈ�Ì@È�ÐàëyÇÎÌ�ÜaäYÌ�ý,ÈvÑYÈvÙn ∈ N

Ñ*dzÈÝùÒÈ�ÐYÏ�È

Gn := {x ∈ G| ω(x) ≥ n},

Ù�Ë"È�Ì3å�ÕqÍÎ×@È�ÐÊëyÇÎÌIÈ�ÇÎÐ.ÈÝÕqÖ.Ù3×@È�ÇÎÏ�È�Ð.ÑYÈ���È�×3×@È�É�ËNÐ�|AТ×@È�Ì3Ï�Ì3ÛYÓYÓ9È�ÐÊÉ�ËNÐGé|IÔ"Ï�È/{�È�åYÌ3×/ÑYÈ��.ÐYdzÈ�Ì3×IÈ�ÇÎÐ.ÈÝÙ@ËNͳï�å.È���È�×3×@È�ÕqÛ.ï�åÊÈ�ÇÎÐ.ÈÝø?ÇÎÍÎ×3Ì3dzÈ�Ì3ÛYÐYÏ�É�Ç Õ

ω(x) := max{n| x ∈ Gn}.

ê�Ðúßk�-ò{âQëyÇÎÌ@Ñ6É�ËNÐÊÈ�ÇÎÐ.È�ÌIø?ÇÎÍÎ×3Ì3dzÈ�Ì3ÛYÐYÏ�òvÛ.Ù�ìq×3òvÍÎdzï@åÊÏ�ÈdÜgËNÌ@ÑYÈ�Ì3×^~.Ñ.Õ$�ω([x, y]) ≥ ω(x) + ω(y)

ÜoäYÌIÕqÍÎͳÈx, y ∈ G

Ï�ÇÎÍÎ×^~Yë�ÕNÙÏ�ͳÈ�dzï�å¢Ö9ÈvÑYÈ�ÛY×@È�Ð.ÑÊdzÙ3×yÔ"ÇÎ×[Gn, Gm] ⊆ Gn+m.

áçÇÎÌÐ.È�ÐYÐ.È�ÐÊÈ�ÇÎÐ.È�Ù@ËNͳï@å.ÈÝø?ÇÎÍÎ×3Ì3dzÈ�Ì3ÛYÐYÏ 6�����J=)d! % éá�ÇÎÌlå�ÕqÍÎ×@È�Ð�Ð.Ë�ï@åØÜoÈvÙ3×^~¢Ñ.Õ$��ÇÎÐ�Ñ*dzÈvÙ@È�Ô ø�ÕqÍÎͳÈ

GnÜaäYÌlÕqÍÎͳÈ

nÈ�ÇÎÐ Á/ËNÌ3Ô�ÕqÍÎ×@È�ÇÎͳÈ�ÌlÉ�ËNÐ

GdzÙ@×^~�ÛYÐ.Ñ�òvÛ.Ù�ìq×3ò�ö

ÍÎdzï�åGn/Gn+1

ÇÎÔè�!È�Т×3Ì3ÛYÔ É�ËNÐG/Gn+1

ÍÎdzÈ�Ï�×{é9Æ/ÈvÙ/ë�È�ÇÎ×@È�Ì@È�ÐúÙ3ÇÎÐ.ÑèÙ3Û,{±ò{ÈvÙ�Ù3ÇÎÉ�È�ÈAÛ.ËN×3dzÈ�б×@È�Ðúò{È�Т×3Ì�ÕqͳÈ�Ìø?ÇÎÍÎ×3Ì3dzÈ�Ì3ÛYÐYÏ�È�ÐèÕqÖ!È�ͳÙ@ï�åQé

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pö�õ/Ì3ÛYÓYÓ!È�é¢á�ÇÎÌ�ÔH��ï@å±×@È�Ð�ÛYÐ.Ù�È�ÇÎÐ.È

ø?ÇÎÍÎ×3Ì3dzÈ�Ì3ÛYÐYÏÚÉ�ËNÐGÖ9ÈvÙ@ï@å�ÕF¦-È�ÐQé�Æ�ÕqòvÛX{�ËNÐ.Ù3×3Ì3ÛYdzÈ�Ì@È�ÐèëyÇÎÌIòvÛYÐ�ìNï�å.Ù3×�È�ÇÎÐ.È�|AÔ"Ï�È�ÖYÛYÐYÏ�Ù3Ö�ÕNÙ3dzÙ=ÑYÈ�Ì

0ÇÎÐàÑYÈ�Ì

õ=Ì3ÛYÓYÓ!È�Ð�ÕqÍÎÏ�È�ÖYÌ�Õ*é-í/dzÈ�ͳÈ�ÞIÛ.Ù@Ù�ÕqÏ�È�ÐÒäYÖ9È�ÌÝõ/Ì3ÛYÓYÓ!È�Ð:�.ÍÎ×3Ì3dzÈ�Ì3ÛYÐYÏ�È�ÐÂ{[�NÐYÐ.È�ÐèëyÇÎÌ=Ñ.ÕqÐYÐèÇÎÐ ÑYÈ�Ì�õ/Ì3ÛYÓYÓ9È�Ð�ÕqÍ öÏ�È�ÖYÌ�Õ"Ö!È�å�ÕqÐ.ÑYÈ�ÍÎÐQéá�ÇÎÌIÑYÈ��.ÐYdzÈ�Ì@È�ÐÒÑ*dzÈ�ûaÉ�È�Ì3É�ËNÍÎͳÙ3×�ìqÐ.Ñ*ÇÎÏ�×@È&ÿ/õ/Ì3ÛYÓYÓ!È�Ð�ÕqÍÎÏ�È�ÖYÌ�Õ�Ñ*ÛYÌ@ï@å

Fp[[G]] = lim←− Fp[G/N ],

ë�ËNÖ!È�Ç�ÑYÈ�Ì=ÓYÌ@Ëqý3È/{±×3ÇÎÉ�È��-ÇÎÔ�ÈvÙ=äYÖ9È�Ì/Ñ*Ç³È Ë$¦-È�Ð.È�Ð@ÁAËNÌ3Ô�ÕqÍÎ×@È�ÇÎͳÈ�Ì=É�ËNÐGÏ�È�ÖYÇÎͳÑYÈ�×/ëyÇÎÌ@Ñ-é9êuÐàÇÎåYÌ=å�ÕqÖ9È�ÐàëyÇÎÌ

Ñ.ÕNÙIêuÑYÈ{ÕqÍI(G) = (g − 1| g ∈ G).ø�È�Ì3Ð.È�ÌAÙ@È�×3ò{È�Ð6ëyÇÎÌIn(G) := (I(G))n.ÁAÛYÐ å�ÕqÖ9È�Ð6ëyÇÎÌIÑYÈ�Ð6ÜgËNÍÎÏ�È�Ð.ÑYÈ�ÐX�*Õq×3òØûaÉ�Ï�Í�é-ßk�/ö�õ/â+~��*Õq×3ò�»�é©��ÿdé

�Q���=���=�nm��KJ�â:�^å,�������G�������@���98:%;�=</#¢�/)d6F��U A[J+�S13)dE Z

pZ�W2).U�*[*,� o ¿ ! �P�§O��5%M8��/�ð8 �R� p 8��Q! %K� In(G)

���5�0��"47A��^O�UT�BA$I^Od!$I.��I�8B��)0�5�

Fp[[G]] oá�ÇÎÌIÑYÈ��.ÐYdzÈ�Ì@È�ÐÒÑ*Ç³È �"!$IcI^���,#T!GU�I�Z=Ës��%MJ=)��R��)�UT�BA Ñ*ÛYÌ@ï�å

G(n) := {g ∈ G| g − 1 ∈ In(G)}.

ø�äYÌAÑ.ÕNÙD�±×3Û.Ñ*ÇÎÛYÔ ÇÎåYÌ@È�Ì�z�ÇÎÏ�È�Ð.Ù@ï@å�ÕcÜo×@È�ÐÊÖ9È�ëlÈ�dzÙ@È�ÐÊëyÇÎÌòvÛYÐ�ìNï@å.Ù3×AÑ.ÕNÙyÜgËNÍÎÏ�È�Ð.ÑYÈ7�QÈ�Ô"Ô�Õ*é�Q���=���=����� ö �B�Â�hâ��HÇu! È ËsNT)�! %5%K�

g, h ∈ GA �5%MJ+Ê

[g, h] − 1 = (gh− 1)(g−1h−1 − 1) + (gh− 1) + (g−1h−1 − 1)

ÇlO È ËtNT)�! %5%É�g ∈ G(n)

UB�08g!G%�%É�h ∈ G(m)

A �5%MJ+Êgh − 1 ≡ g − 1 + h− 1 (

4HE^8 UT%MEIn+m(G)).

Çu< È[G(n), G(m)] ⊆ G(n+m) oÇÆ8 È ËtNT)�! %5%É�

g ∈ G(n)

A �5%;J+Êgp

k∈ G(pkn).A �CB�� Þ ä[� ûgÕ�ÿ�Ì@Èvï@åYÐ.È�×AÔ�ÕqÐ ÛYÐYÔ"ÇÎ×3×@È�ÍÎÖ�ÕqÌ/ÕqÛ.Ù{éYø?äYÌWûaÖ�ÿlÖ9È�Ì@Èvï@åYÐ.È�ÐàëyÇÎÌ

gh− 1 = (g − 1)(h − 1) + g − 1 + h− 1 ≡ g − 1 + h− 1 (Ô�Ë�Ñ*ÛYͳË

In+m(G)).

ø?äYÌWûoï&ÿlÌ@Èvï�åYÐ.È�ÐÊëyÇÎÌÔ�Ë�Ñ*ÛYͳËIn+m(G)

�[g, h] − 1 = (gh − 1)(g−1h−1 − 1) + (gh − 1) + (g−1h−1 − 1)

≡ ((g − 1) + (h− 1))((g−1 − 1) + (h−1 − 1)) + g − 1 + h− 1 + g−1 − 1 + h−1 − 1

≡ (g − 1)(g−1 − 1) + (h− 1)(h−1 − 1) + g − 1 + h− 1 + g−1 − 1 + h−1 − 1

= 0

Fp[[G]]å�Õq×=ülå�ÕqÌ�Õ${±×@È�Ì3dzÙ3×3ÇK{

p~YÑ.Õqå.È�ÌIdzÙ3×

gpk− 1 = (g − 1)p

k

È�ÇÎÐfz�ͳÈ�Ô�È�Т×AÉ�ËNÐIp

kn(G)~Yë�ÈvÙ3å�ÕqÍÎÖ(ûoÑ�ÿlÏ�ÇÎÍÎ×{é

2

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�Q���=���=�:���KJ�â:�^å,�������G���5�0�g���98:%;�R<�#Â��)d6F��UGAGJ+�_13)dE Z

pZ�W2).U�*[*,� o ¿ !G�P�Â��I�JD8 �R�_�"!$I.In���,#:! U�I�Z=Ës�5%;J=)��R�/).UT�BA�/���0�ª6F����J=)d! %K�ªËs�5%;J=)��R�/).UT�BA$-sUT�98g8 �V�

G(n)

O/��%K8����¥���5�0����47A��^O/UB�BA InOd!$I��yI�8���)1CFE �

G oA �CB�� Þ ä[� Æ/dzÈ�È�Ì@Ù3×@Èt£�È�å�ÕqÛYÓY×3ÛYÐYÏÜgËNÍÎÏ�×QÛYÐYÔ"ÇÎ×3×@È�ÍÎÖ�ÕqÌ�ÕqÛ.ÙQÑYÈ�Ô>É�ËNÌ�ÕqÛ.Ù@Ï�È�å.È�Ð.ÑYÈ�Ð��QÈ�Ô"Ô�Õ*é ��ÛYÌQòvëlÈ�ÇÎ×@È�ÐÖ9È�×3Ì�ÕNï@å±×@È�ÐàëyÇÎÌÑ*dzÈ�ÞIÖYÖYÇÎͳÑ*ÛYÐYÏ

φ : G(n)/G(n+1) → In(G)/In+1(G), g 7→ g − 1.Æ/dzÈvÙ@ÈdzÙ3×�È�ÇÎÐ�æ/ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�Û.ÙlÐ�ÕNï@åg��È�ÇÎÍ0ûaÖ�ÿ0ÑYÈvÙs�QÈ�Ô"Ô�ÕNÙ&é¢ø.È�Ì3Ð.È�ÌldzÙ3×φÐ�ÕNï@åØÆ=È��.ÐYÇÎ×3dzËNÐØÉ�ËNÐ

G(n+1)ÇÎÐNý3È/{�×3ÇÎÉ!éYÆ�Õqå.È�ÌdzÙ3×G(n)/G(n+1)

È�Ð.Ñ*ÍÎdzï@å�~�ëlÈvÙ3å�ÕqÍÎÖG(n+1)

ÇÎÐGË$¦9È�РdzÙ@×{é��È�Ç-ÜoÈ�Ì3Ð.È�Ì

U �GÈ�ÇÎÐÊË$¦-È�Ð.È�Ì�Á/ËNÌ3Ô�ÕqÍÎ×@È�ÇÎͳÈ�ÌIÛYÐ.Ñ

ψU : Fp[[G]]→ Fp[G/U ]Ñ*dzÈWÐ�Õq×3äYÌ3ÍÎdzï@å.È���Ì@Ëqý3È/{�×3dzËNÐQé�Æ�Õ�ÑYÈ�̳��È�Ì3ÐàÉ�ËNÐψUË$¦-È�ÐÊdzÙ3×^~�Ï�ÇÎÖY×AÈvÙ/È�ÇÎÐ.È�ÐÒêuÐ.ÑYÈ��

n~.Ù@Ë�Ñ.Õ$�

In(G)ÇÎÔ��È�Ì3ÐàÈ�Т×3å�ÕqÍÎ×@È�ÐÊdzÙ@×{é.ø?äYÌIÕqÍÎͳÈ

g ∈ G(n)å�ÕqÖ9È�Ð ëyÇÎÌAÕqͳÙ@Ë"Ñ*dzÈWõ=ͳÈ�dzï@åYå.È�ÇÎ×

ψU (g − 1) = 0 = ψU (g) − 1.Æ�Õqå.È�ÌdzÙ@×G(n)ÇÎÐUÈ�Т×3å�ÕqÍÎ×@È�Ð�~�ÛYÐ.Ñ Ñ*dzÈ

G(n)Ù3×@È�ÇÎÏ�È�Ð6òvÛYÌ

1ÕqÖQé

2

á Þ �fâ7"än�^� Þ ���B�<;��që � �B�0�^$âPß5F� Þ ���

áçÇÎÌAÑYÈ��.ÐYdzÈ�Ì@È�ÐèÇÎÔ ø.ËNÍÎÏ�È�Ð.ÑYÈ�ÐÒÜaäYÌ/È�ÇÎÐ.È���Ì@Ëqöpö�õ=Ì3ÛYÓYÓ!È

GÈ�ÇÎÐ.ÈÝëlÈ�ÇÎ×@È�Ì@È�ò{È�б×3Ì�ÕqͳÈ�ø?ÇÎÍÎ×3Ì3dzÈ�Ì3ÛYÐYÏP~9Ñ*dzÈÝÏ¢ÕqÐYò

È�б×@Ù@ï@å.È�dzÑYÈ�Ð.ÑYÈWí�È�Ì3ëlÈ�Ð.Ñ*ÛYÐYÏ �.Ð.ÑYÈ�Ð ëyÇÎÌ@Ñ-é��È�Ç�òvÛYÐ�ìNï@å.Ù@×�ÕqÍÎÍÎÏ�È�Ô�È�ÇÎÐqÈ�ÇÎÐ.È

pö+��ËN×@È�ÐYòÚË�ÑYÈ�Ì

0éQÆ/Ç³È !�O.I/J+���MAB���98��

qZ+������J=)d! %')Q����#:� ëyÇÎÌ@ÑúÇÎÐ.Ñ*Û,{�×3ÇÎÉ

ÑYÈ��.ÐYdzÈ�Ì3×/Ñ*ÛYÌ@ï@åG(1,q) := G

ÛYÐ.ÑG(n+1,q) := (G(n,q))q[G(n,q), G].�Q���=���=�TÅ��KJ�â:�^å,�������

G�/���0�����08:%'�R<�#���)Q6���UGAGJ+�713)dE Z

pZ.W2)�U^*�*,� oÇÆ! È �����

K ⊆ G�/���0��!�ORAB�.I/<�#B%ME$I.In���������9J+��)=AG).U�*[*,� oõl ��)ªI^�/Jy6��/�X�5�98 UB&FJ=�5C

K(1) := KUT�98

K(n+1) := (K(n))q[K(n), G].¿ ! ���\A �5%MJP��NB)�!G%�%É�n ∈ N

K(n) ⊆ G(n),L E�OQ��� G(n)

8 �V�nZÆJ+�SW2).U�*[*,�H���¥8B��)��"!$IcI^���,#T! UBI.ZRËs��%MJ=).�V��)�UB�TAS��I�J o p �,InOc�.I/E �08���)Q�_I/J+���KA����r8G�R� K(n)UB�98g8[! 4��=Jx! U,<�#f8 �V�

G(n,q)�Q<�#BJt6/UT)

1!�O oÇlO È �����

q = ps���5�0�

pZ=1?E[J+���T6 o ¿ ! �P�X��I�J

G(n,q) ⊆ G��������Ed�D���0������J+�/)=A )�U�*[*,� o ¿ �V� G(n,q)O/��%K8����h! %'I/E\���5�0����47A��^O�UT�BA$I^Od!$I��yI�8B��)

1 oA �CB�� Þ ä[� ûgÕ�ÿ�ê�Ôi��È�Ô"Ô�Õ ûR¨*éK��éK�G�&ÿ�å�ÕqÖ9È�Ð ëyÇÎÌyÏ�ÈvÙ@È�å.È�Ð�~.Ñ.Õ$�(G(n))

q ⊆ G(nq)ÛYÐ.Ñ

[G(n), G] ⊆ G(n+1)Ï�ÇÎÍÎ×{é�á�ÇÎÌIò{È�ÇÎÏ�È�ÐàÑ*dzÈ7£�È�å�ÕqÛYÓY×3ÛYÐYÏ�ТÛYÐÊÇÎÐ.Ñ*Û,{�×3ÇÎÉ9é�ø?äYÌn = 1

dzÙ3×IÙ3dzÈ{�Í ÕqÌ{é.Æ=È�ÌIêuÐ.Ñ*Û,{±×3dzËNÐ.Ù�Ù@ï@åYÌ3ÇÎ×3×AÜoËNÍÎÏ�×ÕqÛ.Ù

K(n+1) = (K(n))q[K(n), G] ⊆ (G(n))q[G(n), G] ⊆ G(nq)G(n+1) ⊆ G(n+1).Æ/dzÈvÙ�Ï�ÇÎÍÎ×lÜaäYÌq = ps

é�Æ/È�Ì2£�È�ëlÈ�dzÙlÜoäYÌq = 0

È�Ì3Ï�ÇÎÖY×�Ù3dzï@å6ÕqÛ.Ù�Ï�ͳÈ�dzï@å.È�ÌyðIÈvï�åYТÛYÐYÏ"Ñ*ÛYÌ@ï@åÚáúÈ�Ï�Í ÕNÙ@Ù�È�ÐÚÕqÍÎͳÈ�Ì��È�Ì3Ô�ÈG~.ÇÎÐÊÑYÈ�Ð.È�ÐàÈ�ÇÎÐqÕqÛ*Üo×3Ì3ÇÎ×3×{é

ûaÖ�ÿ�áçÇÎÌ�Ö!È�бÛY×3ò{È�Ð6ëydzÈvÑYÈ�Ì3ÛYÔîê�Ð.Ñ*Û,{�×3dzËNÐÚÐ�ÕNï@åné±Æ=È�Ì�ê�Ð.Ñ*Û,{�×3dzËNÐ.Ù�ÕqÐ*Ü�ÕqÐYÏ"dzÙ3×x{�Í ÕqÌ{é�ø�äYÌyÑYÈ�ÐS��ï@åYÌ3ÇÎ×3×

Ù�È�×3ò{È�Ð ëyÇÎÌòvÛYÐ�ìNï@å.Ù@×H := G(n,q) éYÆ�ÕqÐYРdzÙ@× H ÕqͳÙË$¦-È�Ð.È7|AТ×@È�Ì3Ï�Ì3ÛYÓYÓ9ÈÝÐ�ÕNï�åf�QÈ�Ô"Ô�ÕàûR¨*éK��驨�ÿ�È�Ð.Ñ*ÍÎdzï@åÈ�Ì3ò{È�ÛYÏ�×^~�ÛYÐ.Ñ

H/Hq[H,H]dzÙ3×�È�Ð.Ñ*ÍÎdzï@åQéYÆ�Õqå.È�Ì�dzÙ3×�ÕqÖ9È�ÌyÕqÛ.ï@å

G(n,q)/G(n+1,q) = H/Hq[H,G]È�Ð.Ñ*ÍÎdzï�åQé

2

ùÒÕqÐÂ{NÕqÐYÐÃëlÈ�ÇÎ×@È�Ì3åYÇÎÐ(ò{È�ÇÎÏ�È�Ð�~�Ñ.Õ$�ÒÑ*dzÈØÕqÖ.Ù3×@È�ÇÎÏ�È�Ð.ÑYÈqöu�!È�б×3Ì�ÕqÍÎÌ@È�ÇÎå.È�×�Õq×@Ù�ìNï@åYÍÎdzï�å(È�ÇÎÐ.È�ò{È�Т×3Ì�ÕqͳÈÚø?ÇÎÍ ö

×3Ì3dzÈ�Ì3ÛYÐYÏ�Ñ.ÕqÌ@Ù3×@È�ÍÎÍÎ×{é.ø?äYÌIÑYÈ�Ð ø�ÕqÍÎÍq = p

dzÙ@×Ñ*dzÈÝðIÈvï�åYТÛYÐYÏ�ò�ék£=éYÇÎÐúß}Æ=Æ=ù@�¢â�Ñ*ÛYÌ@ï@åYÏ�ÈdÜoäYåYÌ3×{éá�ÇÎÌë�ËNÍÎͳÈ�Ð ÇÎÔ ø.ËNÍÎÏ�È�Ð.ÑYÈ�ÐÒÙ@ï@åYÌ@È�ÇÎÖ9È�Ð��

Gn := G(n,p) é

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êuÐ6Ñ*dzÈvÙ@È�Ô ÞAÖ.Ù@ï@åYÐYÇÎ×3×yÓYÌ�ìNÙ�È�Т×3dzÈ�Ì@È�ÐÊëyÇÎÌ?{�ÛYÌ3ò�Ñ*dzÈ=ëydzï@å±×3ÇÎÏ�Ù3×@È�ÐàÈ�ͳÈ�Ô�È�б×�ÕqÌ@È�ÐàðIÈvÙ@ÛYÍÎ×�Õq×@È�ÑYÈ�ÌY��å.ÈvËNÌ3dzÈ�ÑYÈ�ÌÓ9ËN×@È�ÐYòvÌ@È�dzï@å.È�ÐÒÛYÐ.Ñ6ÛYÐYÇ ÜgËNÌ3Ô�È�ÐÒõ/Ì3ÛYÓYÓ9È�Ð�~YëydzÈÝÙ3dzÈ=ÇÎÐúß}Æ=Æ=ù@�¢â�È�Т×uëydzïQ{�È�ÍÎ×/ëyÇÎÌ@Ñ-éá�ÇÎ̯�,�*dzÈ�Ì@È�Ð�È�ÇÎÐ.È?��Ì3ÇÎÔ"ò&ÕqåYÍ

p~Në�È�ͳï@å.ÈyëyÇÎÌ�ÑYÈ�Ìtz�ÇÎÐ*Ü�ÕNï@åYå.È�ÇÎ×lå�ÕqÍÎÖ!È�Ì�ÕqͳÙ0ÛYÐYÏ�È�Ì�ÕNÑYÈAÕqÐYÐ.È�åYÔ�È�Ð�ëlËNÍÎͳÈ�ÐQé��È�Ç

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ÆAdzÈëydzï@å¢×3ÇÎÏ�Ù@×@È�Ð6ÕqÛ.Ù�ÑYÈ�ÌlÆ=È��.ÐYÇÎ×3dzËNÐ�ÜgËNÍÎÏ�È�Ð.ÑYÈ�Ð z�ÇÎÏ�È�Ð.Ù�ï@å�ÕcÜo×@È�ÐØÙ3ÇÎÐ.Ñ"òvÛ.Ù�ÕqÔ"Ô�È�ÐYÏ�ÈdÜgÕ$��×�ÇÎÔ ÜgËNÍÎÏ�È�Ð.ÑYÈ�Ð�*Õq×3ò�ûaɱÏ�Í�éQß}Æ=Æ=ù@�¢â+~,��å.ÈvËNÌ@È�Ô ­*é©°:~P��Ì@ËNÓ9Ë�Ù3ÇÎ×3dzËNÐf­*éûT~���å.ÈvËNÌ@È�Ô ­*é©��ÿdé�Q���=���=�Bó��KJ�â:�^å,�������

G = < a1, . . . , ad >�/���0��/�98:%;�=<�#g�/)d6F��U A[J+�x*:EGJ+�/�T6/)Q���R<�#:�}W2).U�*[*,�.-¯UT�98HOQ�Q6F���=</#��0�

Gi8 �V�

iZuJ+� W2).U�*[*,��8��/)�!�O.I�J+�/�MA��/�98����

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ÇÆ! È ËtNT)}! %5%K�k ≥ 0

UT�98�! %5%É�i ≥ 1

A �5%;JGi+k = Gp

k

i

-?���,I^OQ�.InE �98���)Q�}�yI/JGi+1 = Φ(Gi)

8G�R���)d!GJRJ=������Z����J+��)RA )�U^*�*,��CFE �Gi o

ÇlO ÈGi = Gp

i−1= {xp

i−1| x ∈ G} = < ap

i−1

1 , . . . , api−1

d >.

