¼ ½¿¾0À5¼ Áÿ½ÅÄ - Université du Luxembourgmath.uni.lu/~wiese/other/diplom.pdf ·...
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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$��Ñ*dzÈvÙ�ÈÝÆ/ÇÎÔ�È�Ð.Ù3dzËNÐ ÇÎÐ É�dzÈ�ͳÈ�Ð ø�ìqÍÎͳÈ�ÐS{±Í³È�ÇÎÐ.È�ÌIÏ�ͳÈ�dzï@å
1dzÙ@×^~
ë�ËNÌ�ÕqÛ.ÙAÙ@dzï@åÚÜoäYÌd(G)+ 6= 1
Ù@ËqÜoËNÌ3×È�Ì3Ï�ÇÎÖY×^~�Ñ.Õ$��ТÛYÌd(G) ≤ 3
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
ké
• 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�ÐÒÈ�ÇÎÐ.ÈvÙD�9ÕqåYÍK{[�NÌ3Ó!È�Ì@Ù
ké
•ê�ÐSÙ@È�Ç"{�È�ÇÎÐ.È�äYÖ9È�Ì
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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
ké
• 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
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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@×
mÑYÈ�Ì�ùàË�Ñ*ÛYÍ ∏
p∈S p~�ÛYÐ.Ñ
U (np)Ö!È�ò{È�dzï�åYÐ.È�×lÑ*dzÈIåP�Nå.È�Ì@È�Ð z�ÇÎÐ.Ù@È�ÇÎÐYå.È�ÇÎ×@È�ÐQé�ø?äYÌlÑYÈ�Ì@È�Ð�Æ=È��.ÐYÇÎ×3dzËNÐ6ÕqÐ�ÛYÐ.È�Ð.Ñ*ÍÎdzï@å.È�Ð��±×@È�ÍÎͳÈAÙ3dzÈ�å.È�!éPG²*é• 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
�-dzÈ�Ï�×�È�ÇÎÐ�ü�ù ö+�D�NÌ3Ó!È�Ì�òvÛYÏ�Ì3ÛYÐ.ÑYÈG~¢Ù@Ë/dzÙ3×�Ñ*dzÈvÙ@È�ÆAÛ�ÕqÍÎÇÎ×�ìq×b{�ËNÔ"Ó�Õq×3ÇÎÖ9È�Í.Ô"ÇÎ×�ÑYÈ�Ì�ÞD{±×3dzËNÐ"ÑYÈ�Ìb{�ËNÔ"ÓYͳÈ��YÈ�Ð�AËNÐNý�ÛYÏ¢Õq×3dzËNÐQé�êuÔ ø�ÕqÍÎÍ�ûoê3ÿ/ÜgËNÌ@ÑYÈ�Ì3Ð(ëyÇÎÌWòvÛ.Ù�ìq×3òvÍÎdzï�å�~0Ñ.Õ$�f{�È�ÇÎÐ.È\��Ì3ÇÎÔ�Ù3×@È�ÍÎͳÈ6ÕqÛ.ÙSÇÎÐ÷ÑYÈ�Ì�z�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ
k|k+ ò{È�Ì,Ü�ÕqÍÎͳÈ�éÞIÛ.ÙÑYÈ�Ô ÆAÛ�ÕqÍÎÇÎ×�ìq×@Ù�Ù�Õq×3òÝëlÈ�Ì@ÑYÈ�ÐàëyÇÎÌIÈ�ÇÎÐ.È��±ÛYÌoý3È/{±×3dzËNÐ