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k ���08 UG6n�R�/)�J¯�������?�9UB)5¡n��&FJ=�=EG� CFEG�Gi/Gi+1

! U.�Gi+k/Gi+k+1

��NB)�!G%�%É�k ≥ 0UB�98g! %5%K�

i ≥ 1 oÇÆ8 ÈG��I�J38[!$ID13)dE^8 UB&FJ?8���)2*9)dE�6nwF&F%'��I/<�#,���`W2).U�*[*,�/�

< ai >-x8 o # o G = < a1 > · · ·< ad > oÇl� È ËtNT)����5�0��!�ORA���I/<�#�%KE$I.In���0������J+��)RA )�U^*�*,�

H ≤ GA �5%MJ+Ê

d(H) ≤ d(G) oá�ÇÎÌIÑYÈ��.ÐYdzÈ�Ì@È�ÐÒÑYÈ�Ðf« ! �BA È�ÇÎÐ.È�ÌÓYÌ@Ë�È�Ð.Ñ*ÍÎdzï@å.È�Ðèõ=Ì3ÛYÓYÓ!ÈÝÑ*ÛYÌ@ï�å

rk(G) = sup{d(H) | H ≤ GË$¦9È�Ð.È�|Iб×@È�Ì3Ï�Ì3ÛYÓYÓ!È

}.

z�ÙÏ�ÇÎÍÎ×�ÜgËNÍÎÏ�È�Ð.ÑYÈ�ÌÜoÛYÐ.Ñ.ÕqÔ�È�Т×�ÕqͳÈ�̪��Û.Ù�ÕqÔ"Ô�È�ÐYå�ÕqÐYÏÒûaÉ�Ï�Í�éQß}Æ=Æ/ù@�¢â+~,��å.ÈvËNÌ@È�Ô ­*éK�^­�ÿdé�Q���=���=�:ô��KJ�â:�^å,�������

G�/���0�?12)lE Z

pZ.W2)�U�*[*,� o ¿ ! �P�\#T![J G ���98:%;�=</#:��� « !G�BA7A����0! U\8�! �P�,- L ���P� G ���08 Z%;�R<�#f��)d6F��UGAGJ3�yI/JtUT�98g�/���0��El�D�/�0�?*:EGJ+���T6n)d�/�=<�#,�X����J+�/)=A )�U�*[*,�HOQ��I.�=Jy6nJ oÁ/ÕNï�åX�*Õq×3òØûR¨*éK��éK�/®±ÿÝûoï&ÿldzÙ@×�ÜaäYÌIÈ�ÇÎÐ.È�È�Ð.Ñ*ÍÎdzï�åàÈ�Ì3ò{È�ÛYÏ�×@È�Ó!ËN×@È�ÐYòvÌ@È�dzï�å.È�õ/Ì3ÛYÓYÓ9È

GÑ*dzÈÝÞAÖYÖYÇÎͳÑ*ÛYÐYÏ

Gi/Gi+1 � Gi+1/Gi+2, x 7→ xp

È�ÇÎÐ6Ù3ÛYÌoý3È/{�×3ÇÎÉ�È�Ì�æ/ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3ÔWÛ.Ù^~YÑ-é�åQé�Ñ*dzÈ/êuÐ.Ñ*ÇÎò{ÈvÙ(Gi : Gi+1)

Ù3×@È�ÇÎÏ�È�Ð ÕqÖQé*ÞAͳÙ@ËWÈ��*dzÙ3×3dzÈ�Ì3×yÈ�ÇÎÐkÔ"ÇÎ×

(Gk : Gk+1) = (Gk+i : Gk+i+1)ÜoäYÌAÕqÍÎͳÈ

i ≥ 0é�²? ¢ÛYÇÎÉ�ÕqͳÈ�Т×=Ñ.ÕqòvÛÊÍÎdzÈdÜgÈ�Ì3×D�?ËN×@È�ÐYòvdzÈ�Ì3ÛYÐYÏ6Ô"ÇÎ×

xpi È�ÇÎÐ.È�Ð

ê�Ù@ËNÔ�ËNÌ3ÓYåYdzÙ3ÔWÛ.ÙòvëydzÙ@ï�å.È�ÐGk/Gk+1

ÛYÐ.ÑGk+i/Gk+i+1

éYÆAdzÈvÙ�ÈvÙÈ�Ì3å.È�ÖY×Ô�ÕqÐ òvÛYÌIÆ=È��.ÐYÇÎ×3dzËNÐ��z�ÇÎÐ.È�Ó!ËN×@È�ÐYòvÌ@È�dzï�å.È�õ/Ì3ÛYÓYÓ!ÈWå.È�ÇK��× UB�P� �nE ).4 ~�ëlÈ�ÐYÐ Ù3dzÈ�È�Ð.Ñ*ÍÎdzï@å È�Ì3ò{È�ÛYÏ�×=dzÙ3×^~�ÛYÐ.ÑX�?ËN×@È�ÐYòvdzÈ�Ì3ÛYÐYÏ6Ô"ÇÎ×piÜoäYÌIÕqÍÎͳÈ

i ∈ NÈ�ÇÎÐ.È�ÐÊê�Ù@ËNÔ�ËNÌ3ÓYåYdzÙ3ÔWÛ.Ù

G1/G2∼= Gi+1/Gi+2

ÇÎÐ.Ñ*ÛYòvdzÈ�Ì3×{éá�dzï�å¢×3ÇÎÏ�È�z�ÇÎÏ�È�Ð.Ù@ï@å�ÕcÜo×@È�ÐãÙ3ÇÎÐ.Ñ÷òvÛ.Ù�ÕqÔ"Ô�È�ÐYÏ�ÈdÜ�Õ$��×�ÇÎÔñÜgËNÍÎÏ�È�Ð.ÑYÈ�ÐÎ�*Õq×3òôûaɱÏ�Í�éyß}Æ=Æ=ù@�¢â+~b��Ì@ËNÓ!Ë�Ù@ÇÎ×3dzËNЮ.é ®P~���å.ÈvËNÌ@È�Ô¬®.éjþ�ÿdé

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pZ.W2)�U^*�*,� o ¿ ! �P�fI.�5�98g�z=/UT��C$! %K�/��J+ÊÇ+� È

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d(H) = d(G)��NT)�! %�%��El�D���0�/�XUT�987*:EGJ+�/�T6/)Q���R<�#:��� �"��J+��)RA ).U�*[*,�/�

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pö�õ/Ì3ÛYÓYÓ!ÈG~YÑ.ÕqÐYÐÚÏ�ÇÎÍÎ×

d(A) = d(B)éYÆ/dzÈvÙIÈ�Ì3Í ÕqÛYÖY×AÑ*dzÈÝÆ=È��.ÐYÇÎ×3dzËNÐdim(G) := d(H)

ÜoäYÌÈ�ÇÎÐ.ÈÝË$¦-È�Ð.È=ÛYÐYÇ ÜoËNÌ3Ô�È�|Iб×@È�Ì3Ï�Ì3ÛYÓYÓ!ÈHÉ�ËNÐ

G.

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Æ/dzÈ��9ÕqåYÍdim(G)

Ð.È�ÐYÐ.È�ÐÊëyÇÎÌÑ*dzÈ�¿ �54}���,I.�RE � É�ËNÐGé

ø�ÕqÍÎͳÙGÉ�ËNÐ6È�Ð.Ñ*ÍÎdzï�å.È�ÔîðAÕqÐYÏP~±ÛYÐ.Ñ

N �GÈ�ÇÎÐÚÕqÖYÏ�ÈvÙ�ï@åYͳË�Ù@Ù@È�Ð.È�ÌYÁ/ËNÌ3Ô�ÕqÍÎ×@È�ÇÎͳÈ�ÌydzÙ3×^~±Ñ.ÕqÐYÐ�Ï�ÇÎÍÎ×�ûoÙ3dzÈ�å.È

ß}Æ=Æ=ù@�¢â+~P��å.ÈvËNÌ@È�Ôi®.é©��ÿ��dim(G) = dim(N) + dim(G/N).

Æ/È�Ì2��Û.Ù�ÕqÔ"Ô�È�ÐYå�ÕqÐYÏ�òvÛpöuÕNÑ*dzÙ@ï�å.È�ÐØÕqÐ�ÕqÍK��×3dzÙ@ï@å.È�Ð�õ=Ì3ÛYÓYÓ!È�Ð�ëyÇÎÌ@Ñ Ï�È�Ï�È�Ö9È�Ð�Ñ*ÛYÌ@ï�å"ÜoËNÍÎÏ�È�Ð.ÑYÈvÙ�Ö!È�Ì@È�ÇÎ×@Ù

ÇÎÐàÑYÈ�ÌYz�ÇÎÐYͳÈ�ÇÎ×3ÛYÐYÏ�È�Ì3ë�ìqåYТ×@ÈvÙ³��å.ÈvËNÌ@È�Ô2ûaÉ�Ï�Í�é-ß}Æ=Æ/ù@�¢â+~P��å.ÈvËNÌ@È�Ô �*éK�&ÿdé�Q���=���=�Pø9�����"����$�B�`��e3������JVE.*:E %KEQA ��I/<�#,�vW2)�U^*[*P�

G#T!GJ?8 �V���9J=)�U�&FJ=UT)��/���0�/)

pZl![8 ��I/<�#,���¥! �9! %;wGJ=�yIn<�#:���W2)�U�*[*,�ªA����0! Uf8[! ���,- L ���P�fI.�V���/���0��El�D�/�0�.-2���08:%'�R<�#���)Q6���UGAGJ+��*:E[J+���T6/)Q���R<�#:�X�"��J+��)RA )�U^*[*P�_����J5#T� %MJ o

ÆAdzÈWËNÖ!È�ÐÒÈ�ÇÎÐYÏ�ÈdÜaäYåYÌ3×@È�Æ/ÇÎÔ�È�Ð.Ù3dzËNÐèÙ3×3ÇÎÔ"Ô�×/Ô"ÇÎ×AÑYÈ�Ì/ÆAÇÎÔ�È�Ð.Ù@dzËNÐèÕqͳÙpöuÕNÑ*dzÙ@ï@å.È ÕqÐ�ÕqÍK��×3dzÙ@ï@å.È"õ/Ì3ÛYÓYÓ9È

äYÖ9È�Ì@È�ÇÎÐQé

ÛýÎÛªý���Ý �p����%�àâàâì�d����ÿ����ô ��9ÿ½Ý�� �bÿ½������)����Ý��½� �9ÿ½Ý��

}d�h}�� �!��¶-��$z64,/�Æ&�6s��&��! 8�<ó/,/3s�d$z64,<� #!��ûà#���&�3 �S�C.0&�� äå�C������&��áçÇÎÌ�ë�ËNÍÎͳÈ�ÐØÇÎÐ�Ñ*dzÈvÙ@È�Ô ÞAÖ.Ù@ï�åYÐYÇÎ×3×�ÕqÖ9È�ͳÙ@ï@å.È�õ/Ì3ÛYÓYÓ9È�Ð�ÛYТ×@È�Ì@Ù@Û.ï@å.È�Ð�~YÕqÛ*Ü�ÑYÈ�Ð.È�ÐØÈ�ÇÎÐ.ÈAê�бÉ�ËNÍÎÛY×3dzËNÐ�~±Ñ*dzÈIëyÇÎÌÔ"ÇÎ×

σÖ9È�ò{È�dzï@åYÐ.È�Ð�~�ËNÓ!È�Ì3dzÈ�Ì3×{é��ÛYÐ�ìNï@å.Ù3×/Ù@È�×3ò{È�Ð ëyÇÎÌ�ÜoäYÌIÈ�ÇÎÐ.È�ûgÕNÑYÑ*ÇÎ×3ÇÎÉ6Ï�ÈvÙ@ï@åYÌ3dzÈ�Ö9È�Ð.È&ÿÕqÖ9È�ͳÙ@ï�å.ÈWõ/Ì3ÛYÓYÓ9È

A

A+ := {a ∈ A | σa = a}ÛYÐ.Ñ

A− := {a ∈ A | σa = −a}.

�Q���y���=�P� ö �B�¥�¥â��������V���A-BUT�98

C!BOQ��%'I/<�#:� W2)�U^*�*,���,-s! U��78����0���S>�UB%MJ=�Ã*9%'��&$!GJ=�RE �f4��=J

2���5�\mDU:JVE Z4HE )+*�#���I.4�UBI�yI/J o ¿ �V� p ��CFE %;UTJ=�RE � σ E.*,�/).�V��)Q��! U.� A - B UT�98

C o ¿ ! �P�gAB��%;J+�/�0ÊÇÆ! ÈA+ = {1

2 (a+ σa) | a ∈ A}UT�98

A− = {12 (a− σa) | a ∈ A}.ÇlO È

A = A+ ⊕A− oÇu< È¥p I�Jf : A→ B

���5�σZÆ����CF! )��=! �9J+��)MH7E 4HE 4HE )+*�#���I.4�UBIc-�I/E_��I�J

f(A±) ⊆ B±-bUT�98}! U/ab��)l8B��4ÍA��/%;J+���

Ker(f)± = Ker(f : A± → B±)UT�98

Im(f)± = Im(f : A± → B±) oÇÆ8 Èhp I�J0 → A → B → C → 0

���5�0�H�Q][!$&FJ+�����>=/UP���T6_4��=JσZ+�5�PCF! )��R! ��J+���ºH�E 4HE 4HE )V*9#��yI�4}���,-tI/E I����08! U:</#S8G�R�7���>=nU,���:6����

0→ A± → B± → C± → 0�d][!$&$J oÇl� È¥¿ �V�_W2)�U^*�*,�

Hom(A,B)C ��)cI^�.#:��� L �5)D4��=Jt8���)YA�� L ( #��P%;�=<�#,��� σ Z=m�&$J=�=E �rÇ B In�/��#��V��)����5��J=).�5C^�R! %É��)

σZ+>XE^8 UT% È -38 o # o (σf)(a) := f(σa) o ¿ ! ���gA �5%;J+Ê

Hom(A,B)± = Hom(A±, B).

ÇÃ� È(A∨)± = (A±)∨ oÇ5A È(A⊗B)+ = (A+ ⊗B+)⊕ (A− ⊗B−) oÇR# È(A⊗B)− = (A− ⊗B+)⊕ (A+ ⊗B−) oA �CB�� Þ ä[� Æ=ÕNÙÌ@Èvï@åYÐ.È�×IÔ�ÕqÐÊÛYÐYÔ"ÇÎ×3×@È�ÍÎÖ�ÕqÌIÐ�ÕNï@åQé

2

�Q���y���y�0�KJ�â:�^å,�������C�/���0�ª���08:%'�R<�#:�ª!BOQ��%'I/<�#:�_W2).U�*[*,�DCFEG4 eb]�*:E �0���9J+���

p 6= 2-�! U��D8��/)³8 �V� p �PCFE %;UTJ=�RE �

σE.*,��)��V��)�J o ¿ ! ���@�d]G�yI/J=�R��)/J?������� σ ZÆ���PC$! ).�R! ��J+�7����)�%É�RA UT�BA

C = A1 ⊕ · · · ⊕Ad+ ⊕B1 ⊕ · · · ⊕Bd− ,

L EBOQ��� Ai = A+i

-Bj = B−

j

-|Ai| = |Bj| = p

-d+ = dimFpC

+UT�98

d− = dimFpC−A���%MJ+��� o

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p

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å�ÕqÍÎÖ!È�ÇÎÐ*Ü�ÕNï@å�~�Ñ.Õ2 6= p

É�ËNÌ�ÕqÛ.Ù@Ï�ÈvÙ@È�×3òv×dzÙ@×{é�Æ�Õqå.È�Ì Ç³Ù3×"ÕqÛ.ï@åèý3ÈvÑYÈ�Ì

Fp[< σ >]ö�ùàË�Ñ*ÛYÍlå�ÕqÍÎÖ9È�ÇÎÐ*Ü�ÕNï@å�~0ëlÈvÙ@å�ÕqÍÎÖ

CÇÎÐ÷È�ÇÎÐ.ÈÚÑ*ÇÎÌ@È/{±×@ÈS�±ÛYÔ"Ô�È�É�ËNÐ

ÇÎÌ3Ì@ÈvÑ*ÛYòvÇÎÖYͳÈ�ÐÒùàË�Ñ*ÛYÍÎÐ ò{È�Ì,Ü�ìqÍÎÍÎ×^~�ÕqÛ*Ü�ÑYÈ�Ð.È�ÐσÈ�Т×uëlÈvÑYÈ�Ì�ÕqͳÙê�бÉ�È�Ì@Ù3dzËNÐÊË�ÑYÈ�ÌIÕqͳÙIê�ÑYÈ�Т×3ÇÎ×�ìq×IËNÓ9È�Ì3dzÈ�Ì3×{é

á ìqÌ@ÈÑ*dzÈ�z�ͳÈ�Ô�È�Т×3ò&ÕqåYÍ9È�ÇÎÐ.ÈvÙ�ÇÎÌ3Ì@ÈvÑ*ÛYòvÇÎÖYͳÈ�ÐFp[< σ >]

ö�ùàË�Ñ*ÛYͳÙ�Ï�ÌQ� ��È�ÌlÕqͳÙp~�Ù@Ë=ò{È�Ìl��È�ͳÈIÈ�Ì�òvÛYÐ�ìNï�å.Ù3×

ÕqͳÙWÕqÖ!È�ͳÙ@ï�å.È�õ/Ì3ÛYÓYÓ9È ÇÎÐôÑ*dzÈ"Ñ*ÇÎÌ@È/{�×@È �±ÛYÔ"Ô�È�É�ËNÐ òvÛZ/pZ

dzÙ@ËNÔ�ËNÌ3ÓYå.È�Ð(õ/Ì3ÛYÓYÓ9È�ÐQéQÞAÛ*Ü�Ñ*dzÈvÙ�È�Ðh{NÕqÐYÐÔ�ÕqÐÒÕqÖ!È�ÌIÑ*dzÈ

σö�ÞD{±×3dzËNÐàÙ@Ë"È�ÇÎÐYÌ3dzï@å±×@È�Ð�~�Ñ.Õ$��Ñ*dzÈ�Ñ*ÇÎÌ@È/{�×@È��±ÛYÔ"Ô�È

σö�ÇÎбÉNÕqÌ3Ç Õqб×AdzÙ@×{é

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Gö�ùÒ˱Ñ*ÛYÍ�éYáúÈ�ÇÎ×@È�Ì/Ù@È�ÇQÈ�ÇÎÐ.È�È�Ð.Ñ*ÍÎdzï�å.È�õ/Ì3ÛYÓYÓ9È

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AÑYÈ�Ì�ÕqÌ3×IËNÓ!È�Ì3dzÈ�Ì3×^~�Ñ.Õ$�"ÜaäYÌIÕqÍÎͳÈ

σ ∈ ∆Ñ*dzÈÝæAËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ@Ô�È�Ð

φσ : G→ G, g 7→ σ−1gÛYÐ.Ñ

θσ : A→ A, a 7→ σaÈ�ÇÎÐÊÉ�È�Ì3×3Ì�ìqÏ�ÍÎdzï@å.ÈvÙ³�?Õ�ÕqÌAÇÎÔ �±ÇÎÐYÐ.ÈÝÑYÈ�ÌY£�È�Ô�È�Ìd{±ÛYÐYÏèûl��éK��é ®±ÿ�ÖYÇÎͳÑYÈ�ÐQé.Æ�ÕNÙyå.È�ÇK��×^~.Ñ.Õ$�

θσ(φσ(g).a) = g.θσ(a)ÖYòvëÝé

σ.(

(σ−1g).a)

= g.(σ.a)ûl�&ÿ

ÜoäYÌWÕqÍÎͳÈa ∈ A

ÛYÐ.ÑÃÕqÍÎͳÈg ∈ G

Ï�ÇÎÍÎ×{é�Æ�Õqå.È�Ì�ÑYÈ��.ÐYdzÈ�Ì3×�Ñ*dzÈ�ø.ÈvÙ3×3ͳÈ�Ï�ÛYÐYÏÒÕqÛ*ÜÑYÈ�Ð¥�AË {�È�×3×@È�ÐÃÉ�ËNÐGÔ"ÇÎ×

áúÈ�Ì3×@È�ÐàÇÎÐA

(σf)(g1| . . . |gn) := σf(σ−1g1| . . . |σ−1gn)È�ÇÎÐ.È�Ðèæ/ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�Û.Ù

Hn(G,A) → Hn(G,A)é9ê�Ð.Ù3Ö9ÈvÙ@ËNÐ.ÑYÈ�Ì@È�dzÙ3×AÑYÈ�Ì/ðAÕqÐ.ÑYËNÓ9È�Ì�Õq×@ËNÌ

∆ö�ÇÎбÉNÕqÌ3Ç Õqб×^~

Ñ-é.åQéYÑ.Õ$�"ÜaäYÌIÕqÍÎͳÈnö+�AË {�È�×3×@È�Ð

f : Gn → AÏ�ÇÎÍÎ×∂(σf) = σ(∂f).

ÞAÛ.ÙD�*Õq×3òØûl��éK��éjþ�ÿyÈ�Ì3å�ÕqÍÎ×@È�ÐÊëyÇÎÌ^~.Ñ.Õ$�"í�È�Ì3ÖYÇÎÐ.Ñ*ÛYÐYÏ�Ù3å.ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ@Ô�È�Ð∆ö�ÇÎТÉ�ÕqÌ3Ç ÕqТ×AÙ@ÇÎÐ.Ñ�~YÑ-é åQé

δ(σf) = σ(δf).ÞAÛ*Ü?ÑYÈ�Ðv�AË {�È�×3×@È�ÐÊÙ3dzÈ�å¢×yÔ�ÕqÐ Ù�ËqÜoËNÌ3×^~*Ñ.Õ$�"ëyÇÎÌÕqÛ*Ü?Ñ*dzÈvÙ@È/áúÈ�dzÙ@È�È�ÇÎÐ.È

∆ö�ÞD{�×3dzËNÐ6ÕqÛ*Ü0ÑYÈ�Ðv�IËNå.ËNÔ�ËNͳËNÏ�dzÈdö

Ï�Ì3ÛYÓYÓ9È�ÐHn(G,A)

È�Ì3å�ÕqÍÎ×@È�ÐQéá�ÇÎÌë�È�Ì@ÑYÈ�ÐÊÛYÐ.ÙyÇÎÔîø�ËNÍÎÏ�È�Ð.ÑYÈ�ÐÊÇÎÐÊÈ�ÇÎÐ.È�ÌIÑYÈ�ÌIÖ!È�dzÑYÈ�Ðf�±ÇÎ×3Û�Õq×3dzËNÐ.È�ÐèÑYÈvÙyÐ�ìNï@å.Ù@×@È�ÐX�*Õq×3ò{ÈvÙÖ9È��.Ð.ÑYÈ�ÐQé

�Q���y���ÉÅ��KJ�â:�^å,�}Çu! È ���/�G�������Y*9)lEF���98:%;�=</#:�\W2)�U^*[*P��! U.��8���)�8 �R�����08:%'�R<�#:�gW2)�U�*[*,�

∆E.*,�/).�V��)/J o A I^�����������8G�yI.&�)d�nJ+��!BOQ��%'I/<�#:�vW2).U�*[*,��4��=JtJ=).�5C^�R! %É��)�Y"*,��)d!GJ=�RE �@CFE �

GUT�98

∆ o¿ ! ���@��).#T! %MJ+��� L �5)�������� ∆ZRm�&FJ=�RE ��!GU.��8��/�S��E$#:E 4HE %MEcA �R�VA ).U�*[*,�/�

Hn(G,A) oÇlO È �����F���5�0�}*0)lE��/�98:%;�=<�#,��W2)�U^*[*P�X4��5J_El�D������4�?7E )�4H! %;J+�/��%É��)

G � F-�UB�08

AIn�/�����5�ð8 ��Ic&�)Q�/J+��)

FZV>XE^8GUB% o­l �5)�0�.#�4}���h! �,-s8[!Qa¢8 �V�7)Q�.I�UB%MJ=�R�/)d���08��H�d][! &FJ+�����>=/UP���T6

1→ G→ Fπ−→ ∆→ 16���)5�/�G%�%MJ5-s8 o # o 8�!daÎ�.I����5�0���@H7E 4HE 4HE )+*�#���I.4�UBI

ϕ : ∆→ F4��5J

π ◦ ϕ = idA �ROnJ o ¿ �V�.I�yI/J"6 o L o 8B��)Ë�! %5%Ã- L �/�P� G ���5�0�³13)lEGZ

pZ�W2).U�*[*,��UT�98

∆���5�0�

qZ.W2)�U^*�*,��4��5J

p 6= q�yI/J o¿ ! ����E.*P��).�V��)/J

σ ∈ ∆! U��

G8 UB)d<�#g��E �^¡�UGA[!GJ=�RE �f4��=J

ϕ(σ)UT�98g! U��

A8 UB)d<�#

σ.a := ϕ(σ).a oË���)��0��)g�/)c#T!G%;J+��� L �5) �������_CFEG�j8��/) l ! #�%s8��.I��0<�#B�P�5JRJ+��I�UT�9!�O�#T� �BA �KA��ºY"*,��)d!GJ=�RE �jCFE �∆! U��}8������E #TE 4HE %KEQA �V�RA )�U^*[*P���

Hn(G,A) oA �CB�� Þ ä[� ûgÕ�ÿ�Æ/dzÈ�õ/äYÍÎ×3ÇÎÏG{�È�ÇÎ×IÉ�ËNÐôûl�&ÿlÜgËNÍÎÏ�×IÑ*ÇÎÌ@È/{�×{éûaÖ�ÿ�Æ/dzÈÝÈ�Ì@Ù3×@È�Ð ÞAÛ.Ù@Ù�ÕqÏ�È�Ð Ù3ÇÎÐ.Ñv{�Í ÕqÌ{éYÆ/dzÈWõ/äYÍÎ×3ÇÎÏG{�È�ÇÎ×É�ËNÐ(ûl�&ÿ�Ù3dzÈ�å¢×IÔ�ÕqÐÊÕqÛ.Ù

g.(σ.a) = g.(ϕ(σ).a) = ϕ(σ).(

(ϕ(σ−1) g ϕ(σ)).a) = σ.(

(σ−1.g).a)

.Æ/dzÈ�ͳÈ�×3òv×@È"ÞAÛ.Ù@Ù�ÕqÏ�È�ÜoËNÍÎÏ�×�Ñ.ÕqÌ�ÕqÛ.Ù^~-Ñ.Õ$�

Hi(G, ·)ÇÎТÉ�ÕqÌ3Ç ÕqТ×�ÛYТ×@È�ÌÝÑYÈ�Ì�Þ�{�×3dzËNÐèÉ�ËNÐ

GdzÙ3×�ûaɱÏ�Í�é0ßkÁ�±áãâ+~��Ì@ËNÓ!Ë�Ù3ÇÎ×3dzËNÐr��é©°*驨�ÿ�~YÛYÐ.Ñ ý3È=òvëlÈ�Ç���ï�åYÐYÇÎ×3×@È

ϕ,ψ : ∆→ FÙ3×@È�×@Ù?{�ËNÐNý,ÛYÏ�dzÈ�Ì3×IÙ3ÇÎÐ.Ñ-é

2

ê�Ô ø.ËNÍÎÏ�È�Ð.ÑYÈ�ÐèÖ!È�×3Ì�ÕNï�å¢×@È�ÐèëyÇÎ̳�AËNÐ.Ù@Èn ±Û.È�ÐYò{È�ÐÊÜoäYÌAÑ*dzÈ�ÞIÖYÖYÇÎͳÑ*ÛYÐYÏ�È�ÐÒòvëydzÙ@ï�å.È�ÐèÑYÈ�ÐX�IËNå.ËNÔ�ËNͳËNÏ�dzÈdöÏ�Ì3ÛYÓYÓ9È�ÐQéYáçÇÎÌIÙ@È�×3ò{È�Ð Ù3×@È�×@ÙÉ�ËNÌ�ÕqÛ.Ù^~.Ñ.Õ$��È�ÇÎÐ.È

∆ö�ÞD{�×3dzËNÐÊÕqÛ*Ü0ÑYÈ�Ð��AËNå.ËNÔ�ËNͳËNÏ�dzÈ�Ï�Ì3ÛYÓYÓ!È�ÐèÉ�ËNÌ3ÍÎdzÈ�Ï�×{é

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�Q���y���Kó�� ö �B�¥�¥â�� p I�Jφ : G→ H

���5�∆Z+�5�PCF!G).�R! ��J+��)�H7EG4_EG4_EG)V*�#B�yI�4�UBIc-�8���)24��5J��/���0�/4

∆ZÆ����CF! )��=! ��ZJ+�/��H7EG4_EG4_EG)V*�#B�yI�4�UBI

θ : B → ACFE 4

HZ+>XE^8 UT%

B�5�f8��/�

GZV>�E^8 UT%

AC$��)/J=)d�^A %;�=<�#v��I�J��54Ì�9���P���ª8���)

L³��4}��).&�UT�BArÇÎN o N oq[ È -3I/EgI����08S8 �V��)d��I.UT%;J=�V��)Q���98����ÇH�E 4HE 4HE )V*9#��yI�4}���(φ, θ) : Hn(H,B) → Hn(G,A)��OQ���^�n! %5%ÃI

∆ZÆ���PC$! ).�R! ��J o

A �CB�� Þ ä[� ��È�Çσ ∈ ∆

é*áúÈ�Ï�È�Ð(

ψ(σf))(

g1| . . . |gr)

= θ(

σf(

σ−1(φ(g1))| . . . |σ−1(φ(gr))

)

)

= σθ(

f(

φ(σ−1g1)| . . . |φ(σ−1gr))

)

=(

σ(

ψ(f))

)

(

g1| . . . |gn)

,

ë�ËNÖ!È�Ç�ëyÇÎÌòvÛYÌIÞAÖ,{±äYÌ3òvÛYÐYÏψ := (φ, θ)

Ï�ÈvÙ�È�×3òv×å�ÕqÖ!È�Ð�~�È�Ì3Ï�ÇÎÖY×IÙ3dzï@åàÑ*dzÈ7£�È�å�ÕqÛYÓY×3ÛYÐYÏ.é2

Æ=ÕNÙ_��È�Ô"Ô�Õ�{[�NÐYÐ.È�ÐóëyÇÎÌ ÇÎÐ.Ù3Ö!ÈvÙ�ËNÐ.ÑYÈ�Ì@È ÕqÛ*Ü/Ñ*dzÈÚê�Ð:Ï�Õq×3dzËNÐóÛYÐ.ÑóÑ*dzÈÚðAÈvÙ3×3Ì3ÇK{±×3dzËNÐ ÕqбëlÈ�Ð.ÑYÈ�ÐQész�ÇÎÐ.ÈÕqÐ�ÕqͳËNÏ�ÈWÞIÛ.Ù�Ù�ÕqÏ�È=Ï�ÇÎÍÎ×�ÜaäYÌAÑ*dzÈ�æAËNÔ�ËNͳËNÏ�dzÈG~�Ù@Ë"Ñ.Õ$�ØÕqÛ.ï@åàÑ*dzÈ7�IËNÌ@ÈvÙ3×3Ì3ÇK{�×3dzËNÐ

∆ö�ÇÎбÉNÕqÌ3Ç Õqб×IdzÙ3×{é

ÞIÛ.Ù=Ù@È�ÇÎÐ.È�Ì=È���ÓYÍÎÇÎòvÇÎ×@È�Ð@£�ÈvÙ@ï�åYÌ@È�ÇÎÖYÛYÐYÏôûaò�ék£/é�ÇÎÐ[ÁIö+���

]~��!é0®���ÿÈ�Ì@Ù@È�å.È�ÐÒëyÇÎÌ^~9Ñ.Õ$��ÑYÈ�̪{�ÕqÐ.ËNÐYdzÙ@ï@å.È

ê�Ù@ËNÔ�ËNÌ3ÓYåYdzÙ3ÔWÛ.ÙGab ∼= H−2(G,Z)

ÜoäYÌÈ�ÇÎÐ.ÈÝÈ�Ð.Ñ*ÍÎdzï�å.È�õ/Ì3ÛYÓYÓ9ÈGÈ�Ö9È�Ð*ÜgÕqÍÎͳÙ

∆ö�ÇÎТÉ�ÕqÌ3Ç ÕqТ×AdzÙ3×{é�õ/ͳÈ�dzï@å.ÈvÙIÏ�ÇÎÍÎ×yÜoäYÌÑ*dzÈÝêuÙ�ËNÔ�ËNÌ3ÓYåYdzÈ

AG/NGA ∼= H0(G,A).

ÞIÛ.ÙAÑYÈ�Ì�£�ÈvÙ@ï@åYÌ@È�ÇÎÖYÛYÐYÏÚÑYÈvÙ/ülÛYÓ*ö+��Ì@Ë�Ñ*Û,{�×@ÈvÙ=ÕqÛ*Ü�ÑYÈ�Ð��AË {�È�×3×@È�Ð÷ûaò�é�£/é.ÇÎÐÃßkÁ�±áãâ+~YÓQé9­�þ�ÿ�Ù@ï@åYÍÎdzÈ/��È�ÐëyÇÎÌAÙ@È�ÇÎÐ.È

∆ö�ê�бÉNÕqÌ3Ç ÕqÐYò

σ(a ∪ b) = (σa) ∪ (σb).

ÞAÛ*Ü�Ñ*dzÈvÙ@ÈY�±ÇÎ×3Û�Õq×3dzËNÐ"ëyÇÎÌ@Ñ Ñ.ÕNÙ�ÜoËNÍÎÏ�È�Ð.ÑYÈ2�QÈ�Ô"Ô�Õ:~�ÑYÈvÙ�Ù@È�Ð_£�È�ë�È�dzÙ�Ù3dzï@å"Ù@ËqÜgËNÌ3×0Ñ*ÛYÌ@ï@åHÁ=ÕNï@åYÌ@Èvï@åYÐ.È�ÐØÈ�Ì3Ï�ÇÎÖY×^~ÞAТë�È�Ð.Ñ*ÛYÐYÏ}�.Ð.ÑYÈ�ÐQé�Q���y���yô0� ö �B�¥�¥â��������

∪ : A×B → C���5�0�

ZZlO��5%'�5�0�d!G)d�ªmO^O��5%K8 UB�TAg8B��)�!�OQ��%'I/</#:���¢W2)�U^*�*,���

A,B,C-!GU.��8B���0���@8 �V�}W2)�U�*[*,�

∆E�*,��)��R��)/J o p I/J ∪ ∆

Z+�5�PCF! )��=!G��J5-¯I/EHI.�5�98 8G�R�7���08 UG6n�R�/)�J+���IH�E 4HE 4HE )V*9#��yI�4}���A→ Hom(B,C), a 7→ (b 7→ a ∪ b)

UT�98A→ C, a 7→ a ∪ b��NT)t�n��I�J+�.I

b ∈ B4��5J

σb = b! U:</#

∆Z+�5�PCF! )��=!G��J o

á�ÇÎÌ?ëlËNÍÎͳÈ�Ð ÛYÐ.Ù�ÕqÖ�åYdzÈ�Ì�ÕqÛ*Ü�ÑYÈ�Ð ø�ÕqÍÎÍ∆ ∼= Z/2Z

Ù@Ó!È�òvÇ ÕqÍÎdzÙ3dzÈ�Ì@È�ÐQéqÆ�ÕqÖ!È�Ç�ëlÈ�Ì@ÑYÈ∆É�ËNÐ�ÑYÈ�Ì0ê�бÉ�ËNÍÎÛY×3dzËNÐ

σÈ�Ì3ò{È�ÛYÏ�×{é-Æ/dzÈ�Ì@ÈvÙ3ÛYÍÎ×3dzÈ�Ì@È�Ð.ÑYÈ"Þ�{�×3dzËNÐúÕqÛ*ÜlÑYÈ�Ð@�IËNå.ËNÔ�ËNͳËNÏ�dzÈ�Ï�Ì3ÛYÓYÓ9È�ÐôdzÙ3×�Ñ.ÕqÐYÐ È�Т×uëlÈvÑYÈ�ÌW×3Ì3ÇÎÉ±Ç ÕqÍ�Ë�ÑYÈ�ÌÕqÛ.ï�åàÈ�ÇÎÐ.ÈÝêuТÉ�ËNÍÎÛY×3dzËNÐQéz�ÇÎÐÊë�È�ÇÎ×@È�Ì@È�ÌIëydzï@å±×3ÇÎÏ�È�Ì/æ/ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3ÔWÛ.ÙAÉ�ËNÐf�IËNå.ËNÔ�ËNͳËNÏ�dzÈ�Ï�Ì3ÛYÓYÓ9È�ÐÒdzÙ@×IÑ*dzÈf¼ )l! �PIlA )Q�.IcI��=EG� é.Æ=È�Ì£�È�ë�È�dzÙ�ÇÎåYÌ@È�Ì

∆ö�ê�бÉNÕqÌ3Ç ÕqÐYò�È�Ì,ÜoËNÌ@ÑYÈ�Ì3×�È�ÇÎÐ.È�Ï�È�Ð�ÕqÛ.È�ÞIÐ�ÕqÍK�*Ù@È"ÑYÈ�ÌWÆ=È��.ÐYÇÎ×3dzËNÐQé�á�ÇÎÌ�ëlÈ�Ì@ÑYÈ�Ð(ÇÎåYÐôÇÎÔ ø.ËNÍ ö

Ï�È�Ð.ÑYÈ�ÐàÇÎÐ È�ÇÎÐ.È�Ô �±Ó9È�òvÇ ÕqÍ ÜgÕqÍÎÍ�Ñ*ÛYÌ@ï�å*ÜaäYåYÌ@È�ÐQé�Q���y���a`�� ö �B�¥�¥â��������

p 6= 2UT�98

G��������13)dE Z

pZ�W2).U�*[*,�Â! U���8B��)h8 �R� p �PCFE %;UTJ=�RE �

σE.*,��)��R�/)�J5-�In�/�

H�G���5�SEl�D���0��)�?7EG).4H! %MJ+���5%K��).-

X���5�\J=)��5C^�=!G%K��)

GZV>�E^8 UT%Ã-t! U��³8���4

σJ=).�5C^�R! %0E�*,��)��R��)Q�.-bUT�98_! U��³8���4>fUB%MJ=�©*0%'��&$!GJ=�RE �X4��=J

2���5�vmDU:JVE 4HE )+*�#���I.4�U�I��I�J o¿ ! ���f��I�J38 �R� ��Ì�ÕqÐ.Ù3Ï�Ì@ÈvÙ@Ù3dzËNÐ

JÉA: H1(H,X)G/H → H2(G/H,X)

�/���σZ+�5�PCF! )��=!G��J+��)�H�E 4HE 4HE )V*9#��yI�4�U�I o

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p

A �CB�� Þ ä[� á�ÇÎÌÙ3Ó9È�òvÇ ÕqÍÎdzÙ3dzÈ�Ì@È�Ð Ñ*dzÈ�ÇÎÐúßk�/ö�õ/â+~YÞIÖ.Ù@ï�åYÐYÇÎ×3×D­*éûT~YÖYòvëÝé*ÇÎÐúßkÁ�±áãâ+~,��Ì@ËNÓQé���é©°*éjþ:~YÏ�È�Ï�È�Ö9È�Ð.ÈÈ��*ÓYÍÎÇÎòvÇÎ×@È7£�ÈvÙ@ï�åYÌ@È�ÇÎÖYÛYÐYÏ�ÑYÈ�ÌY��Ì�ÕqÐ.Ù3Ï�Ì@ÈvÙ@Ù3dzËNÐèÕqÛ*Ü?ÛYÐ.Ù@È�Ì@È�Ðàø�ÕqÍÎÍQÛYÐ.Ñ6É�È�Ì,ÜgËNÍÎÏ�È�ÐÊÑ*dzÈ

σö�ÞD{±×3dzËNÐQé

á�ÇÎÌ=Ï�È�å.È�ÐôÕqÛ.Ù=É�ËNÐ È�ÇÎÐ.È�Ô1ö+�IËNòn�T{±ÍÎÛ.Ù

a ∈ H1(H,X)G/HÛYÐ.ÑèÈ�ÇÎÐ.È�Ô í�È�Ì3×3Ì@È�×@È�Ì

a ∈ a~!ë�È�ͳï@å.È�Ì

ÇÎÔ Ù3Ó!È�òvdzÈ�ÍÎͳÈ�Ðúø�ÕqÍÎÍ�ÑYÈvÙ�×3Ì3ÇÎÉ�Ç ÕqͳÈ�ÐôùàË�Ñ*ÛYͳÙXÈ�ÇÎÐÃæAËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3ÔWÛ.Ù�dzÙ@×{é

aëyÇÎÌ@ÑèÐ�ÕNï�åôí�ËNÌ�ÕqÛ.Ù@Ù@È�×3òvÛYÐYÏ

ÛYб×@È�ÌAÑYÈ�ÌY�AËNÐNý�ÛYÏ¢Õq×3dzËNÐÒÔ"ÇÎ×Yz�ͳÈ�Ô�È�Т×@È�Ð ÕqÛ.ÙG�,�*dzÈ�Ì3×^�

a(g−1hg) = a(h)ÜoäYÌIÕqÍÎͳÈ

g ∈ GÛYÐ.ÑÊÕqÍÎͳÈ

h ∈ H.

Æ�Õqå.È�ÌÏ�ÇÎÍÎ×AÙ3Ó!È�òvdzÈ�ÍÎÍV�a(g1g2) = a(g2g1)

ÜaäYÌIÕqÍÎͳÈg1, g2 ∈ G

Ô"ÇÎ×g1g2 ∈ H.

áçÇÎÌ�ë�ìqåYͳÈ�Ð(È�ÇÎÐ.È�Ð���ï@åYÐYÇÎ×3×s : G/H → G

Ô"ÇÎ×s(0) = 0

é�Æ�ÕqÐYÐÃÙ@ï�åYÌ@È�ÇÎÖY×�Ù3dzï�åg ∈ G

È�ÇÎÐ.ÑYÈ�ÛY×3ÇÎÏ ÕqͳÙg = s(γ) · h

Ô"ÇÎ×γ ∈ G/H

ÛYÐ.Ñh ∈ H

éYÆ�ÕqÔ"ÇÎ×IÑYÈ��.ÐYdzÈ�Ì@È�ÐàëyÇÎÌIÈ�ÇÎÐ.È1ö+�IË {�È�×3×@È

b : G→ XÑ*ÛYÌ@ï�å

b(g) = b(s(γ) · h) := a(h) + b(s(γ)).

áçÇÎÌIÙ@È�×3ò{È�ÐàÑ.ÕqÖ!È�Çb(s(0)) = b(0) = 0

é.Æ�Õqå.È�ÌIdzÙ3×Ñ*dzÈ7z�ÇÎÐ.Ù@ï@åYÌ�ìqÐ,{�ÛYÐYÏØÉ�ËNÐbÕqÛ*Ü

HÏ�ͳÈ�dzï@å

aé.ø?äYÌ�ý3ÈvÑYÈ

ÞAÛ.Ù3ë�ÕqåYÍ�ÑYÈ�ÌAÕqÐ.ÑYÈ�Ì@È�ÐÊáúÈ�Ì3×@ÈÝÉ�ËNÐb(s(γ))

Ï�ÇÎÍÎ×

b(g + h) = b(s(γ) · h1 · h) = a(h1 · h) = a(h1) + a(h) = b(g) + a(h)

ÜoäYÌIÕqÍÎͳÈg ∈ G

ÛYÐ.Ñ ÕqÍÎͳÈh ∈ H

é*áçÇÎÌÑYÈ��.ÐYdzÈ�Ì@È�ÐàÑ*dzÈÝø.ÛYÐ,{±×3dzËNÐϕa : G/H → X, γ 7→

1

2a(

s(σγ)−1 · σs(γ))

,

ÜoäYÌÑ*dzÈ�Ï�ÇÎÍÎ×ϕa(σγ) = a

(

s(γ)−1 · σs(σγ))

= (σa)(

σs(γ)−1σs(γ))

= −(σa)(

s(σγ)−1 · σs(γ))

= −ϕσa(γ).

Æ�Õ"Ñ*dzÈ��Ì�ÕqÐ.Ù3Ï�Ì@ÈvÙ@Ù3dzËNÐÒÈ�ÇÎÐ æAËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ@Ô�Û.Ù/dzÙ3×yÛYÐ.ÑÚÐ�ÕNï@åàí�ËNÌ�ÕqÛ.Ù@Ù@È�×3òvÛYÐYÏ�Ñ*dzÈ�Ñ*ÇÎÌ@È/{±×@È��!È�Ì3ͳÈ�Ï�ÛYÐYÏH1(H,X)G/H = (H1(H,X)G/H )+ ⊕ (H1(H,X)G/H )−

È��*dzÙ3×3dzÈ�Ì3×^~"{[�NÐYÐ.È�Ð÷ëyÇÎÌWÛYÐ.ÙWÇÎÔ2ø�ËNÍÎÏ�È�Ð.ÑYÈ�ÐóÕqÛ*ÜIÑ*dzÈ�ø�ìqÍÎͳÈσa = a

˱ÑYÈ�Ìσa = −a

òvÛYÌ3ä.ïc{±òvdzÈ�å.È�ÐQé0áçÇÎÌÙ�ï@åYÌ@È�ÇÎÖ9È�ÐàÑ.ÕqÐYÐ Ù3ÛYÏ�Ï�ÈvÙ3×3ÇÎÉ σa

a = ±1é,z�ÙIÏ�ÇÎÍÎ×IÑ.ÕqÔ"ÇÎ×

ϕσa =σa

aϕa.

áçÇÎÌIÑYÈ��.ÐYdzÈ�Ì@È�РбÛYÐb(s(γ)) :=

σa

aϕa(γ).

Æ�ÕqÐYÐ Ï�ÇÎÍÎ×(σb)(g) =

σa

ab(g)ÜoäYÌIÕqÍÎͳÈ

g ∈ G,

ë�ÕNÙÔ�ÕqÐÊÑYÈ�ÌAðIÈvï@åYбÛYÐYÏ(σb)(g) = b

(

σs(γ))

+ (σa)(h) = b(

s(σγ)s(σγ)−1σs(γ))

+ (σa)(h)

= b(

s(σγ))

+ a(

s(σγ)−1 · σs(γ))

+ (σa)(h) = b(

s(σγ))

+ 2ϕa(γ) + (σa)(h)

=σa

aϕa(σγ) + 2ϕa(γ) + (σa)(h) = −

σa

aϕσa(γ) + 2ϕa(γ) + (σa)(h)

= ϕa(γ) + (σa)(h) =σa

ab(g).

È�б×3ÐYÇÎÔ"Ô�×{é!Æ�ÕqÔ"ÇÎ×dzÙ3×yÏ�È�ò{È�ÇÎÏ�×^~�Ñ.Õ$��Ñ*dzÈ���Û.ËNÌ@Ñ*ТÛYÐYÏa 7→ b

ÇÎТÉ�ÕqÌ3Ç ÕqТ×AÛYТ×@È�ÌσdzÙ@×{é

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á�ÇÎÌIÈ�Ì3å�ÕqÍÎ×@È�ÐÊТÛYÐÊÈ�ÇÎÐ.È�Ð2ö+�AËNòn�B{�È�Í

f : G×G→ XÑ*ÛYÌ@ï@åÊÑ*dzÈ�ø.ÈvÙ3×3ͳÈ�Ï�ÛYÐYÏ

f(g1, g2) := (∂b)(g1, g2) = b(g1) + b(g2)− b(g1g2).áçÇÎÌ�å�Õq×3×@È�ÐôÖ9È�Ì@È�ÇÎ×@Ù�Ï�ÈvÙ@È�å.È�Ð�~�Ñ.Õ$�ÊÑYÈ�ÌÝðAÕqÐ.ÑYËNÓ9È�Ì�Õq×@ËNÌσö�ÇÎТÉ�ÕqÌ3Ç ÕqТ×�dzÙ3×^~9ëlËNÌ�ÕqÛ.ÙWÙ3dzï@åÃÈ�Ì3Ï�ÇÎÖY×^~QÑ.Õ$�àÕqÛ.ï@å

Ñ*dzÈ���Û.ËNÌ@Ñ*ТÛYÐYÏa 7→ f

ÛYТ×@È�ÌσÇÎТÉ�ÕqÌ3Ç ÕqТ×IdzÙ@×{é

Æ/È�Ì��IËNòn�T{�È�ÍfdzÙ3×WÙ�ËS{�ËNÐ.Ù3×3Ì3ÛYdzÈ�Ì3×^~0Ñ.Õ$�èÈ�Ì�ТÛYÌ É�ËNÐ÷ÑYÈ�ÐÂÁ/È�Ö!È�Ð,{�Í ÕNÙ@Ù@È�Ð

s(γ) + HÕqÖYå�ìqÐYÏ�×^~�ë�ÕNÙ

Ô�ÕqÐ ÜoËNÍÎÏ�È�Ð.ÑYÈ�ÌIðAÈvï@åYбÛYÐYÏØÈ�Т×3ÐYÇÎÔ"ÔW×^�f(g1 · h1, g2 · h2) = b(g1 · h1) + b(g2 · h2)− b(g1 · h1 · g2 · h2)

= f(g1, g2) + a(h1) + a(h2)− a(h1 · h2)) = f(g1, g2).��ËNÔ"ÇÎ×IÈ�Ì3å�ÕqÍÎ×@È�Ð ëyÇÎÌIÕqͳÙ@Ë"È�ÇÎÐ.È�Ð2ö+�AËNòn�B{�È�Í

f ∈ H2(G/H,X)é�Á=ÕNï@åôßkÁ�±áãâ+~,��Ì@ËNÓQé���é©°*éjþ:~YëyÇÎÌ@Ñ6Ñ*dzÈ��Ì�ÕqÐ.Ù3Ï�Ì@ÈvÙ@Ù3dzËNÐ6Ñ*ÛYÌ@ï�åÚÑ*dzÈ7��Û.ËNÌ@Ñ*бÛYÐYÏ

a 7→ fÏ�È�Ï�È�Ö9È�Ð�~�ëlÈ�ͳï�å.È�Ù3dzï@å6ÕqͳÙ@Ë ÕqͳÙ

σö�ÇÎбÉNÕqÌ3Ç Õqб×yå.È�Ì�ÕqÛ.Ù3Ï�ÈvÙ3×@È�ÍÎÍÎ×

å�Õq×{é2|IÐ.Ù@È�Ì@ÈÝÖYdzÙ3å.È�Ì3ÇÎÏ�È�Ð�ÐAÖ!È�Ì3ͳÈ�Ï�ÛYÐYÏ�È�Ð�Ï.dzÈ/��È�ÐÊÈ�ÇÎÐ6ÇÎÐÊÑYÈ�Ð

�Q���y���5ø��KJ�â:�^å,�������G���5�0�b*9)dE����98,%'�R<�#:��W2)�U�*[*,�.-

H�G���5�S!�ORA��.In<�#�%KE$IcI^���0��)�?7EG).4H! %MJ+���5%K��)DUB�08

X���5�

GZ+>XE^8 UT% o ¿ ! �P�vA ��%MJ+ÊÇÆ! Èh¿ �V�DËsNT�^��Z ¼ ��).4}��Z+���>=/UP���T6

0→ H1(G/H,XH ) owpsr−→ H1(G,X) thu v−→ H1(H,X)G/H ¢ £−→ H2(G/H,XH ) oqpsr−→ H2(G,X)�yI/J3�d][! &FJ oÇlO È¥p I�JH �G

El�D���P-G�/���0�³13)dE Z

pZ�W2).U�*[*,�x��NT)

p 6= 2-X�/����J=).�5C^�R! %��)

GZ+>XE^8 UT%Ã-2! U.��8B��4 >�UT%;J=�Ã*9%;��Z&$!GJ=�RE �f4��=J

2����� p I/EG4_EG)V*�#B�yI�4�UBIª�yI/J5-tUB�98gE.*,�/).�V��)/J28G�R� p �PCFE %;U:J=�=E �

σ! U.�

GÇVUT�98 J=).�5C^�R! %"! U��

XÈ -I/E}�yI/Jx8 �V�7���>=/UP���T6�! UBI}ÇÆ! È

σZÆ���PC$! ).�R! ��J oA �CB�� Þ ä[� ø?äYÌÚûgÕ�ÿÝòvÇÎ×3dzÈ�Ì@È�Ð(ëyÇÎÌ�ßkÁ�±áãâ+~���Ì@ËNÓQé3��é©°*é©°*é�Æ/dzÈ�ÞIÛ.Ù�Ù�ÕqÏ�ÈàûaÖ�ÿ�ÜoËNÍÎÏ�×"ÕqÛ.Ù��QÈ�Ô"Ô�Õ÷ûR¨*驨*é ®±ÿ

ÕqÐYÏ�È�ë�È�Ð.ÑYÈ�×=ÕqÛ*Ü�Ñ*dzÈÝêuÐ:Ï�Õq×3dzËNÐ ÛYÐ.Ñ Ñ*dzÈ�ðAÈvÙ3×3Ì3ÇK{±×3dzËNÐÊÛYÐ.Ñ ÕqÛ.Ù��QÈ�Ô"Ô�Õ ûR¨*驨*é©°�ÿdé2

Æ/È�Ì�ÜgËNÍÎÏ�È�Ð.ÑYÈ��*Õq×3ò=Ü�Õ$��×�ëydzï�å¢×3ÇÎÏ�È�É�ËNÐ6ÛYÐ.Ù�Ö9È�ÐP�N×3ÇÎÏ�×@È�ðIÈvÙ@ÛYÍÎ×�Õq×@È=òvÛYÌσö�ÞD{±×3dzËNÐÊÕqÛ*Ü0ÑYÈ�ÐS�IËNå.ËNÔ�ËNͳËqö

Ï�dzÈ�Ï�Ì3ÛYÓYÓ9È�ÐÊòvÛ.Ù�ÕqÔ"Ô�È�ÐQé�Q���y���aj��KJ�â:�^å,�������V���

AUT�98

B!�OQ��%'I/</#:��*0)lE��/�98:%;�=<�#,��W2).U�*[*,�/�,-³C ��)cI^�.#:���Î4��5Jª8���)�Y"*,��)d!GJ=�=EG�¤���5�0��)p ��CFE %;UTJ=�RE �

σ- L E�OQ��� L �5) A = A+

UT�98B = B−

�nE )d8���)�� oÇÆ! È ËtNT)�! %5%É�7J=)���C��=! %É���h8 ��Ic&�)Q�/J+���BZV>XE�8 UB%;�

X4��=JsJ=)��5C^�=!G%K��)

σZRm�&FJ=�RE �fUT�98\! %5%K�

iA �5%;J+Ê

Hi(A,X) = H i(A,X)+.ÇlO È ËtNT)�! %5%É�7J=)���C��=! %É���h8 ��Ic&�)Q�/J+���BZV>XE�8 UB%;�

X4��=JsJ=)��5C^�=!G%K��)

σZRm�&FJ=�RE �\AG��%MJ+Ê

H1(B,X) = H1(B,X)−.Çu< È¥p I�JB�/���0�³6/w$&�%;��I/<�#:�ª13)dE Z

pZ.W2)�U^*�*,�3��NB)

p 6= 2-38�! �P�vA���%MJ+���

H2(A,Z/pZ) = H2(A,Z/pZ)+UT�98

H2(B,Z/pZ) = H2(B,Z/pZ)−.

A �CB�� Þ ä[� ûgÕ�ÿ�ÛYÐ.ÑôûaÖ�ÿlÛYÐ.Ñ Ñ*dzÈÝÈ�Ì@Ù@×@È�ÞIÛ.Ù@Ù�ÕqÏ�È=É�ËNÐÃûoï&ÿ�Ù3ÇÎÐ.Ñ\{�Í ÕqÌ{éø�äYÌyûoï&ÿ9Ö9È�×3Ì�ÕNï@å±×@È�Ð"ëyÇÎÌ�Ñ.ÕNÙ�z�Ð.ÑYÈ�ÑYÈ�Ì�ø?äYÐ*Üaö+��È�Ì3Ô�Èdöu��Èn ±Û.È�ÐYòIÕqÛ.Ù¯�*Õq×3ò�ûR¨*驨*éû�ÿ-ÖYòvÏ�Í�éqÑYÈ�Ì0Æ=ÕqÌ@Ù@×@È�ÍÎÍÎÛYÐYÏ

1→ R→ Zp → B → 1�H1(R,Z/pZ)B

¥ ¦−→ H2(B,Z/pZ)

¸kµ�^−→ H2(Zp,Z/pZ).áçÇÎÌ�Ù@È�×3ò{È�ÐèÑ*dzÈ�ÞD{�×3dzËNÐàÉ�ËNÐ

σÕqÛ*Ü

ZpÜoËNÌ3×^~9Ñ-é åQé

Zp = (Zp)−é!Æ�ÕqÐYÐÒÏ�ÇÎÍÎ×=Ë$¦-È�Ð.Ù3dzï@å±×3ÍÎdzï@å

R = R− é9Æ�ÕZpÜaÌ@È�Ç-dzÙ3×^~*Ï�ÇÎÍÎ×

H2(Zp,Z/pZ) = 0~YÛYÐ.Ñ ëyÇÎÌÈ�Ì3å�ÕqÍÎ×@È�ÐàÙ@ËNÔ"ÇÎ×IÑYÈ�Ð

σö�ÇÎбÉNÕqÌ3Ç Õqб×@È�ÐÒê�Ù@ËNÔ�ËNÌ3ÓYåYdzÙ3ÔWÛ.Ù

H1(R,Z/pZ)B → H2(B,Z/pZ).Æ�Õ Ð�ÕNï�åÃûaÖ�ÿyÕqÖ9È�ÌH1(R,Z/pZ) = H1(R,Z/pZ)−

Ï�ÇÎÍÎ×^~*ÜgËNÍÎÏ�×Ñ*dzÈÝÞAÛ.Ù@Ù�ÕqÏ�È�é2

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p

}d�h}��h�����:�z,�(p(8ä-�����0��&���1264$I¶-��$'6 ,/�Æ&�6 �0&��" G�<ó/,/3 ��$z64,/�

áçÇÎÌIÙ3×@È�ÍÎͳÈ�Ð6òvÛYÐ�ìNï�å.Ù3×AÈ�ÇÎÐYÇÎÏ�ÈÝëydzï@å¢×3ÇÎÏ�È��IËNÔ"ÔWÛY×�Õq×@ËNÌ3Ì@È�Í Õq×3dzËNÐ.È�Ð òvÛ.Ù�ÕqÔ"Ô�È�ÐQé�Q���y���ak�� ö �B�¥�¥â��������

G���5�0�hW2).U�*[*,�SUB�08ÂOQ�Q6F���R<�#��0�

G(n)8 �V�\CFEG�¤8����

nZÃ�n![<�#:������E 4�4�U:JV!GJVE )Q����/)d6F��U A[J+�X����J+�/)=A )�U�*[*,� oÇÆ! È ËtNT)�! %5%É�

x, y, z ∈ GA �5%;J+Ê

[x, yz] = [x, y][x, z][[x, y], z].ÇlO È ËtNT)�! %5%É�x, y ∈ G

UB�98\! %5%É�n ∈ N

A �5%;J+Ê(xy)n ≡ xnyn[x, y]

n(n−1)2 (

4HE^8 UT%KEG(3)).

Çu< È ËtNT)�! %5%É�x ∈ G(n)

-t!G%�%É�y ∈ G

UB�08 ! %5%É�a, b ∈ N

A �5%MJ+Ê[xa, yb] ≡ [x, y](ab) (

4HE^8 UT%MEG(n+2)).

��È�ÇpÈ�ÇÎÐ.È�ÛYÐYÏ�È�Ì�ÕNÑYÈ���Ì3ÇÎÔ"ò&ÕqåYÍ�é�áçÇÎÌ?å�ÕqÖ9È�Ð"Ï�ÈvÙ@È�å.È�Ð�~�Ñ.Õ$�

dimFpH1(G)ÕqͳÙ0Ô"ÇÎÐYÇÎÔ�ÕqͳÈYz�Ì3ò{È�ÛYÏ�È�Ì3ò&ÕqåYÍ

É�ËNÐGÇÎб×@È�Ì3ÓYÌ@È�×3dzÈ�Ì3×=ëlÈ�Ì@ÑYÈ�Ð�{NÕqÐYÐQé9õ=È�Ð�ÕqÛ.È�Ì/Ï�ÇÎÖY×AÑ*dzÈ�Æ/ÇÎÔ�È�Ð.Ù3dzËNÐèÑYÈvÙ³��ÍÎÛ.Ù,ö�ÖYòvëÝé�ÑYÈvÙ/ùÊÇÎбÛ.Ù3×@È�ÇÎͳÙIÉ�ËNÐ

H1(G)Ñ*dzÈ�Ô"ÇÎÐYÇÎÔ�ÕqͳÈ7z�Ì3ò{È�ÛYÏ�È�Ì3ò&ÕqåYÍ?ÑYÈvÙY��ÍÎÛ.Ù,ö�ÖYòvëÝé.ÑYÈvÙùàÇÎТÛ.Ù@×@È�ÇÎͳÙIÑYÈ�ÌIÞAÖ!È�ÍÎdzÙ@dzÈ�Ì3ÛYÐYÏ"É�ËNÐ

GÕqÐQé

�Q���y���=�nm�� ö �B�Â�hâ������/�G���5�0�����98:%;�R<�#f��)d6F��UGAGJ+�13)dE Z

pZ�W2).U�*[*,� o ¿ ! �P�\A ��%MJ+Ê

d(G)± = dimFp H1(G)± = d((G/G2)

±) = d((Gab)±).

A �CB�� Þ ä[� Æ/dzÈÒÈ�Ì@Ù3×@Èèõ=ͳÈ�dzï@åYå.È�ÇÎ×6ëlËNÍÎͳÈ�Ð ëyÇÎÌ ÕqͳÙÚÆ=È��.ÐYÇÎ×3dzËNÐçÉ�ËNÐd(G)±

É�È�Ì@Ù@×@È�å.È�ÐQé3z�ÙØÏ�ÇÎÍÎ×ÚÐ�ÕNï�å��È�Ô"Ô�ÕÊûR¨*驨*éK�&ÿHom((G/G2)

±,Z/pZ) = Hom(G/G2,Z/pZ)±,ë�ËNÌ�ÕqÛ.Ù"Ñ*dzÈ�òvëlÈ�ÇÎ×@È õ/ͳÈ�dzï@åYå.È�ÇÎ× ÜgËNÍÎÏ�×{é�áúÈ�Ï�È�ÐHom(Gab,Z/pZ) = Hom(G/G2,Z/pZ)

È�Ì3å�ÕqÍÎ×@È�Ð÷ëyÇÎÌÑ*dzÈ�Ñ*Ì3ÇÎ×3×@È�ÕqÛ.ÙÑYÈ�Ô Æ/Ç ÕqÏ�Ì�ÕqÔ"Ô

Gab = (Gab)+ ⊕ (Gab)−

y

y

y

G/G2 = (G/G2)+ ⊕ (G/G2)

−,ÇÎÐàÑYÈ�Ô ÕqÍÎͳÈ�0ÜgÈ�ÇÎͳÈ��±ÛYÌoý,È/{�×3dzËNÐ.È�ÐàÙ@ÇÎÐ.Ñ-é

2

�Q���y���=����� ö �B�Â�hâ������/�G���5�0�7���08:%'�R<�#\�/)d6F��U A[J+�D13)dE Z

pZ.W2)�U^*�*,�.-x! U��³8��/)7�/���0� p �PCFE %;UTJ=�RE �

σE.*,�/).�V��)/J5-InEÂ8�!da

d(G)+ = 0A ��%MJ o ¿ !G�P�¤E.*P��).�V��)/J σ !GU.� G(n)/G(n+1)

!G%ÃI p 8�����J=�=JV�GJ5-b�n! %�%'InA���)d![8����yI/J5-UB�98!G%ÃI p �PC$��).I.�RE �,-¯�n! %�%'I

nUT�BA��/)l![8B�S��I�J o ¿ !�OQ����OQ�c6����R<�#��0� G(n) L �V�d8���)�UT4 8[!$I}e3)Q6���UGA �P��Iv8B��)

nZ�n![<�#:�����E 4�4�U:JV!GJVE )Q��� o

A �CB�� Þ ä[� Æ/È�Ðf£�È�ë�È�dzÙyÜoäYåYÌ@È�Ð6ëyÇÎÌÓ9È�Ìê�Ð.Ñ*Û,{�×3dzËNÐ Ð�ÕNï@ånéTÁ=ÕNï@åÊí�ËNÌ�ÕqÛ.Ù@Ù@È�×3òvÛYÐYÏ�dzÙ@×

G/Gp[G,G] = (G/Gp[G,G])−.

ÁAÛYÐ Ö9È�×3Ì�ÕNï@å±×@È�ÐÊëyÇÎÌIÑ*dzÈ=Ð�Õq×3äYÌ3ÍÎdzï�å.È���Ì@Ëqý3È/{�×3dzËNÐ(G/[G,G])+ � (G/Gp[G,G])+ = (G/Φ(G))+,

ÑYÈ�Ì@È�Ðf£lÇÎͳÑÚ×3Ì3ÇÎÉ�Ç ÕqÍ-dzÙ3×{éYÆ/dzÈ=ÞIÐYò&ÕqåYÍ�ÑYÈ�ÌYz�Ì3ò{È�ÛYÏ�È�Ð.ÑYÈ�ÐÊÉ�ËNÐ(G/[G,G])+

dzÙ3×yÙ@ËNÔ"ÇÎ×0~YÛYÐ.Ñ

G/[G,G]dzÙ3×

Ï�ͳÈ�dzï�å(G/[G,G])−

~Yë�ÕNÙÑYÈ�ÐÊêuÐ.Ñ*Û,{±×3dzËNÐ.Ù�ÕqÐ*ÜgÕqÐYÏ�Ñ.ÕqÌ@Ù@×@È�ÍÎÍÎ×{é��È�Ç�Ñ*dzÈÝÞAÛ.Ù@Ù�ÕqÏ�È=ТÛYÐ ÜaäYÌnÖ9È�Ì@È�ÇÎ×@ÙyÖ!È�ëydzÈvÙ�È�ÐQéYá ìqåYͳÈ

x ∈ G(n) ÛYÐ.Ñ y ∈ G é.Æ�ÕqÐYÐ6dzÙ3×σ[x, y] ≡ [σx, y−1] (

Ô�˱Ñ*ÛYͳËG(3)).

Page 64: ¼ ½¿¾0À5¼ Áÿ½ÅÄ - Université du Luxembourgmath.uni.lu/~wiese/other/diplom.pdf · 2007-03-08 · ë ËNÖ!È Ç ˙ ÕqÛ*Ü A+ ×3Ì3ÇÎÉ Ç ÕqÍ ÛYÐ.ÑØÕqÛ*Ü

ùÒË�Ñ*ÛYͳËG(n+1) dzÙ3× σx È�Т×uëlÈvÑYÈ�Ì x Ë�ÑYÈ�Ì x−1 ý,È6Ð�ÕNï�åãÑYÈ�ÔS~�ËNÖ n Ï�È�Ì�ÕNÑYÈ Ë�ÑYÈ�Ì�ÛYÐYÏ�È�Ì�ÕNÑYÈÊdzÙ3×{é�ÞIÛ.Ù��È�Ô"Ô�ÕÊûR¨*驨*驱�ÿ�ûoï&ÿ�È�Ì3å�ÕqÍÎ×@È�ÐÊëyÇÎÌ

[x, y−1] ≡ [x, y]−1 (Ô�˱Ñ*ÛYͳË

G(n+2))ÛYÐ.Ñ

[x−1, y−1] ≡ [x, y] (Ô�Ë�Ñ*ÛYͳË

G(n+2)).

Æ�ÕqÌ�ÕqÛ.ÙÜoËNÍÎÏ�×ÑYÈ�ÌIêuÐ.Ñ*Û,{�×3dzËNÐ.Ù@Ù@ï@åYÌ3ÇÎ×3×{é2

�Q���y���=�:��� A �T�h�B8F��������������G���5�0����98:%;�R<�# ��)d6F��UGAGJ+�Y13)dE Z

pZ.W2)�U^*[*P�.-b!GU.�ª8���)ª�/���0� p �PC$E %'U:J=�RE �

σE.*P��Z)��V��)�J o ¿ ! �P�X��I�Jx8G�R���d][! &FJ+�����>=/UP���T6

0→ (pGab)∨ → H2(G)→ pH

2(G,Qp/Zp)→ 0.

!GU�I�0![Jy6}Ç OTo N o$# È σ ZÆ���PC$! ).�R! ��J oA �CB�� Þ ä[� Æ�Õ�Ñ*dzÈvÙ@È���Èn ¢Û.È�ÐYò ÕqÛ.ÙAÈ�ÇÎÐ.È�ÌAÍ ÕqÐYÏ�È�ÐÒÈ��.Õ${�×@È�Ð���Èn ±Û.È�ÐYòWòvÛÒÈ�ÇÎÐ.È�ÌI×3Ì3ÇÎÉ�Ç ÕqͳÈ�Ì3ëlÈ�dzÙ@È

σö�ÇÎÐYÉNÕqÌ3Ç ö

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2z�ÙÏ�ÇÎÍÎ×ÑYÈ�ÌyÜgËNÍÎÏ�È�Ð.ÑYÈ��±Ó9È�òvÇ ÕqÍ ÜgÕqÍÎÍ�ÑYÈ�Ì ��NB�P���/J5#�Z=Ë�E )�4H�/% é�Q���y���=�TÅ��KJ�â:�^å,� ÇÆ! È �����V���

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B���5�vOQ6cA % o G UT�98 H J=)���C��=! %É��)2>XE�8 UB% o ¿ ! �P�A ��%MJ+Ê

H2(G×H,B) = H2(G,B)⊕H2(H,B)⊕ (H1(G,B)⊗H1(H,B)).

ÇlO È �����V���Gi��NB)

i = 1, . . . , n���98:%;�=</#:�fW2).U�*[*,����UT�98

B���5�`OQ6QA % o ! %5%É��) Gi J=)��5C^�=!G%K��)H>XE^8 UT% o ¿ ! ���A ��%MJ+Ê

H2(

n∏

i=1

Gi, B) =

n⊕

i=1

H2(Gi, B)⊕⊕

i<j

(H1(Gi, B)⊗H1(Gj , B)).

�����08@8 �R�XW2)�U�*[*,���Gi!�Oc��%ÃIn<�#ÂUT�98@E.*P��).�V��)/J����5�0� p �PCFEG%'U:J=�=EG�

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A �CB�� Þ ä[� ûgÕ�ÿ�Æ/dzÈvÙ�ÜgËNÍÎÏ�×ÝòvÛYÔ £�È�dzÙ@ÓYdzÈ�ÍlÕqÛ.Ù�ßkÁ�±áãâ+~"z¯�-éQê@êdéK��éû�é�ûaÖ�ÿAÜgËNÍÎÏ�×ÝÇÎÐ.Ñ*Û,{±×3ÇÎÉôÕqÛ.Ù�ûgÕ�ÿ=ÛYб×@È�Ì£�È�бÛY×3òvÛYÐYÏ�É�ËNÐH2((G1 ×G2)×G3, B) = H2(G1 ×G2, B)⊕H2(G3, B)⊕ (H1(G1 ×G2, B)⊗H1(G3, B))

=

3⊕

i=1

H2(Gi, B)⊕⊕

i<j

(H1(Gi, B)⊗H1(Gj , B))

2ÆAdzÈD�=äYÐYÐ.È�×3å*ö�ø.ËNÌ3Ô�È�Í-ëlÈ�Ì@ÑYÈ�Ð�ëyÇÎÌ�Ö!È�бÛY×3ò{È�Ð�~¢ÛYÔ Ñ*dzÈIÆ/ÇÎÔ�È�Ð.Ù3dzËNÐ�É�ËNÐH2(G/G2)

± òvÛØÖ!ÈvÙ@×3ÇÎÔ"Ô�È�ÐQé�Q���y���=�Bó��KJ�â:�^å,�������

G�/���0�����08:%'�R<�#���)Q6���UGAGJ+�713)dE Z

pZ.W2)�U^*�*,� oÇÆ! È �����

d = d(G) o ¿ ! �P�vA ��%MJ+ÊdimFp H

2(G/G2) =d(d+ 1)

2.

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G-tI/E�A �5%;J¯A����0! U,�/)

dimFp H2(G/G2)

+ = d+ +

(

d+

2

)

+

(

d−

2

) UB�08dimFp H

2(G/G2)− = d− + d+ · d−,

L E�OQ��� d+ = d(G)+UT�98

d− = d(G)−AB�.In�/Jy6^J L UT)d8�� o

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p

A �CB�� Þ ä[� ûgÕ�ÿlÜoËNÍÎÏ�×IÕqÛ.ÙWûaÖ�ÿ�~YÇÎÐ.ÑYÈ�ÔGÔ"ÇÎ×IÑYÈ�Ìy×3Ì3ÇÎÉ�Ç ÕqͳÈ�Ð

σö�ÞD{�×3dzËNÐ6É�È�Ì@Ù@È�å.È�Ð ëyÇÎÌ@Ñ-é

ÞIÛ.ÙD�*Õq×3ò�ûR¨*驨*驨�ÿ�È�Ì3å�ÕqÍÎ×@È�ÐÊëyÇÎÌIÈ�ÇÎÐ.Èσö�ÇÎбÉNÕqÌ3Ç Õqб×@È��!È�Ì3ͳÈ�Ï�ÛYÐYÏ

G/G2∼= A1 ⊕ · · · ⊕Ad+ ⊕B1 ⊕ · · · ⊕Bd− ,ë�ËNÖ!È�ÇAÑ*dzÈ

AiÛYÐ.Ñ

Bjõ/Ì3ÛYÓYÓ9È�Ð ÑYÈ�ÌgÒAÌ@Ñ*бÛYÐYÏ

pÙ3ÇÎÐ.Ñ�~?ÜoäYÌ�Ñ*dzÈ

Ai = A+i

ÛYÐ.ÑBj = B−

j

Ï�ÇÎÍÎ×{é�Æ/dzÈ�=äYÐYÐ.È�×3å*öu�!È�Ì3ͳÈ�Ï�ÛYÐYÏÚÉ�ËNÐH2(G/G2)

å�Õq×Ñ.ÕqÐYÐàÑ*dzÈWõ=ÈvÙ3×�ÕqÍÎ×

H2(G/G2) ∼=

d+⊕

i=1

H2(Ai) ⊕⊕

i<j

H1(Ai)⊗H1(Aj) ⊕

i<j

H1(Bi)⊗H1(Bj)

⊕d+⊕

i=1

d−⊕

j=1

(H1(Ai)⊗H1(Bj)) ⊕

d−⊕

j=1

H2(Bj),

ë�ËNÖ!È�ÇAÐ�ÕNï@å �*Õq×3ò(ûR¨*驨*é©��ÿσÕqÛ*Ü�ÑYÈ�ÌØÈ�Ì@Ù3×@È�Ð��!È�ÇÎͳÈÒÕqͳÙ�êuÑYÈ�б×3ÇÎ×�ìq×ØÛYÐ.ÑçÕqÛ*ÜÝÑYÈ�Ì�òvëlÈ�ÇÎ×@È�Ð ÕqͳÙ�êuТÉ�È�Ì@Ù3dzËNÐ

ËNÓ9È�Ì3dzÈ�Ì3×{é"��Ë&ëlËNåYÍlÑ*dzÈ_z�Ì3ò{È�ÛYÏ�È�Ð.ÑYÈ�Ð*ö=ÕqͳÙ�ÕqÛ.ï�åôÑ*dzÈ"ðIÈ�Í Õq×3dzËNÐ.È�ÐYò&ÕqåYÍlÉ�ËNÐZ/pZ

dzÙ3×1~9ëlÈvÙ3å�ÕqÍÎÖôÙ@dzï@åúÑ*dzȣ�È�å�ÕqÛYÓY×3ÛYÐYÏ�Ñ*ÛYÌ@ï@åàÞAÖYò&ìqåYͳÈ�ÐÊÑYÈ�ÌIÆ/ÇÎÔ�È�Ð.Ù3dzËNÐ.È�ÐàÈ�Ì3Ï�ÇÎÖY×{é

2

}d�h}��a¨�����,!$'&��0©�z&�6 ��.�&��Z�z,�(p(¸äå�C������&���1Á6i$Q¶-��$z64,/�ô&�6s��&��" 8�<ó/,/3s�d$z64,/�

��È�ÇpÈ�ÇÎÐ.È�ÛYÐYÏ�È�Ì�ÕNÑYÈ\��Ì3ÇÎò&ÕqåYÍ�é?áçÇÎÌ�{�ËNÔ"Ô�È�Ð÷ТÛYÐ÷òvÛ÷È�ÇÎÐ.È�Ô2ðAÈvÙ3ÛYÍÎ×�Õq×^~�Ñ.ÕNÙWÜoäYÌWÑ*dzÈg£�È�×3Ì�ÕNï@å¢×3ÛYÐYÏúÑYÈ�Ì

õÝÕqͳËNdzÙ,ö�õ/Ì3ÛYÓYÓ9È�Ð É�ËNÐÒü�ùÊö+���NÌ3Ó9È�Ì3ÐÊÉ�ËNÐ�ÁAÛY×3ò{È�Ð Ù@È�ÇÎÐ ëyÇÎÌ@ÑôûaÉ�Ï�Í�é-ß á�â+~,��Ì@ËNÓ!Ë�Ù@ÇÎ×3dzËNÐX¨*éK�&ÿdé�Q���y���=�:ô��KJ�â:�^å,�������

G�������_���98:%;�=</#��/)d6F��U A[J+��*:EGJ+���:6/)Q���=</#:�713)dE Z

pZ.W2)�U^*[*P�.-?!GU.� L �/%M<�#,��)_�/���0� p �PC$E %'UTZJ=�RE �

σE.*,�/).�V��)/J o p I�J d(G+) = 0

-28[! �P�X�yI/JG!�OQ��%'I/</# oA �CB�� Þ ä[� áúÈ�Ï�È�Ð

d(G+) = 0È�Ì3å�ÕqÍÎ×@È�ÐØëyÇÎÌ�ÕqÛ.Ùs��È�Ô"Ô�ÕØûR¨*驨*éK�G�&ÿ�~�Ñ.Õ$�

σÕqͳÙlê�бÉ�È�Ì@Ù3dzËNÐ�ÕqÛ*Ü

G/[G,G]ÛYÐ.ÑàÕqͳÙêuÑYÈ�б×3ÇÎ×�ìq×/ÕqÛ*Ü[G,G]/[G,G,G]

ËNÓ9È�Ì3dzÈ�Ì3×{éP�QÈ�×3òv×@È�Ì@ÈvÙAÇÎÔ"ÓYÍÎÇÎòvdzÈ�Ì3×IÇÎÐ.Ù@Ö!ÈvÙ@ËNÐ.ÑYÈ�Ì@È[G,G]/H = ([G,G]/H)+,ë�ËNÖ!È�Ç

H = [G,G]p[G,G,G]Ï�ÈvÙ@È�×3òv×ëyÛYÌ@ÑYÈ�é��È�Ç

x ∈ Gé!Æ�ÕqÐYÐÒÏ�ÇÎÖY×�ÈvÙAÈ�ÇÎÐ

r ∈ [G,G]Ô"ÇÎ×

σx = x−1ré.áçÇÎÌAÖ9È�Ì@Èvï@åYÐ.È�РбÛYÐèÑ*dzÈ

σö�ÞD{±×3dzËNÐ ÕqÛ*Ü

ÕqÛ*ÜGpH/H

�σxp = (σx)p = (x−1r)p ≡ x−p (

Ô�˱Ñ*ÛYͳËH),ë�ËNÖ!È�Ç�Ñ*dzÈÝðIÈvï�å.È�ÐYÌ@È�Ï�È�Í�ûaÖ�ÿ�É�ËNÐS�QÈ�Ô"Ô�ÕàûR¨*驨*驱�ÿ�Ö9È�ТÛY×3òv×IëyÛYÌ@ÑYÈ�é�ÞIͳÙ@Ë Ï�ÇÎÍÎ×

GpH/H = (GpH/H)−.Æ=ÕGÓ!ËN×@È�ÐYòvÌ@È�dzï�åÊdzÙ3×^~.È�Ì3å�ÕqÍÎ×@È�ÐÊëyÇÎÌIÑ*dzÈÝêuÐ,{±ÍÎÛ.Ù3dzËNÐ

([G,G]/H)+ = [G,G]/H ⊆ GpH/H = (GpH/H)−,ë�ËNÌ�ÕqÛ.ÙÜgËNÍÎÏ�×^~YÑ.Õ$�[G,G]/H

×3Ì3ÇÎÉ±Ç ÕqÍ�dzÙ3×{é��È�×3òv×Ô�ÕqÐK = K(1) = [G,G]

ÛYÐ.ÑK(n+1) = (K(n))p[Kn, G]

~YÑ-é åQéK(2) = H

~*Ù@Ë�ÜoËNÍÎÏ�×IÕqÛ.Ù��*Õq×3òûR¨*éK��éK�^­�ÿ�ûgÕ�ÿ�~!Ñ.Õ$�6Ñ*dzÈ

K(i) Èvï@å±×=òvÛYÌ 1ÕqÖ.Ù3×@È�ÇÎÏ�È�ÐQé�áúÈ�Ï�È�Ð

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0→ H1(G/G2)∼−→ H1(G)

0−→H1(G2)

G ¥ ¦−→ H2(G/G2)

¸kµ�^−→ H2(G)ûaÉ�Ï�Í�éP�*Õq×3ò�ûR¨*驨*éû�ÿ3ÿlÛYÐ.Ñv�*Õq×3ò�ûR¨*驨*éK�/®±ÿ�È�Ì@Ù@È�å.È�ÐÚëyÇÎÌyÙ@ËqÜgËNÌ3×^~*Ñ.Õ$��Ö9È�Ç9È�ÇÎÐ.È�Ì�È�Ð.Ñ*ÍÎdzï�åÊÈ�Ì3ò{È�ÛYÏ�×@È�Ð6Ó9ËN×@È�ÐYòvÌ@È�Ç ö

ï�å.È�Ðf��Ì@Ëqöpö�õ=Ì3ÛYÓYÓ!È

GÙ3×@È�×@Ù

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(

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+ d(G) − dim(G).

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di := (Gi : Gi+1)é.Æ�Õ

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|H/Hn| ≤ |G/Gn|éÁIÛYÐ å�ÕqÖ9È�Ð6ëyÇÎÌAÕqÖ9È�ÌIÕqÛ.ï@åÒÈ�ÇÎÐ.È7��Ì@Ëqý3È/{±×3dzËNÐ

π : H � G,

Ñ.ÕØÑ*dzÈ�ðIÈ�Í Õq×3dzËNÐ.È�ÐèÉ�ËNÐHÕqÛ.ï�åÒÇÎÐ

GÈ�Ì,ÜaäYÍÎÍÎ×=Ù3ÇÎÐ.Ñ-é!ÞIÛ,��È�Ì@ÑYÈ�Ô Ï�ÇÎÍÎ×

π(Hn) = Gné�Æ=ÈvÙ3ë�È�Ï�È�Ð È�Ì3å�ÕqÍÎ×@È�Ð

ëyÇÎÌIÉ±Ç Õπê�Ù@ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�È�Ð

H/Hn∼= G/Gn.z�ÇÎÐSz�ͳÈ�Ô�È�Т×IÕqÛ.ÙyÑYÈ�Ô ��È�Ì3ÐÚÉ�ËNÐ

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1dzÙ3×{é�Æ=Õqå.È�Ì

dzÙ@×HòvÛ

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p

�Q���y���=�Pø9��ñ}�9F�0ß5ß5â,B�������G���5�0�v���98,%'�R<�#Â��)Q6���UGAGJ+�\UT�P� �/EG).4}��13)dE Z

pZ�W2).U�*[*,� o ¿ ! �P�j��I�J H1(G2)

G =H1(G2)

UB�98g8 �V�7���>=/UP���T60→ H1(G2) ¢ £−→ H2(G/G2) owpsr−→ H2(G)→ 0��I�J2�d][!$&$J o

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ÇÎÐ*Ü)) = dimFp H

2(G/G2)− dimFp H1(G2)

G ≥ d(G) +

(

d(G)

2

)

− d(G2) ≥

(

d(G)

2

)

.

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2(G) =(

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) ~�ë�ÈvÙ3å�ÕqÍÎÖ Ñ*dzÈD£�È�å�ÕqÛYÓY×3ÛYÐYÏ�È�ÐÚÜgËNÍÎÏ�È�ÐQé2

æIdzÈ�Ì�ÕqÛ.Ù2{[�NÐYÐ.È�Ð6ëyÇÎÌyбÛYÐ Ñ*dzÈ/Æ/ÇÎÔ�È�Ð.Ù3dzËNÐ.È�Ð ÑYÈvÙ?��ÍÎÛ.Ù,ö�ÛYÐ.ÑÚÑYÈvÙyùÊÇÎбÛ.Ù3×@È�ÇÎͳÙ�É�ËNÐH2(G)

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dimFp H2(G)+ =

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2

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+

(

d(G)−

2

) UT�98dimFp H

2(G)− = d(G)+ · d(G)−.

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pZ.W2)�U�*[*,� o p I�J d(G) = 1

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-x8[! ����#T![JGab

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A �CB�� Þ ä[� ê�Ù3×d(G) = 1

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(

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= 0dzÙ3×{é9Æ=Õqå.È�Ì�dzÙ@×

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G = GabdzÙ@ËNÔ�ËNÌ3ÓYåÊòvÛ

Zpé

ê�Ô ø�ÕqÍÎÍd(G) = 2

dzÙ3×�ÑYÈ�Ì�ðAÈ�Í Õq×3dzËNÐ.È�ÐYÌ�ÕqÐYÏr(G) =

(

22

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= 1~±ÛYÐ.ÑS�*Õq×3ò ûR¨*éK��éû�ÿ�ÍÎdzÈdÜgÈ�Ì3×�Ñ.ÕNÙ2z�Ì3Ï�È�ÖYÐYdzÙ{é

2

Ý $å[�B�"���T�/;�� Þ �`ìs�P�^�B�"å[$� Þ5÷ ���T� ´ $�,ëpënê\$�"ì"ìs�B�

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σE.*,��)��V��)�J o ¿ ! ���\A �5%MJP��NB)�!G%�%É� i ∈ Z

Êd(G)± ≥ d(Gi)

±.

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σö�ÇÎТÉ�ÕqÌ3Ç ÕqТ×@È_�±ÛYÌoý,È/{�×3dzËNÐ

G/G2 � Gi/Gi+1 = Gi/(Gi)2.��È�Ô"Ô�ÕÊûR¨*驨*éK�^²�ÿ�ÍÎdzÈdÜgÈ�Ì3×ТÛYÐàÑ.ÕNÙ�z�Ì3Ï�È�ÖYÐYdzÙ{é2ÁIÛYÐX{�ËNÔ"Ô�È�ÐèëyÇÎÌAòvÛèÈ�ÇÎÐ.È�Ô ëydzï�å¢×3ÇÎÏ�È�ÐúðAÈvÙ3ÛYÍÎ×�Õq×=ÕqÛ.Ù�ß á�âyû=��Ì@ËNÓ9Ë�Ù3ÇÎ×3dzËNÐ@¨*é©­�ÿyäYÖ9È�Ì=Ñ*dzÈ�£�È�òvdzÈ�å¢ÛYÐYÏ

ÑYÈ�ÌD��ÍÎÛ.Ù,ö�ÛYÐ.Ñ ùàÇÎТÛ.Ù@×@È�ÇÎͳÈ�É�ËNÐH1(G)

~!ÕqͳÙ@Ë É�ËNÐd(G)+

òvÛd(G)−

é�Q���y���y�����KJ�â:�^å,�������

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σ o ¿ ! �P��A��/%;J+����ÊÇÆ! È (

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≤ d(G)+ + dimFp pH2(G,Qp/Zp)+,ÇlO È

d(G)+ · d(G)− ≤ d(G)− + dimFp pH2(G,Qp/Zp)−.p I/J

GUB��� �nE )�4 4��=J3�/�98:%;�=<�#,��)³mOQ��%;��I.�V��).UT�BA$-?8[! �P�\A �5%;Jx���hOQ���R8�����Ë3� %5%K�/�rW2%É���R<�#[#:���=J o

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A �CB�� Þ ä[� ÞIÛ.ÙÑYÈ�Ô z�Ð.ÑYÙ3×3ä.ïQ{6ÑYÈ�Ìσö�ÇÎбÉNÕqÌ3Ç Õqб×@È�ÐÒÈ��.Õ${±×@È�Ðf��Èn ±Û.È�ÐYòØûaÉ�Ï�Í�é9�*Õq×3òØûR¨*驨*éû�ÿ3ÿ

0→ H1(G2)G ¥ ¦−→ H2(G/G2)

¸kµ�^−→ H2(G)

ÛYÐ.Ñ�£�È�Ô�È�Ìd{�ÛYÐYÏàûR¨*驨*éK�^¨�ÿÈ�Ì3å�ÕqÍÎ×@È�Ð ëyÇÎÌIÑ*dzÈ7|AÐYÏ�ͳÈ�dzï@å±ÛYÐYÏdimFp H

2(G/G2)± ≤ d(G2)

± + dimFp (pGab)± + dimFp pH

2(G,Qp/Zp)±.

Æ�ÕGÓ9ËN×@È�ÐYòvÌ@È�dzï@åèdzÙ3×^~�Ï�ÇÎÍÎ×IÐ�ÕNï@å@�QÈ�Ô"Ô�ÕÒûR¨*驨*驨G²�ÿ

d(G2)± ≤ d(G)±

é�ÞAÛ.Ùª�*Õq×3ò6ûR¨*éK��驱�ÿÙ�ï@åYÍÎdzÈ/��È�ÐÒëyÇÎÌë�È�ÇÎ×@È�ÌAÑ*dzÈ7|AÐYÏ�ͳÈ�dzï@å±ÛYÐYÏ

dimFp (pGab)± = d(G)± − rkZp(G

ab)± ≤ d(G)±,

Ù�Ë"Ñ.Õ$��ëyÇÎÌdimFp H

2(G/G2)± ≤ 2d(G)± + dimFp pH

2(G,Qp/Zp)±

Ö9È/{�ËNÔ"Ô�È�ÐQéx��È�×3ò{È�ÐóëyÇÎÌ"бÛYÐ Ñ*dzÈ6ÆAÇÎÔ�È�Ð.Ù@dzËNÐóÉ�ËNÐH2(G/G2)

± È�ÇÎÐ�~?Ñ*dzÈÚÇÎÐÎ�*Õq×3òôûR¨*驨*éK�/®±ÿWÖ9È�Ì@Èvï@åYÐ.È�×ëyÛYÌ@ÑYÈG~�Ù@Ë�ÜgËNÍÎÏ�×Ñ.ÕNÙðIÈvÙ3ÛYÍÎ×�Õq×{é£�È�бÛY×3ò{È�Ð6ëyÇÎÌ?�IËNÌ@ËNÍÎÍ ÕqÌWûR¨*驨*éK��»�ÿ�ÛYÐ.Ñd(Gi)

± = d(G)±~�Ù@Ë�È�Ì3å�ÕqÍÎ×@È�Ð6ëyÇÎÌ�ÇÎÔ ø�ÕqÍÎͳÈ�È�ÇÎÐ.È�Ì�ÛYÐYÇ ÜgËNÌ3Ô�È�Ð

õ=Ì3ÛYÓYÓ!È�Ô"ÇÎ×AÈ�Ð.Ñ*ÍÎdzï@å.È�ÌAÞAÖ9È�ÍÎdzÙ3dzÈ�Ì3ÛYÐYÏ�õ/ͳÈ�dzï�åYå.È�ÇÎ×@È�ÐQé2z�Ù?dzÙ3×�É�ËNÐ êuТ×@È�Ì@ÈvÙ@Ù@ÈG~qø�ìqÍÎͳÈ�òvÛ�{�È�ÐYÐ.È�Ð�~qÇÎÐ�ÑYÈ�Ð.È�Ð

pH2(G,Qp/Zp)−

×3Ì3ÇÎÉ±Ç ÕqÍ*dzÙ3×{écÆAdzÈvÙ?×3Ì3ÇM¦9×�òvÛYÔ"ÇÎÐ.ÑYÈvÙ@×ÜoäYÌ

d(G)+ = 1ÛYÐ.ÑÚÜoäYÌ

d(G)− = 0òvÛ�~*Ü�ÕqÍÎͳÙ

GÛYÐYÇ ÜoËNÌ3Ô Ô"ÇÎ×IÈ�Ð.Ñ*ÍÎdzï@å.È�Ì/ÞIÖ9È�ÍÎdzÙ3dzÈ�Ì3ÛYÐYÏ�dzÙ3×{é

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d(G) = 3È�ÇÎÐ.È£�ÈvÙ@ËNÐ.ÑYÈ�Ì3å.È�ÇÎ×WÑ.ÕqÌ^~!ÑYÈ�ÐYÐúÑYÈ�Ìz�Ì3ò{È�ÛYÏ�È�Ð.ÑYÈ�ÐYÌ�ÕqÐYÏàdzÙ3×=Ï�ͳÈ�dzï@åôÑYÈ�Ô ðAÈ�Í Õq×3dzËNÐ.È�ÐYÌ�ÕqÐYÏ.éQÆ�ÕqÌ�ÕqÛ.Ù�È�Ì3å�ìqÍÎ×�Ô�ÕqÐz�ÇÎÐ.Ù@ï@åYÌ�ìqÐ,{�ÛYÐYÏ�È�Ð÷ÜaäYÌ Ñ*dzÈ}��ÍÎÛ.Ù,ö/ÛYÐ.Ñ(ùàÇÎТÛ.Ù3×@È�ÇÎͳÈ�é0Æ�ÕNÙ�ëyÇÎÌ@ÑÃÉ�ËNÐ(Ï�Ì@Ë ��È�ÔñêuТ×@È�Ì@ÈvÙ�Ù@È"ÜoäYÌWÛYÐ.Ù�Ù@È�ÇÎÐ�~?Ñ.Õ

Ñ*dzÈÓ!ËN×@È�ÐYòvÌ@È�dzï�å.È�Ð6õ�ÕqͳËNdzÙ3ö�õ/Ì3ÛYÓYÓ!È�Ð�É�ËNÐ�ü�ù ö+�D�NÌ3Ó!È�Ì3Ð�~±Ñ*dzÈyëyÇÎÌ�Ö9È�×3Ì�ÕNï@å±×@È�ÐØë�È�Ì@ÑYÈ�Ð�~�z�Ì3ò{È�ÛYÏ�È�Ð.ÑYÈ�ÐYÌ�ÕqÐYÏ{�ͳÈ�ÇÎÐ.È�ÌIË�ÑYÈ�ÌyÏ�ͳÈ�dzï@å�­�Ö!ÈvÙ3ÇÎ×3ò{È�ÐQé�Q���y���y�����KJ�â:�^å,�������

G�/���0��UT�P� �/E )�4}� 12)lE Z

pZ.W2)�U�*[*,�.-_! U.��8B��)X���5�0� p �PCFE %;U:J=�=E �

σE.*P��).�V��)/J o exI\I^���

d(G) ≤ 3-tUT�98\8 �R�³mOQ��%;��I.�V��).UT�BAgCFEG�

GIn���2���98:%;�=</# o ¿ !G�P�\AG��%MJ

d(G)+ = 1UT�98

d(G)− = 2

E�8���)d(G)+ = 3

UB�08d(G)− = 0.

A �CB�� Þ ä[� áúÈ�Ï�È�Ð_�IËNÌ@ËNÍÎÍ ÕqÌIûR¨*驨*éK�^±�ÿ�{�ÕqÐYÐWбÛYÌ�ÑYÈ�Ì0ø�ÕqÍÎÍd(G) = 3

ÕqÛ*Üa×3Ì@È�×@È�ÐQéNÆ�Õ3 = r(G) = d(G) =

dimFppGab = dimFpH

2(G)ÕqÛ.Ù�ÑYÈ�Ð í�ËNÌ�ÕqÛ.Ù@Ù@È�×3òvÛYÐYÏ�È�ÐèÜgËNÍÎÏ�×^~-È�Ì3å�ÕqÍÎ×@È�Ð ëyÇÎÌ�ÕqÛ.Ù£�È�Ô�È�Ìd{±ÛYÐYÏôûR¨*驨*éK�^¨�ÿ

Ñ*dzÈσö�ÇÎбÉNÕqÌ3Ç Õqб×@ÈWê�Ù@ËNÔ�ËNÌ3ÓYåYdzÈ

(pGab)∨ ∼= H2(G)

ÛYÐ.ÑÊÑ.Õqå.È�ÌAÕqÛ.ÙY�IËNÌ@ËNÍÎÍ ÕqÌ ûR¨*驨*éK�^��ÿd(G)− = dimFpH

2(G)− = d(G)+ · d(G)−,

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p 6= 2 o ¿ �R�_&$E 4D*9%É�d]��_��E ��¡.UGA[!GJ=�RE �j��� k|k+OQ��Z6F���R<�#��0��� L �5) L �R�Q8���)74��=J σ o ¿ !G�P�\AG��%MJ+ÊÇÆ! È ËtNT)�! %5%É�

i ∈ Z�yI/Jx8��/) p I/E 4HE )+*�#���I.4�UBIHi(G(LS(p)|k), ES(LS(p))) ∼= H2−i(G(LS(p)|k),Qp/Zp)

∨.! U�I ¼ #,�dE )Q��4 Ç�N oPO:oPO<; È ÇlO È σ Z+�5�PCF! )��=!G��J oÇlO È ËtNT)�! %5%É�i ∈ Z

�yI/Jx8��/) p I/E 4HE )+*�#���I.4�UBIHi(G(kS(p)|k), ES(kS(p))) ∼= H2−i(G(kS(p)|k),Qp/Zp)

∨.! U�I ¼ #,�dE )Q��4 Ç�N oPO:oañ>= È σ ZÆ���PC$! ).�R! ��J oA �CB�� Þ ä[� z�ÇÎÐ.È�ÌØÞAÐ�ÕqÍK��Ù�È6ÑYÈvÙH£�È�ëlÈ�dzÙ@ÈvÙ�É�ËNТ��å.ÈvËNÌ@È�Ô ûl��éK��é©­$®±ÿ�È�Т×3ÐYÇÎÔ"Ô�×�Ô�ÕqÐ�~�Ñ.Õ$�ôÙ@dzï@å Ö9È�dzÑYÈ

ê�Ù@ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�È�ÐÒòvÛ.Ù�ÕqÔ"Ô�È�Ð.Ù@È�×3ò{È�ÐúÕqÛ.ÙAí�È�Ì3ÖYÇÎÐ.Ñ*ÛYÐYÏ�Ù3å.ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�È�Ð�~QÑYÈ�Ô £�ÇÎͳÑYÈ�ÐàÉ�ËNÐX��ËNб×3Ìoý3ÕqÏ�ÇÎÐ*öÆ/Û�ÕqͳÈ�Ð�~*ÑYÈ�Ô ðIÈ�òvÇÎÓYÌ@ËNòvÇÎ×�ìq×@Ù3å.ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ@Ô�Û.ÙAÛYÐ.Ñ�ÑYÈ�Ô æ/ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�Û.ÙIÕqÛ.ÙyÑYÈ�Ôi�*Õq×3ò=É�ËNÐvÁ/Õ${�Õcö�¢ÕqÔ�Õcö+�?Õq×@È�é�í�ËNÐÒÕqÍÎÍQÑ*dzÈvÙ@È�Ð å�ÕqÖ9È�Ð ëyÇÎÌÑ*dzÈ

σö�ê�бÉNÕqÌ3Ç ÕqÐYòWÖ!È�ëydzÈvÙ�È�ÐQé

2

}d�h���h���Xê����? 8�0&�#!3Ó(Z�0��� î<$'�z#!.03 �d34#!�S�8&���5��C�0�0��&¹ó/,/�2.@0ß(+*-���C��&��C�êuÐ Ñ*dzÈvÙ@È�Ô ÞAÖ.Ù@ï@åYÐYÇÎ×3×ØÙ3×@È�ÍÎͳÈ�Ð ëyÇÎÌ�ðIÈvÙ@ÛYÍÎ×�Õq×@È6äYÖ!È�Ì�Ñ*dzÈ

Sö�ê�ÑYÈ{ÕqÍK{�Í ÕNÙ@Ù@È�ÐYÏ�Ì3ÛYÓYÓ9È6ÛYÐ.ÑóÑ*Ç³È ê�ÑYÈ{ÕqÍK{±Í ÕNÙ@Ù�È�Ð*ö

Ï�Ì3ÛYÓYÓ9ÈAÔ�˱Ñ*ÛYͳË�È�ÇÎÐ.È�Ô ùÒË�Ñ*ÛYÍmòvÛ.Ù�ÕqÔ"Ô�È�ÐQé*Æ�ÕqÖ!È�Ç!ËNÌ3dzÈ�Т×3dzÈ�Ì@È�ÐÚëyÇÎÌ�ÛYÐ.Ù�ë�È�ÇÎ×@ÈvÙ3×3Ï�È�å.È�Ð.Ñ6ÕqÐàß áôÕqâ�é±Æ=ÕNÙ

æ=ÕqÛYÓY×3Ì@ÈvÙ3ÛYÍÎ×�Õq×AdzÙ3×È�ÇÎÐ.È�í�È�Ì@Ù3dzËNÐÊÑYÈvÙ��QÈvËNÓ!ËNͳÑ*×@Ù�ï@å.È�Ð��±ÓYdzÈ�Ï�È�ÍÎÛYÐYÏ�Ù�Ù�Õq×3ò{ÈvÙ{é�Q���ÉÅ��ak��KJ�â:�^å,�������

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pZÆJ+�/�_e3�5�,#:�/�5J5I L UB)Q6���%;� µp(k) ����J5#T!G%;J+� o·l �5)�OQ�/J=)d![<�#BJ+�/��/���0�3A[!G%ME ��IcIn<�#:�?e3) L ���=J+��)�UB�BA L|k+

-�CFE �S8���) L ��)��nE )d8���)��,-�8[!daÂ8 �R��WY! %KE ��IlA )�U�*[*,� H := G(L|k)!BOQ��%'I/<�#UT�98 CFE 4 eb].*:E ������J+���

p 6= 2In��� o �����,�^��)��0��)

B ⊆ k×/(k×)p�/���0���"��J+��)RA ).U�*[*,��4��=J

L = k(B1p ),

L �/%M<�#,���9![<�#�8���) ¼ #:�dE )��R�ª��UT4�4}��).I/</#:��)ª��( )+*,��) ÇVC.A % o 6 o L oBA ?Z=�ª� ¼>C -b��!GJy6 pdpdp o N oq[ È �d] ��I�J=�V��)/J o¿ ! ����8��Óbt�P�V��)�JH ×B → µp(k), < h, b >:=

h(b1/p)

b1/p�/���0���P�R<�#BJ=Zl! UBIlA��d!G)�J+�/J+��1?![! )�UB�TA$-38 �V� �=ÛYÔ"Ô�È�Ì,ö+�?Õ�ÕqÌ3ÛYÐYÏ - L ��%K<�#:� σ Z+�5�PCF! )��R! ��Jt��I�J5-x8 o # o< hσ, bσ >= (< h, b >)σ,

L EBOQ��� σ L �R�d8B��).UT4 8 �R�&$E 4D*9%É�d]����E ��¡.UGA[!GJ=�RE ����� k|k+Oc�Q6����R<�#���� o

A �CB�� Þ ä[� ��ÛYÐ�ìNï@å.Ù3×/Ù@È�dzÈ�Ðh ∈ H

ÛYÐ.Ñb1, b2 ∈ B

éYÆ=ÕqÐYÐÊdzÙ3×

< h, b1b2 >=h(b

1/p1 b

1/p2 )

b1/p1 b

1/p2

=h(b

1/p1 )

b1/p1

h(b1/p2 )

b1/p2

=< h, b1 >< h, b2 > .

��È�dzÈ�ÐãëlÈ�ÇÎ×@È�Ìh1, h2 ∈ H

ÛYÐ.Ñb ∈ B

é0áúÈ�ÐYÐiÙ@ËèÏ�È�ë�ìqåYÍÎ×�dzÙ3×^~�Ñ.Õ$�

h2(b1/p) = ζ ipb

1/p Ï�ÇÎÍÎ×^~�ëlËNÖ9È�Çζp ∈ µp(k)

È�ÇÎÐ.È�ÓYÌ3ÇÎÔ"ÇÎ×3ÇÎÉ�Èpö�×@Èz�ÇÎÐYå.È�ÇÎ×@Ù3ëyÛYÌ3ò{È�Í0Ù@È�ÇV~YÑ.ÕqÐYÐÊÈ�Ì3å�ÕqÍÎ×@È�ÐÊëyÇÎÌIÑ*dzÈWõ/ͳÈ�dzï�åYå.È�ÇÎ×h1 ◦ h2(b

1/p)

h2(b1/p)=h1(ζ

ipb

1/p)

ζipb1/p

=h1(b

1/p)

b1/p.

Æ�ÕqÔ"ÇÎ×IÈ�Ì3Ì@Èvï�åYÐ.È�ÐàëyÇÎÌ

< h1h2, b >=h1 ◦ h2(b

1/p)

b1/p=h1 ◦ h2(b

1/p)

h2(b1/p)

h2(b1/p)

b1/p=< h1, b >< h2, b > .

Page 72: ¼ ½¿¾0À5¼ Áÿ½ÅÄ - Université du Luxembourgmath.uni.lu/~wiese/other/diplom.pdf · 2007-03-08 · ë ËNÖ!È Ç ˙ ÕqÛ*Ü A+ ×3Ì3ÇÎÉ Ç ÕqÍ ÛYÐ.ÑØÕqÛ*Ü

áúÈ�ÐYÐiÙ@Ë Ï�È�ë�ìqåYÍÎ×AdzÙ3×^~YÑ.Õ$�

σ(b1/p) = ζ ipb1/p Ï�ÇÎÍÎ×^~.Ö!È/{�ËNÔ"Ô�È�Ð ëyÇÎÌAÑ*Ç³È σ ö�ê�бÉNÕqÌ3Ç ÕqÐYò

< hσ, bσ >=σ−1 ◦ h ◦ σ(σ(b1/p))

σ(b1/p)= σ

h(ζ ipb1/p)

ζipb1/p

= (< h, b >)σ,

ë�ËNÖ!È�Çσ ∈ G(L|k+)

È�ÇÎÐ.È�ø.ËNÌ3×@Ù@È�×3òvÛYÐYÏ�É�ËNÐσÖ!È�ò{È�dzï�åYÐ.È�×{é

áúÈ�ÇÎ×@È�ÌIò{È�ÇÎÏ�È�Ð ëyÇÎÌ^~.Ñ.Õ$��Ñ*dzÈ��?Õ�ÕqÌ3ÛYÐYÏØÐYdzï@å¢×,öuÕqÛ.Ù@Ï�È{ÕqÌ3×@È�×/dzÙ@×{é�õ/ÇÎÍÎ×yÜoäYÌÈ�ÇÎÐh ∈ H

< h, b1/p >= ζ ip = 1ÜoäYÌAÕqÍÎͳÈ

b ∈ B,

Ñ.ÕqÐYÐ6ËNÓ9È�Ì3dzÈ�Ì3×h×3Ì3ÇÎÉ�Ç ÕqÍ-ÕqÛ*Ü

BÛYÐ.Ñ�Ñ.ÕqÔ"ÇÎ×ÕqÛ*Ü

L|ké±ø�ËNÍÎÏ�ÍÎdzï@å6dzÙ3×

h = idké�ê�Ù3×�ÛYÔ"Ï�È/{�È�åYÌ3×�ÜoäYÌ�È�ÇÎÐ

b ∈ B

< h, b1/p >=h(b1/p)

b1/p= 1ÜaäYÌIÕqÍÎͳÈ

h ∈ H,

Ñ.ÕqÐYÐÊdzÙ3×b1/p ∈ L

ÛYÐ.Ñ6Ù@ËNÔ"ÇÎ×b ∈ Lp

é2

�Q���ÉÅ��=�nm�� ö �B�Â�hâ������/�S��������ÇV4H(QA %;�=</#:��) L ���yI^�ª%É�Q��)Q� È >����BA��ªCFE � 13)���4�I/J+��%�%É�������5�0�.IY�"! #�%Ã& ( )V*P��)cI k o�������^��)��0��)

a ∈ k×8���)d! )�J5-Y8[!da

K := k(a1/n)�/���0�H! U�ab��).#T! %ÉO�CFEG�

SUB��C$��)Q6 L ���KAGJ+��e2) L ���=J+��)�UB�TAvCFE �

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(a) ∈ J SKÍ ÕqÛY×@È

aOk,S = pα11 · · · p

αss

Ô"ÇÎ×pi 6∈ S

é*áúÈ�Ï�È�ÐÊÑYÈ�Ì�|AТÉ�È�Ì3òvëlÈ�ÇÎÏ�×3å.È�ÇÎ×�ÕqÛ,��È�Ì3å�ÕqÍÎÖÊÉ�ËNÐSÈ�Ì3Ï�ÇÎÖY×IÙ3dzï@åàÑ*dzÈ��!È�Ì3ͳÈ�Ï�ÛYÐYÏ�ÇÎÐ

J SL

aOL,S = (P1,1 · · ·P1,r1)α1 · · · (Ps,1 · · ·Ps,rs)

αs ,

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piÍÎdzÈ�Ï�È�ÐQéGÁAÛYÐ"dzÙ3×

a1/n ∈ L~qëlÈvÙ@å�ÕqÍÎÖ

nÑ*dzÈ��9ÕqåYͳÈ�Ð

αiÜaäYÌ

i = 1, . . . , s×@È�ÇÎÍÎ×{éa := p

α1/n1 · · · pαs/n

sdzÙ@×Ù@ËNÔ"ÇÎ×IÈ�ÇÎÐÊê�ÑYÈ{ÕqÍ-Ô"ÇÎ×IÑYÈ�Ð Ï�È�ëyäYÐ.Ù@ï�å¢×@È�ÐXz�ÇÎÏ�È�Ð.Ù@ï@å�ÕcÜo×@È�ÐQé2

�Q���ÉÅ��=�����KJ�â:�^å,�������k���5� � >�Z=��( )+*,��) oÇÆ! È �����

S���5�0�XÇV4H(QA %;�R<�#:��) L ����In�g%É�Q��)Q� È >@���BA�� CFEG�Â13).�54�I�J+��%5%É���jCFE � k+ o p �`8���)_e3) L �/�5J+��)�UT�BA k|k+6���)5�/!G%�%É�7& ���5�0�����

SA��/%K�RAB���0�713)��54�I/J+��%5%K� o ¿ ! �P�vA �5%;J+Ê

(ES(k))− = µ(k),

L E�OQ��� µ(k)8 �R�7>��/�BA���8���)7�5�

kA���%É�RA��������Xe3�5�,#:���=J5I L UT)Q6���%;�¥OQ�Q6��/�=<�#B�0� oÇlO È �����

m���5�f>XE^8 UT%�CFE �

k+ o ¿ ! �P�vA �5%;J+Ê(O×

k,m)− ⊆ µ(k).

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ÿ

O×k,m ↪→ E(k).

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(ES(k))−È�б×3å�ÕqÍÎ×@È�Ð(dzÙ@×{é���È�Ç

x ∈ (ES(k))−é�Æ�Õ{�È�ÇÎÐ.È���Ì3ÇÎÔ�Ù3×@È�ÍÎͳÈ

p ∈ Sò{È�Ì,ÜgìqÍÎÍÎ×^~YÏ�ÇÎÍÎ×�ÜoäYÌIÑ*dzÈvÙ@È

vp(x−1) = vp(x) = vp(x) = vp(x),

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ÕqͳÙ�Ëvp(x) = 0

~.ëlÈvÙ3å�ÕqÍÎÖxÖ!È�Ì@È�ÇÎ×@ÙIÇÎÐ

E(k)−ÍÎdzÈ�Ï�×{éPz�ÙdzÙ3×yÐ�ÕNï@åàí�ËNÌ�ÕqÛ.Ù@Ù@È�×3òvÛYÐYÏ

xx = 1éz�ÇÎÐ.È=Ï¢ÕqÐYò{È��9ÕqåYÍ

x ∈ E(k)~YÑYÈ�Ì@È�Ð ÞAÖ.Ù@ËNÍÎÛY×3Ö9È�×3Ì�ÕqÏ

1dzÙ@×^~±Ç³Ù@×�È�ÇÎÐ.Ȫz�ÇÎÐYå.È�ÇÎ×@Ù@ëyÛYÌ3ò{È�Í�é�ê�Ù3×lÐ�ìqÔ"ÍÎdzï�å

f ∈Z[X]

Ñ.ÕNÙùàÇÎÐYÇÎÔ�ÕqÍÎÓ!ËNÍK��Ð.ËNÔîÉ�ËNÐx~*ë�È�ͳï@å.ÈvÙ/ÑYÈ�ÐÒõ/Ì�ÕNÑ

nå�ÕqÖ9ÈG~.Ù@Ë å�ÕqÖ9È�ÐÊÕqÍÎͳÈÝùàÇÎÐYÇÎÔ�ÕqÍÎÓ!ËNÍK��Ð.ËNÔ�ÈÝÑYÈ�Ì�?ËN×@È�ÐYò{È�ÐÚÉ�ËNÐ

xÈ�ÇÎÐ.È�ÐÚõ/Ì�ÕNÑ}{±Í³È�ÇÎÐ.È�ÌlÏ�ͳÈ�dzï@å

né�ÆAdzÈD�IË�ÈED�òvdzÈ�Т×@È�Ð6Ñ*dzÈvÙ@È�Ì3��ËNÍK��Ð.ËNÔ�ÈAÙ@ÇÎÐ.Ñ�Ï¢ÕqÐYò{ȳ�9ÕqåYͳÈ�Ð�~

Ñ*dzÈØÕqÖ!È�Ì�Ñ*ÛYÌ@ï�ånÖ9È�×3Ì�ÕqÏ�Ù3Ô�ì$��ÇÎÏÊÖ9ÈvÙ@ï�åYÌ�ìqÐ,{±×�Ù3ÇÎÐ.Ñ-éQÆ�Õ6ÈvÙ�бÛYÌWÈ�Ð.Ñ*ÍÎdzï@å(É�dzÈ�ͳÈ�Ù@ËNͳï@å.È�Ì���ËNÍK��Ð.ËNÔ�È�Ï�ÇÎÖY×^~

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ϕ : J Sk+ → (J Sk )+, a 7→ aOk,S .

z�ÇÎÐÊêuÑYÈ{ÕqÍa ∈ (J Sk )+

å�Õq×IÈ�ÇÎÐ.È7��Ì3ÇÎÔ"ò{È�Ì3ͳÈ�Ï�ÛYÐYÏa = Pe1

1 · · ·Perr ·Q

f11 Q1

f1 · · ·Qfss Qs

fs Ô"ÇÎ×ei, fj ∈ Z,

ë�ËNÖ!È�ÇPi = Pi

ÜoäYÌi = 1, . . . , r

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P = P˱ÑYÈ�Ì

ϕ(p) = QQÔ"ÇÎ×È�ÇÎÐ.È�Ôi��Ì3ÇÎÔ"dzÑYÈ{ÕqÍPÖYòvëÝé

QÉ�ËNÐ

ké*êuÙ3×�åYÇÎÐYÏ�È�Ï�È�Ð

pÈ�ÇÎÐ.ÈvÙyÑYÈ�ÌÈ�Ð.Ñ*ÍÎdzï@å6É�dzÈ�ͳÈ�Ð6ÇÎÐ

k|k+ É�È�Ì3òvëlÈ�ÇÎÏ�×@È�Ð��Ì3ÇÎÔ"dzÑYÈ{ÕqÍ³È É�ËNÐk+ ~!Ñ.ÕqÐYÐÒÈ�Ì3å�ÕqÍÎ×@È�ÐèëyÇÎÌ ϕ(p) = P2 Ô"ÇÎ×=È�ÇÎÐ.È�Ô ��Ì3ÇÎÔ"dzÑYÈ{ÕqÍ P É�ËNÐ k ~�ëlÈ�ͳï�å.ÈvÙ P = PÈ�Ì,ÜoäYÍÎÍÎ×{éYÆ�ÕqÌ�ÕqÛ.ÙyÙ@ï@åYÍÎdzÈ/��È�ÐÊëyÇÎÌ^~�Ñ.Õ$��ÑYÈ�Ì?�AË {�È�Ì3Ð É�ËNÐ

ϕÑYÈ�Ðvz¯�*Ó!ËNÐ.È�б×@È�Ð

2ÛYÐ.Ñ6Ù@ËWɱdzÈ�ͳÈz�Ì3ò{È�ÛYÏ�È�Ìyå�Õq×^~

ëydzÈ�ÈvÙyÇÎÐk|k+ É�È�Ì3òvëlÈ�ÇÎÏ�×@È���Ì3ÇÎÔ"dzÑYÈ{ÕqͳÈWÕqÛ,��È�Ì3å�ÕqÍÎÖàÉ�ËNÐ S Ï�ÇÎÖY×{éÆAÇÎÌ@È/{�×=ÑYÈ�Ì=Æ=È��.ÐYÇÎ×3dzËNÐ È�Т×3Ð.È�åYÔ�È�Ð ëyÇÎÌ^~!Ñ.Õ$�

(PSk )+ = PSk+

Ï�ÇÎÍÎ×{é�Æ/ÛYÌ@ï�å�ÐAÖ9È�Ì3Ï¢ÕqÐYÏ�òvÛèÑYÈ�РÈAÛ.ËN×3Ç öÈ�б×@È�ÐÊÔ�Ë�Ñ*ÛYͳË

PSk+

È�Ì3å�ÕqÍÎ×@È�ÐÊëyÇÎÌAÕqÛ.ÙϕÑYÈ�Ð æAËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ@Ô�Û.Ù

φ : ClS(k+) = J Sk+/PSk+ → (J Sk )+/PSk+ = (ClS(k))+,

ë�È�ͳï@å.È�Ì=Ñ.Õqå.È�ÌAÕqÛ.ï@åàÈ�ÇÎÐ.È�Ð��IË {�È�Ì3ÐàÉ�ËNÔ¬z¯�*Ó!ËNÐ.È�б×@È�Ð2Ö!ÈvÙ@ÇÎ×3òv×{é��È�Ç

aÈ�ÇÎÐSz�ͳÈ�Ô�È�Т×=ÑYÈvÙY��È�Ì3Ð.ÈvÙIÉ�ËNÐ

φé�Æ=ÕqÐYÐÊdzÙ3×

aOk,S = αOk,SÔ"ÇÎ×È�ÇÎÐ.È�Ô

α ∈ PSkéYÆ�Õqå.È�ÌÏ�ÇÎÍÎ×

(1) = Ok,S =aOk,S

aOk,S=α

αOk,S .

αα

dzÙ@×ÝÈ�ÇÎÐhz�ͳÈ�Ô�È�Т×WÉ�ËNÐ(ES(k))−

ÛYÐ.ÑúÙ�ËNÔ"ÇÎ×�Ð�ÕNï@år�*Õq×3òÊûR¨*é©­*éK�G�&ÿ=È�ÇÎÐ.È}z�ÇÎÐYå.È�ÇÎ×@Ù3ëyÛYÌ3ò{È�Í�é?õ=ÇÎÖY×ÝÈvÙ�È�ÇÎÐ.Èz�ÇÎÐYå.È�ÇÎ×@Ù3ëyÛYÌ3ò{È�Íζ ∈ µ(k)

ÑYÈ�Ì�ÕqÌ3×^~¢Ñ.Õ$� αα = ζ2 Ï�ÇÎÍÎ×^~¢Ñ.ÕqÐYÐ�ÜoËNÍÎÏ�× αζ−1 = αζ−1

ÛYÐ.Ña = αOk,S

é¢Æ/ÈvÙ@å�ÕqÍÎÖdzÙ@×ÑYÈ�Ì���È�Ì3Ð É�ËNÐ

φÇÎе(k)/(µ(k))2

È�Т×3å�ÕqÍÎ×@È�Ð�~.ë�È�ͳï@å.ÈvÙAdzÙ@ËNÔ�ËNÌ3ÓYå òvÛZ/2Z

dzÙ@×{éûaÖ�ÿ�Æ=È�Ì�£�È�ëlÈ�dzÙÉ�È�Ì3Í ìqÛ*Üo×É�ËNÍÎÍK{�ËNÔ"Ô�È�ÐèÕqÐ�ÕqͳËNÏ.é

2

�Q���ÉÅ��=�TÅ���ñ}�9F�0ß5ß5â,B�������k�/��� � >�Z=��( )+*,��)�UB�98

p���5�0�UT�BA��/)l![8B�713)���4�6^! #�% o

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ÇÆ! È �����S���5�0�XÇV4H(QA %;�R<�#:��) L ����In�g%É�Q��)Q� È >@���BA�� CFEG�Â13).�54�I�J+��%5%É���jCFE � k+ o p �`8���)_e3) L �/�5J+��)�UT�BA k|k+6���)5�/!G%�%É�7& ���5�0�����

SA��/%K�RAB���0�713)��54�I/J+��%5%K� o ¿ ! �P�fI.�5�98g8 �V�HW2)�U^*�*,���

ClS(k+)(p) ∼= (ClS(k)(p))+�yInE 4HE )V*9# oÇlO È �����m���5�f>XE^8 UT%�CFE �

k+ o ¿ ! �P�fI.�5�98g8 �R�}W2)�U^*[*P���Clmk+(p) ∼= (Clmk (p))+�yInE 4HE )V*9# o

A �CB�� Þ ä[� Á=ÕNï@å��*Õq×3ò6ûR¨*é©­*éK�^¨�ÿå�ÕqÖ!È�ÐÒÖ!È�dzÑYÈWÑYËNÌ3×AÖ9È�×3Ì�ÕNï@å±×@È�×@È�Ð æAËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�È�Ð2ö�õ/Ì3ÛYÓYÓ9È�ÐèÕqͳÙ��È�Ì3ÐÊÛYÐ.ÑS�IË {�È�Ì3ÐQé�Æ=Õ Ñ*dzÈvÙ�È�Ï�ͳÈ�dzï@åYò{È�ÇÎ×3ÇÎÏÚÕqÛ.ï@å

pö�õ=Ì3ÛYÓYÓ!È�Ð Ù3ÇÎÐ.Ñ�~.Ô�ä.Ù@Ù@È�ÐàÙ3dzÈ=×3Ì3ÇÎÉ±Ç ÕqÍ�Ù@È�ÇÎÐQé

2

�Q���ÉÅ��=�Bó��KJ�â:�^å,�������k���5� � >�Z=��( )+*,��) o­l �5)�OQ�/J=)d![<�#BJ+����6 L ���"Ë3� %5%K� oÇÆ! È �����

S���5�0�XÇV4H(QA %;�R<�#:��) L ����In�g%É�Q��)Q� È >@���BA�� CFEG�Â13).�54�I�J+��%5%É���jCFE � k+ o p �`8���)_e3) L �/�5J+��)�UT�BA k|k+6���)5�/!G%�%É�S& ���5�0�f���

SA���%É�RA�������13)���4�I/J+��%�%É� o ËsNT) E := ES(k)

A���%MJ+���ð8[!G�P�E− = µk

UB�98(E :

E+E−) ≤ 2.ÇlO È �����m���5�f>XE^8 UT%�CFE �

k+ o ËsNT) E := O×k,m

AB��%;J+�/�¥8[! �P�E− ⊆ µk

UT�98(E : E+E−) ≤ 2.

A �CB�� Þ ä[� Æ/Ç³È È�Ì@Ù3×@È�£�È�å�ÕqÛYÓY×3ÛYÐYÏ6dzÙ@×/Ñ*Ç³È ÞAÛ.Ù@Ù�ÕqÏ�ÈWÉ�ËNÐh�*Õq×3ò ûR¨*é©­*éK�G�&ÿdé!áçÇÎÌ/Ö9È�×3Ì�ÕNï@å±×@È�ÐúÑ*Ç³È È��YÕ${�×@È��Èn ±Û.È�ÐYò1→ E+E− → E

φ−→ E−/(E−)2,Ô"ÇÎ×

φ(x) := xx(E−)2

é�ùèÕqÐúÖ!È{ÕNï�å¢×@ÈG~?Ñ.Õ$�φ(x) = x

x ∈ (E−)2~�ÕqͳÙ�Ë

x = xζ2 ~QÇÎÔ"ÓYÍÎÇÎòvdzÈ�Ì3×^~?Ñ.Õ$� xζ−1È�ÇÎФz�ͳÈ�Ô�È�Т×�É�ËNÐE+ dzÙ3×{é�Æ�Õqå.È�Ì�È��*dzÙ3×3dzÈ�Ì3×ØÈ�ÇÎÐ.ÈX�!È�Ì3ͳÈ�Ï�ÛYÐYÏ x = (xζ−1)ζ

ÇÎÐãÈ�ÇÎÐ.È�ÐE+ öWÛYÐ.ÑãÈ�ÇÎÐ.È�Ð

E− ö�ÞIб×@È�ÇÎÍ�éYáúÈ�Ï�È�Ð |µ(k)/µ(k)2| ≤ 2ÜoËNÍÎÏ�×Ñ*dzÈ�òvë�È�ÇÎ×@È�£�È�å�ÕqÛYÓY×3ÛYÐYÏ.é

2ÁIÛYÐÃå�ÕqÖ!È�ÐúëyÇÎÌ ÕqÍÎͳÈ"í�ËNÌ3Ö9È�Ì@È�ÇÎ×3ÛYÐYÏ�È�ÐÃÖ9ÈvÈ�Ð.ÑYÈ�×^~QÛYÐ.Ñ¥{[�NÐYÐ.È�ÐÃÑ*dzÈ"É�ËNÐÃÛYÐ.ÙÝÖ9È�ÐP�N×3ÇÎÏ�×@È�í�È�Ì@Ù3dzËNÐ(ÑYÈvÙ��ÈvËNÓ!ËNͳÑ*×@Ù@ï�å.È�Ð@�±ÓYdzÈ�Ï�È�ÍÎÛYÐYÏ�Ù�Ù�Õq×3ò{ÈvÙÖ9È�ëlÈ�dzÙ@È�ÐQé�Q���ÉÅ��=�:ô��KJ�â:�^å,�������

p 6= 2-YUT�98vI^���

k�/��� � >�Z=��( )+*,��).-D8���)}8G�R�fW2)�U^*[*P�

µp(k)8���)

pZuJ+���he3���P#:���=J5I.Z

L UT)Q6���%;�¥����J5#T! %MJ+� o­l �5)�OQ�/J=)d![<�#TJ+���X6 L �/�"Ëx�G%�%É� oÇÆ! È �����S���5�0�XÇV4H(QA %;�R<�#:��) L ����In�g%É�Q��)Q� È >@���BA�� CFEG�Â13).�54�I�J+��%5%É���jCFE � k+ o p �`8���)_e3) L �/�5J+��)�UT�BA k|k+6���)5�/!G%�%É�D& ���5�0�³�5�

SA���%É�RA����0�ª13)��54�I/J+��%5%K� o ����� L|k 8 �R�D4H!^]G��4H! %É�³UT�PC$��)Q6 L ���KAGJ+����%K�/4H�/��JV! )7!�OQ��%'I/</#:�

pZ=e3) L ���=J+��).UT�BAgCFEG� k -t�5��8B��)�! %5%K�³13)���4�I/J+��%�%É���h! UBI S CFEG%�%96F��).%É�RAGJxI����98 o�l ��)ªI^�/Jy6����

A := ClS(k)(p),I/Eg8[!da

A+ = ClS(k+)(p)A ��%MJ�Ç+C.A % o ��E )dE %�%K! )}Ç OToUñBo N ñ ÈcÈ oÇlO È �����S�/���0�����98:%;�=</#:�}13)��54�I/J+��%5%K����4H�/�BA��gCFEG�

k+- L �/%M<�#,�g& ���5�0�\�9J+��%5%K�gNPOQ��) p ���9J5#T! %MJ+�.-³UT�98XI^���

L|k8G�R� 4H!^] �54H! %É�v! U/a¯��).#T! %ÉO

SUT�PC$�/)d6 L ���MA[J+����%É��4}����JV! )v!�OQ�/%ÃIn<�#:� p ZRe3) L ���5J+�/).UT�BA�CFEG� k o ���/Jy6F�

m =∏

p∈S pUB�08

A := Clmk (p),InE 8�!da

A+ = Clmk+(p)A ��%MJ�Ç+C.A % o ��E )dE %�%K! )}Ç OToUñBo N ñ ÈcÈ o¿ ! �P��%;�V�5�n�/)�J8���) « �Q6n�©*9)dE�6n�5JV�GJ5I.#TE 4HE 4HE )+*�#���I.4�UBI\���5�0� 4��5Jª8���) &$E 4D*9%É�d]����j��EG�^¡.UGA[!GJ=�RE �j&$EG4�*,!GJ=�RO/%K�p InE 4HE )+*�#��V�A/Ap ∼= G(L|k),UT�98v�.I�A �5%;J

d(A)+ ≤ 1 + d(A)−.

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A �CB�� Þ ä[� áúÈ�Ï�È�Ð�ÑYÈ�Ì�ùèÕF�*ÇÎÔ�ÕqÍÎÇÎ×�ìq×ØdzÙ3×�Ñ*dzÈfz�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏL|k+ ÇÎÐ Ö!È�dzÑYÈ�Ðçø�ìqÍÎͳÈ�ÐãÏ¢ÕqͳËNdzÙ@Ù@ï�åQé�ùàÇÎ×

σÖ9È�ò{È�dzï@åYÐ.È�Ð�ëyÇÎÌ�ëydzÈvÑYÈ�Ì3ÛYÔ Ñ*dzÈS{�ËNÔ"ÓYͳÈ��*ÈS�AËNÐNý�ÛYÏ¢Õq×3dzËNÐ�ÇÎÐ

k|k+ é0áçÇÎÌ�Ù@È�×3ò{È�Ð H := G(L|k)é�Æ/dzÈ

σö�ÇÎбÉNÕqÌ3Ç Õqб×@È�êuÙ�ËNÔ�ËNÌ3ÓYåYdzÈ

A/Ap ∼= G(L|k)ÜgËNÍÎÏ�×ÝÕqÛ.Ù³�AËNÌ@ËNÍÎÍ ÕqÌ�ûR¨*é©­*é©°�ÿIÛYÐ.ÑÒÑYÈ�Ð¥�*ìq×3ò{È�Ðãûl��驨*驨:�&ÿIÖYòvëÝé

ûl��驨*驨$®±ÿdéÞIÛ.Ùv�*Õq×3ò÷ûR¨*é©­*驱�ÿ�È�Ì3å�ÕqÍÎ×@È�Ð ëyÇÎÌ�È�ÇÎÐ.ÈX|AТ×@È�Ì3Ï�Ì3ÛYÓYÓ9È

B ⊆ k×/(k×)pÔ"ÇÎ×

L = k(B1/p)ÛYÐ.ÑãÑ*dzÈ�=ÛYÔ"Ô�È�Ì,ö+�0Õ�ÕqÌ3ÛYÐYÏ

H ×B → µp(k).Æ/dzÈ

σö�êuТÉ�ÕqÌ3Ç ÕqÐYò�ÇÎÔ"ÓYÍÎÇÎòvdzÈ�Ì3×AÙ�ËqÜoËNÌ3×^~YÑ.Õ$�

< H+, B+ >= 1 =< H−, B− >

Ï�ÇÎÍÎ×^~�ÑYÈ�ÐYÐ(µp(k))

+ È�Т×3å�ìqÍÎ×AТÛYÌIÑ*Ç³È 1é9��ËNÔ"ÇÎ×IdzÙ3×Ñ*dzÈ7�0Õ�ÕqÌ3ÛYÐYÏH+ ×B− → µp(k)

ÐYdzï�å¢×,öuÕqÛ.Ù3Ï�È{ÕqÌ3×@È�×^~!ÛYÐ.Ñ Ñ.Õqå.È�ÌIÏ�ÇÎÍÎ×d(A)+ = d(A+) = d(H+) = d(B−).

ÞIÛ.Ù2�QÈ�Ô"Ô�ÕÚûR¨*é©­*éK�^²�ÿ�ͳÈvÙ@È�ÐØëyÇÎÌÕqÖ�~�Ñ.Õ$��ÈvÙlòvÛWý,ÈvÑYÈ�Ôb ∈ B

~¢ëlÈ�ͳï@å.ÈvÙÔ�˱Ñ*ÛYͳË(k×)p

Ö!ÈvÙ3×3ÇÎÔ"ÔW×�dzÙ3×^~È�ÇÎÐÒêuÑYÈ{ÕqÍ

a ∈ J SkÇÎÔ ø�ÕqÍÎÍ�ûgÕ�ÿyÛYÐ.Ñ ÇÎÔ ø�ÕqÍÎÍ�ûaÖ�ÿ

a ∈ J mk

ûaëyÇÎÌAÐ.È�åYÔ�È�ÐÒÑ.ÕNÙD£lÇÎͳÑÊÛYТ×@È�Ì=ÑYÈ�ÌD��Ì@Ëqý3È/{�×3dzËNÐJk � J m

k

ÿ�Ï�ÇÎÖY×Ô"ÇÎ×(b) = ap

é9��ËNÔ"ÇÎ×AÈ�Ì3å�ÕqÍÎ×@È�Ð ëyÇÎÌIÈ�ÇÎÐ.È�ÐÊæ/ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�Û.Ùφ : B → pA, b 7→ a.

z�ÙdzÙ3×?{�Í ÕqÌ^~YÑ.Õ$�φëlËNåYͳÑYÈ��.ÐYdzÈ�Ì3×/ÛYÐ.Ñ

σö�ÇÎТÉ�ÕqÌ3Ç ÕqТ×AdzÙ3×^~.Ù@Ë"Ñ.Õ$��Ù3dzï�åÊÈ�ÇÎÐÊæ/ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�Û.Ùψ : B− → (pA)−

È�Ì3Ï�ÇÎÖY×{éê�Ô ø�ÕqÍÎÍ�ûgÕ�ÿ�dzÙ3×�È�ÇÎÐ\z�ͳÈ�Ô�È�Т×

b ∈ BÇÎÔ ��È�Ì3ÐØÉ�ËNÐ

φ~�Ü�ÕqÍÎͳÙ

(b) = (a)pÜoäYÌlÈ�ÇÎÐ

a ∈ ES(k)Ï�ÇÎÍÎ×{é±Æ�Õqå.È�Ì

dzÙ@×ker(ψ) ≤ (ES(k)/(ES(k))p)−.

êuÔîø�ÕqÍÎÍlûaÖ�ÿyÈ�Ì3Ï�È�Ö!È�ÐàÕqÐ�ÕqͳËNÏ�È�ÐIÖ9È�Ì3ͳÈ�Ï�ÛYÐYÏ�È�Ðker(ψ) ≤ (O×

k,m/(O×k,m)p)−.

Æ�ÕqÖ9È�ÇQëyÛYÌ@ÑYÈ��*Õq×3òØûR¨*é©­*éK�G�&ÿ�Ö!È�бÛY×3òv×{éÁIÛYÐ6ÜgËNÍÎÏ�×IÕqÛ.ÙD�*Õq×3òØûR¨*é©­*éK�/®±ÿyÔ"ÇÎ×3×@È�ͳÙIêuÐ.ÑYÈ���Ö9È�×3Ì�ÕNï@å±×3ÛYÐYÏ�È�Ð�~�Ñ.Õ$�(ES(k)/(ES(k))p)− = µ(k)/(µ(k))p

ÖYòvëÝé(O×

k,m/(O×k,m)p)− ⊆ µ(k)/(µ(k))p

Ï�ÇÎÍÎ×{é.á�ÇÎÌyå�ÕqÖ9È�ÐàÕqͳÙ@Ë ÇÎÐ.Ù3Ï�ÈvÙ�ÕqÔ�×AÑ*dzÈ�È��.Õ${�×@È���Èn ¢Û.È�ÐYò1→ ker(ψ)→ B− ψ

−→ (pA)−,

ÛYÐ.ÑÊÑ.Õqå.È�ÌIÈ�Ì3å�ÕqÍÎ×@È�ÐÊëyÇÎÌ

d(B−) ≤ d(ker(ψ)) + d(pA)− ≤ d(Z/pZ) + d(A−) = 1 + d(A)−.

áúÈ�Ï�È�ÐàÑYÈ�ÌIËNÖ!È�ÐÊÏ�È�ò{È�ÇÎÏ�×@È�ÐÒõ/ͳÈ�dzï@åYå.È�ÇÎ×d(B−) = d(A)+

dzÙ@×ÑYÈ�ÌD�*Õq×3òÝÖ!È�ëydzÈvÙ�È�ÐQé2

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SÛYТÉ�È�Ì3òvë�È�ÇÎÏ�×@È�z�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ�È�ÐQé��È�Ç

k|k+ È�ÇÎÐÒü�ùÊö+���NÌ3Ó9È�ÌÛYÐ.Ñ p 6= 2È�ÇÎÐ.È7��Ì3ÇÎÔ"ò&ÕqåYÍ�é

á�ÇÎÌÖ9È�×3Ì�ÕNï@å±×@È�ÐàÑ*dzÈ�Ö9È�dzÑYÈ�Ð6ÜgËNÍÎÏ�È�Ð.ÑYÈ�ÐX�±ÇÎ×3Û�Õq×3dzËNÐ.È�Ð��ûoê3ÿ

• SÙ@È�ÇQÈ�ÇÎÐ.È�ûaÔH�NÏ�ÍÎdzï�å.È�Ì3ëlÈ�dzÙ@ÈWͳÈvÈ�Ì@È&ÿ�ùÒÈ�ÐYÏ�È�É�ËNÐ���Ì3ÇÎÔ�Ù3×@È�ÍÎͳÈ�ÐÊÉ�ËNÐ

k+ é•ê�Ðk|k+ ò{È�Ì,Ü�ÕqÍÎͳÈ{�È�ÇÎÐ.È�ÇÎÐ S Ï�È�ͳÈ�Ï�È�Ð.È���Ì3ÇÎÔ�Ù3×@È�ÍÎͳÈ�é

• LS(p)|kÙ@È�Ç!Ñ*dzÈ/Ô�ÕF�*ÇÎÔ�ÕqͳÈ/ÛYТÉ�È�Ì3òvëlÈ�ÇÎÏ�×@ÈÝÏ¢ÕqͳËNdzÙ@Ù@ï@å.È

pö+z�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ�É�ËNÐ

k~±ÇÎÐ�ëlÈ�ͳï�å.È�ÌIÕqÍÎͳÈ�±×@È�ÍÎͳÈ�ÐàÕqÛ.Ù

SÉ�ËNÍÎÍQò{È�Ì3ͳÈ�Ï�×AÙ3ÇÎÐ.Ñ-é

• G := G(LS(p)|k)Ù@È�ÇQÑ*dzÈWõÝÕqͳËNdzÙ,ö�õ/Ì3ÛYÓYÓ9ÈÝÑ*dzÈvÙ@È�Ì�z�Ì3ë�È�ÇÎ×@È�Ì3ÛYÐYÏ.é

• E := ES(LS(p)) = lim−→LS(p)⊃K⊃k, K|k ´Vµ/¶/·k¸k¹=º E(K)

~�ëlËNÖ9È�ÇE(K) := ES(K)

Ñ*dzÈ/õ/Ì3ÛYÓYÓ9ÈÑYÈ�Ì

Sö+z�ÇÎÐYå.È�ÇÎ×@È�Ð"È�ÇÎÐ.È�Ì?È�Ð.Ñ*ÍÎdzï�å.È�РϢÕqͳËNdzÙ@Ù@ï@å.È�Ð_�D�NÌ3Ó!È�Ì@È�Ì3ë�È�ÇÎ×@È�Ì3ÛYÐYÏ

LS(p) ⊃ K ⊃ kÖ!È�ò{È�dzï�åYÐ.È�×{é

ûoê@ê3ÿ• SÙ@È�ÇQÈ�ÇÎÐ.ÈÝÈ�Ð.Ñ*ÍÎdzï�å.È�ùÒÈ�ÐYÏ�È=É�ËNÐ���Ì3ÇÎÔ�Ù3×@È�ÍÎͳÈ�ÐàÉ�ËNÐ

k+ é•ê�ÐSÙ@È�Ç"{�È�ÇÎÐ.È�äYÖ9È�Ì

pÍÎdzÈ�Ï�È�Ð.ÑYÈH�±×@È�ÍÎͳÈ�ûoÑYÈ�Ì@È�Ð ùÒÈ�ÐYÏ�ÈÝëyÇÎÌ/Ô"ÇÎ×

SpÖ9È�ò{È�dzï@åYÐ.È�Ð�ÿAÈ�Т×3å�ÕqÍÎ×@È�Ð��

Sp ∩ S = ∅é

• kS(p)|kÙ�È�Ç�Ñ*dzÈÝÔ�ÕF�*ÇÎÔ�ÕqͳÈWÕqÛ,��È�Ì3å�ÕqÍÎÖÒÉ�ËNÐ

SÛYТÉ�È�Ì3òvëlÈ�ÇÎÏ�×@È�Ï¢ÕqͳËNdzÙ@Ù�ï@å.È

pö+z�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ�É�ËNÐ

• G := G(kS(p)|k)Ù@È�ÇQÑ*dzÈ�õ�ÕqͳËNdzÙ,ö�õ/Ì3ÛYÓYÓ9È�Ñ*dzÈvÙ@È�Ì�z�Ì3ë�È�ÇÎ×@È�Ì3ÛYÐYÏ.é

• E(K) := O×K,m

~.ëlËNÖ9È�ÇkS(p) ⊃ K

È�ÇÎÐ.È�È�Ð.Ñ*ÍÎdzï@å.ÈWÏ¢ÕqͳËNdzÙ@Ù@ï@å.È����NÌ3Ó9È�Ì@È�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ�È�Ð É�ËNÐkÙ@È�Ç�éYÆ�ÕqÖ!È�Ç-dzÙ@×

mÑYÈ�ÌAùàË�Ñ*ÛYÍ ∏

p∈S pé

• E := O×kS(p),m = lim−→

K

E(K)~?ë�ËNÖ!È�ÇyÑYÈ�Ì"Ñ*ÇÎÌ@È/{±×@Èg�-ÇÎÔ�ÈvÙ äYÖ9È�Ì"Ñ*dzÈ�ÇÎÐ

kS(p)È�б×3å�ÕqÍÎ×@È�Ð.È�Ð

È�Ð.Ñ*ÍÎdzï@å.È�ÐàÏ¢ÕqͳËNdzÙ@Ù@ï@å.È�Ðfz�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ�È�ÐèÉ�ËNÐkÍ ìqÛ*Üo×{é

á�ÇÎÌ�å�ÕqÍÎ×@È�ÐÚÜoÈvÙ@×^~¢Ñ.Õ$� ÇÎÐØÖ9È�dzÑYÈ�Ð6ø�ìqÍÎͳÈ�Ð�Ñ*dzÈ=ÞAÖ!È�ÍÎdzÙ3dzÈ�Ì3ÛYÐYÏ É�ËNÐGÈ�Ð.Ñ*ÍÎdzï@åÚdzÙ3×�ûoÞIÖ.Ù@ï�åYÐYÇÎ×3×��驨*éK�AÖYòvëÝé�*Õq×3ò÷ûl��驨*éû�ÿ3ÿdé�Æ/dzÈSz�Ì3ë�È�ÇÎ×@È�Ì3ÛYÐYÏ�È�Ð

LS(p)|k+ ÛYÐ.Ñ kS(p)|k+ Ù3ÇÎÐ.ÑóÖ9È�dzÑYÈ6Ï¢ÕqͳËNdzÙ@Ù@ï�å�~�Ù@ËúÑ.Õ$�úëyÇÎÌØÈ�ÇÎÐ.ÈÞD{�×3dzËNÐÊÑYÈ�Ì?{�ËNÔ"ÓYͳÈ��*È�ÐX�IËNÐNý,ÛYÏ¢Õq×3dzËNÐÒÕqÛ*ÜGÖ9È/{�ËNÔ"Ô�È�Ð�~YëlÈ�ͳï�å.È�ëyÇÎÌyÔ"ÇÎ×

σÖ9È�ò{È�dzï@åYÐ.È�ÐQé

á�ÇÎÌë�ËNÍÎͳÈ�РТÛYÐàÉ�ËNÌ�ÕqÛ.Ù@Ù@È�×3ò{È�Ð�~.Ñ.Õ$�GÓ!ËN×@È�ÐYòvÌ@È�dzï�å dzÙ3×yÛYÐ.Ñ��IËNÐ.Ù@Èn ±Û.È�ÐYò{È�ÐÊÖ9È�×3Ì�ÕNï@å±×@È�ÐQé

ùÊÇÎ×3åYÇÎÍ ÜgÈ�É�ËNÐ`�*Õq×3ò ûR¨*驨*驨:�&ÿÝë�ËNÍÎͳÈ�Ð÷ëyÇÎÌWÑ*dzÈ�ùf�NÏ�ÍÎdzï�å,{�È�ÇÎ×@È�Ð÷ÜaäYÌd+ := d(G)+

ÛYÐ.Ñd− := d(G)−È�ÇÎÐ.Ù�ï@åYÌ�ìqÐ,{�È�ÐQé!Æ=ÕqòvÛ6ë�È�Ì@ÑYÈ�ÐÊëyÇÎÌIÈ�ÇÎÐ.È�ËNÖ!È�Ì@È���ï�åYÌ�ÕqÐ,{�ÈÝÜoäYÌ

dimFp (pH2(G,Qp/Zp)

−)

Ù@Û.ï@å.È�ÐQéYÞAÛ*Ü�Ñ*dzÈvÙ@È/áúÈ�dzÙ@È/ëlÈ�Ì@ÑYÈ�Ð ëyÇÎÌÕqÛ*Ü�ÑYÈ�Ð ø�ÕqÍÎÍd(G) ≤ 3

Ï�ÈdÜaäYåYÌ3×�ë�È�Ì@ÑYÈ�ÐQé.ÞIÐÚÑ*dzÈvÙ@È�Ì��±×@È�ÍÎͳÈ=Ï�È�å.È�ÐÈ�б×@Ù@ï@å.È�dzÑYÈ�Ð.ÑÒÑ*dzÈ�ÇÎÔi�ÝÕqÓYÇÎ×@È�Í?ê�Ö9È�ëydzÈvÙ@È�Ð.È�ÐàÆ/Û�ÕqÍÎÇÎ×�ìq×@Ù@Ù�ìq×3ò{È�È�ÇÎÐQéê�Ð ÑYÈ�Ì��?Õq×ÍÎdzÈdÜgÈ�Ì3×��IËNÌ@ËNÍÎÍ ÕqÌ ûR¨*é©­*é©��ÿyÕqÐYÏ�È�ëlÈ�Ð.ÑYÈ�×/ÇÎÔ ø�ÕqÍÎÍ

i = 0È�ÇÎÐ.È

σö�ÇÎТÉ�ÕqÌ3Ç ÕqТ×@È��±ÛYÌoý,È/{�×3dzËNÐ

E(k) � H0(G,E) = E(k)/NGE ∼= H2(G,Qp/Zp)∨

ÛYÐ.ÑÊÑ.Õqå.È�ÌÇÎÐ.Ù3Ö9ÈvÙ@ËNÐ.ÑYÈ�Ì@È�Ð�ÕNï@å��*Õq×3òØûR¨*é©­*éK�G�&ÿyÈ�ÇÎÐ.È��±ÛYÌoý3È/{�×3dzËNеp(k) ⊇ (E(k)/p)− � (pH

2(G,Qp/Zp)−)∨.

Æ/dzÈvÙyÇÎÔ"ÓYÍÎÇÎòvdzÈ�Ì3×=ÕqͳÙ@Ë

dimFp (pH2(G,Qp/Zp)

−) ≤

{

0Ü�ÕqÍÎͳÙ

µp(k) 6⊆ k

1Ü�ÕqÍÎͳÙ

µp(k) ⊆ k.

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ÞAÛ.ÙD�*Õq×3òØûR¨*驨*驨:�&ÿyÈ�Ì@Ù@È�å.È�Ð ëyÇÎÌбÛYÐÊÑ*dzÈ7|AÐYÏ�ͳÈ�dzï@å¢ÛYÐYÏd+ · d− ≤ d− + 1.

Æ�Õqå.È�ÌY{[�NÐYÐ.È�ÐàТÛYÌIÑ*dzÈ/ÜgËNÍÎÏ�È�Ð.ÑYÈ�Ðàø�ìqÍÎͳÈÝÕqÛ*Üo×3Ì@È�×@È�Ð��d− = 0

˱ÑYÈ�Ìd+ = 0

˱ÑYÈ�Ìd+ = 1

Ë�ÑYÈ�Ì(d+ = 2

ÛYÐ.Ñd− = 1

ÛYÐ.ѵp(k) ⊆ k).

áçÇÎÌ�ë�ËNÍÎͳÈ�Ðd+ 6= 1

ÕqÐYÐ.È�åYÔ�È�ÐQéb��ËNÔ"ÇÎ×WÈ�Ì3å�ÕqÍÎ×@È�Ð(ëyÇÎÌ�Ñ.ÕqÐYÐ÷ÕqÛ,��È�ÌWÇÎÐôÑYÈ�Ðj��ËNÐ.ÑYÈ�Ì,Ü�ìqÍÎͳÈ�Ðd+ = 0

ÛYÐ.Ñd− = 0

Ñ*dzÈ7|AÐYÏ�ͳÈ�dzï@å¢ÛYÐYÏd(G) ≤ 3

éê�Ôîø�ÕqÍÎÍ

d(G)+ = 0dzÙ3×

GÐ�ÕNï@å��*Õq×3òØûR¨*驨*éK�{þ�ÿyÕqÖ9È�ͳÙ@ï@å�~YÇÎÐÊÛYÐ.Ù@È�Ì@È�Ô ø�ÕqÍÎÍ�ÕqͳÙ�Ë"È�Ð.Ñ*ÍÎdzï@åQé�±ÇÎÐ.Ñ�Ñ*dzÈ

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ûR¨*é©­*éK�{þ�ÿ3ÿÑ*dzÈ7|AÐYÏ�ͳÈ�dzï@å¢ÛYÐYÏd+ ≤ 1 + d−,ë�È�ͳï@å.È�ÑYÈ�Ð ø�ÕqÍÎÍ

d− = 0ÕqÛ.Ù@Ù@ï@åYÍÎdzÈ/��×{éÁIÛYÐúÏ�ÇÎÖY×WÈvÙÝÈ�ÇÎÐ.È�ÐôêuÐ.ÑYÈ��

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Ï�ÇÎÍÎ×d(P )+ ≤ d+ ÛYÐ.Ñ d(P )− ≤ d−.áçÇÎÌë�ËNÍÎͳÈ�ÐÒÕqÐYÐ.È�åYÔ�È�Ð�~!Ñ.Õ$�

d+ 6= 0ÛYÐ.Ñ

d− 6= 0ÛYÐ.Ñ Ù�ËNÔ"ÇÎ×

d(P ) ≤ d(G) ≤ 3Ï�ÇÎÍÎ×{éYá ìqÌ@È

d(P ) < 3~

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ëydzÑYÈ�Ì@Ù@ÓYÌ3dzï@å¢×{é!Æ=ÈvÙ3å�ÕqÍÎÖS{[�NÐYÐ.È�Ð ëyÇÎÌIÛYÐ.ÙIÕqÛ*Ü0ÑYÈ�Ðàø�ÕqÍÎÍd(G) = d(P ) = 3, d+ = d(P )+

ÛYÐ.Ñd− = d(P )−

Ö9ÈvÙ@ï�åYÌ�ìqÐ,{�È�ÐQéÆ=ÕNÙÏ�Ì3ÛYÓYÓ!È�б×3å.ÈvËNÌ@È�×3dzÙ@ï@å.È ðIÈvÙ@ÛYÍÎ×�Õq×ÑYÈvÙ³�*Õq×3ò{ÈvÙ�ûR¨*驨*驨G¨�ÿyÍÎdzÈdÜoÈ�Ì3×бÛYÐÊÈ�ÇÎÐ.È�Ð áçdzÑYÈ�Ì@Ù3ÓYÌ3Û.ï@åQéá�ÇÎÌ�Ü�ÕNÙ@Ù@È�ÐóÑ*dzÈÚÆAdzÙQ{±Û.Ù@Ù@dzËNÐ(òvÛ.Ù�ÕqÔ"Ô�È�ÐóÇÎÔ ÜgËNÍÎÏ�È�Ð.ÑYÈ�Ð`��å.ÈvËNÌ@È�ÔS~�ë�È�ͳï@å.ÈvÙH��å.ÈvËNÌ@È�Ô ­*éK�6ÕqÛ.Ù6ß á�â

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d(G)+ = 1Ë�ÑYÈ�Ì

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d(G)+ = 1Ö!ÈvÑYÈ�ÛY×@È�×�Ð�ÕNï@å��IËNÌ@ËNÍÎÍ ÕqÌ�ûR¨*é©­*éK�^­�ÿQÇÎÔ>ø�ÕqÍÎÍ9ûoê,ÿ�~&Ñ.Õ$�IÑ*dzÈ�õ�ÕqͳËNdzÙ3ö�õ/Ì3ÛYÓYÓ!È

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K := kK+ ~¢Ù@Ë�È�Ì3å�ÕqÍÎ×@È�Ð�ëyÇÎÌlÕqͳÙ@Ë�È�ÇÎÐ.ÈIÈvï@å¢×@ÈÈ�Ð.Ñ*ÍÎdzï�å.È�z�Ì3ë�È�ÇÎ×@È�Ì3ÛYÐYÏØÉ�ËNÐèü�ù ö+�D�NÌ3Ó!È�Ì3ÐQéê�бÉ�ÈvÙ3×3dzÈ�Ì@È�ÐÃëyÇÎÌWÈ�ÇÎÐôëlÈ�ÇÎ×@È�Ì@ÈvÙ�ùèÕqÍ�ÑYÈ�ÐÃÆ/Û�ÕqÍÎÇÎ×�ìq×@Ù@Ù�Õq×3òÚÕqÛ.Ù��AËNÌ@ËNÍÎÍ ÕqÌ�ûR¨*é©­*é©��ÿ�~?Ù@Ë6È�Ì3å�ÕqÍÎ×@È�ÐÃëyÇÎÌWÈ�ÇÎÐ.È

σö�ÇÎбÉNÕqÌ3Ç Õqб×@È��±ÛYÌoý3È/{±×3dzËNÐ��

H2(G,Qp/Zp)∨ ∼= H0(G,E) � H0(G(K|k), E(K)).

Æ�ÕK|kü�ù ö+�D�NÌ3Ó!È�ÌAÙ3ÇÎÐ.Ñ�~YÏ�ÇÎÍÎ×Ð�ÕNï�å��IËNÌ@ËNÍÎÍ ÕqÌ ûR¨*é©­*é ®±ÿ

G(K|k) ∼= G(K+|k+) ∼= G(K|k)+ÛYÐ.Ñ Ñ.Õqå.È�Ì

H0((G(K|k), E(K))/p)− = H0(G(K|k), µ(K)(p))/p,

É�ËNÌ�ÕqÛ.Ù@Ï�ÈvÙ@È�×3òv×µp(k) ⊆ O

×K,m = E(K)

ÇÎÔîø�ÕqÍÎÍlûoê@ê3ÿdé*á�ÇÎÌyå�ÕqÖ9È�ÐÊÙ@ËNÔ"ÇÎ×yÇÎÐ.Ù3Ï�ÈvÙ�ÕqÔ�×AÑ*dzÈ��±ÛYÌoý3È/{�×3dzËNÐ(pH

2(G,Qp/Zp)−)∨ � H0(G(K|k), µ(K)(p))/p.

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pH2(G,Qp/Zp)−

×3Ì3ÇÎÉ�Ç ÕqÍdzÙ@×{é9á�ÇÎÌ�È�Ì3å�ÕqÍÎ×@È�Ð(ÕqͳÙ@Ë È�ÇÎÐ.È�ÐúáçdzÑYÈ�Ì@Ù3ÓYÌ3Û.ï@å�~QÜgÕqÍÎͳÙ�Ñ*dzÈ�õ/Ì3ÛYÓYÓ!È

H0(G(K|k), µ(K)(p))/pÐYdzï@å±×�×3Ì3ÇÎÉ�Ç ÕqÍ

dzÙ@×{é$Á/È�åYÔ�È�Ð ëyÇÎÌ�ò�éF£=é�ÕqÐ�~NÑ.Õ$�µ(K)(p) = µ(k)(p)

Ï�ÇÎÍÎ×^~NÛYÐ.Ñ�Ñ.Õ$�kÑ*dzÈ

pö�×@È�Ð_z�ÇÎÐYå.È�ÇÎ×@Ù3ëyÛYÌ3ò{È�ÍÎÐ�È�б×3å�ìqÍÎ×^~

Ñ.ÕqÐYÐàÈ�Ì3å�ÕqÍÎ×@È�ÐÊëyÇÎÌH0(G(K|k), µ(K)(p))/p = µ(k)(p)/p ∼= Z/pZ.Æ/dzÈvÙ@È�ÐØø�ÕqÍÎÍ�È�Ì3Ì@È�dzï@å.È�ÐØëyÇÎÌ�Ë$¦-È�ТÖ�ÕqÌ�Ñ*ÛYÌ@ï�åØÑ*dzÈø.ËNÌ@ÑYÈ�Ì3ÛYÐYÏP~±Ñ.Õ$��ÞyÑqý,ÛYÐ,{±×3dzËNÐ�É�ËNÐ

µps+1ÇÎÔ ø�ÕqÍÎÍ�ûoê3ÿ�ÐYdzï@å¢×

ÛYбÉ�È�Ì3òvëlÈ�ÇÎÏ�×=ÛYÐ.Ñ ÇÎÐSÉ�ËNÍÎÍ�ò{È�Ì3ͳÈ�Ï�×AÛYÐ.Ñ ÇÎÔ ø�ÕqÍÎÍ�ûoê@ê3ÿ�ÕqÛ,��È�Ì3å�ÕqÍÎÖ

SÐYdzï@å¢×/ÛYТÉ�È�Ì3òvë�È�ÇÎÏ�×=dzÙ3×^~*Ü�ÕqÍÎͳÙ

µps(k)Ñ*dzÈ�õ/Ì3ÛYÓYÓ!È�ÑYÈ�ÌIÇÎÐkÈ�Т×3å�ÕqÍÎ×@È�Ð.È�Ð�z�ÇÎÐYå.È�ÇÎ×@Ù3ëyÛYÌ3ò{È�ÍÎÐÊÉ�ËNÐ

pö+��ËN×@È�ÐYò{ËNÌ@Ñ*бÛYÐYÏÚdzÙ3×{é

ÞIͳÙðAÈvÙ3ÛYÍÎ×�Õq×È�Ì3å�ÕqÍÎ×@È�ÐÊëyÇÎÌÜoËNÍÎÏ�È�Ð.ÑYÈvÙY��å.ÈvËNÌ@È�Ô ÇÎÐ í�È�Ì�ÕqÍÎÍÎÏ�È�Ô�È�ÇÎÐ.È�Ì3ÛYÐYÏ�É�ËNÐôß áãâ+~,��å.ÈvËNÌ@È�Ô�­*驨*é�Q���Kó��y�0�������B��$�B��� ¿ �R� À,E )d! U�I.In�/Jy6nUB�TA���� In���V���Ú8G�R�@��� Ç p+È E^8��/)`Ç pdp+È !G4 L³�RA �5�P�æ8��.I�mO.I/<�#B�P�5JRJ5IAB�RA��^OQ�/�0��� o ¿ ��) � >�Z=��( )+*,��) k|k+

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µp(k) ⊆ O×K,m = E(K) o e3IgI^��� µ(k)(p) = µps(k)

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k(µps+1)|k�yI/Jt�P�R<�#BJsUB��C$��)Q6 L ���KAGJ3UB�08}�5� S CFEG%�%96F��).%É�RAGJ5-�54ÍË�! %5%tÇ pdp+È Ê

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