µ(k)/p ⊇ (E(k)/p)− � (pH2(G,Qp/Zp)
−)∨ûV»�ÿ
ûµ(k)Ù3ÇÎÐ.ÑèÑ*dzÈ�ÇÎÐ
kÈ�Т×3å�ÕqÍÎ×@È�Ð.È�Ð¥z�ÇÎÐYå.È�ÇÎ×@Ù@ëyÛYÌ3ò{È�ÍÎÐ�ÿ=Ù@ï@åYÍÎdzÈ/��È�Ð@{[�NÐYÐ.È�Ð�~!ë�È�ͳï@å.È�Ù@ËqÜgËNÌ3×=Ñ*dzÈWÖ9È�å�ÕqÛYÓY×@È�×@È
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i > 0é�ø�äYÌ
i ≤ 0Ù@È�×3òv×yÔ�ÕqÐ
Hi(G,A) = lim←−U�G ½l¾�´Vµ H
i(G/U,AU ),ûR��ÿ
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ÓYÌ@Ë�È�Ð.Ñ*ÍÎdzï@å.ÈØí�È�Ì@Ù3dzËNÐ(ÑYÈvÙ��*Õq×3ò{ÈvÙWÉ�ËNÐÂÁ=Õ${NÕ^�¢ÕqÔ�Õcö+��Õq×@ÈØÛYÐ.ÑôÑYÈvÙ�Æ/Û�ÕqÍÎÇÎ×�ìq×@Ù@Ù�Õq×3ò{ÈvÙ�É�ËNÐr��ËNÇÎ×@ËNÛ�~�ë�ÕNÙWÇÎÐÑYÈ�Ì"ÞAÌ3Ö!È�ÇÎ×�ßÃ�±âyÙ3×3Û.Ñ*dzÈ�Ì3×WëyÛYÌ@ÑYÈ�é?áçÇÎÌWÖ9È�ÐP�N×3ÇÎÏ�È�ÐÒý,ÈvÑYË�ï@å(ТÛYÌ"Ñ*dzÈØí�È�Ì�ÕqÍÎÍÎÏ�È�Ô�È�ÇÎÐ.È�Ì3ÛYÐYÏôÑYÈvÙH�*Õq×3ò{ÈvÙ�É�ËNÐÁ=Õ${�Õn�¢ÕqÔ�Õcö+��Õq×@ÈG~�ÑYÈ�ÐYÐãÑ*dzÈvÙ@È�Ì�ÍÎdzÈdÜoÈ�Ì3×�ÇÎÔÄ��Û.Ù�ÕqÔ"Ô�È�Ð.Ù3ÓYdzÈ�ÍIÔ"ÇÎ×ØÑYÈ�ÔEðIÈ�òvÇÎÓYÌ@ËNòvÇÎ×�ìq×@Ù3å.ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ@Ô�Û.ÙÛYÐ.ÑÊÑYÈ�Ô æ/ÕqÛYÓY×3dzÑYÈ{ÕqͳÙ�Õq×3òWÑYÈ�Ð Æ/Û�ÕqÍÎÇÎ×�ìq×@Ù@Ù�Õq×3òÚûR°�ÿdé|IÔ Ñ*Ç³È õ=È�Ô�È�ÇÎÐ.Ù�ÕqÔ_{�È�ÇÎ×@È�ÐèÑYÈ�ÌIø�ìqÍÎͳÈØûoê3ÿlÛYÐ.ÑÃûoê@ê3ÿlå.È�Ì�ÕqÛ.Ù3òvÛ.Ù3×@È�ÍÎͳÈ�Ð�~�å�ÕqÖ9È�Ð ëyÇÎÌIÑYÈ�ÐàÆ/Û�ÕqÍÎÇÎ×�ìq×@Ù@Ù�Õq×3òòvÛYÐ�ìNï�å.Ù3×ÚÇÎÐ�ÑYÈ�ÌS�±ÓYÌ�ÕNï@å.ÈèÑYÈ�Ì6ÕqÖ.Ù3×3Ì�Õ${±×@È�Ð��=Í ÕNÙ@Ù@È�Ð,{[�NÌ3Ó9È�Ì3×3å.ÈvËNÌ3dzÈÒÖ!È�ëydzÈvÙ�È�Ð�ÛYÐ.ÑçÙ@ï@åYÍÎdzÈ/��ÍÎdzï�å+ÕqÛ*Ü�Ñ*dzÈ{�ËNÐ,{�Ì@È�×@È�Ð��±ÇÎ×3Û�Õq×3dzËNÐ.È�ÐèÕqÐYÏ�È�ëlÈ�Ð.ÑYÈ�×{éø�äYÌ�Ñ*Ç³È ê�ÑYÈ{ÕqÍK{±Í ÕNÙ@Ù�È�ÐYÏ�Ì3ÛYÓYÓ!È�É�ËNÐôü�ù ö+�D�NÌ3Ó!È�Ì3Ð Ï�ÇÎÍÎ×^~!Ü�ÕqÍÎͳÙ
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g.(a⊗ b) := g.a⊗ g.bÖYòvëÝé
(g.φ)(a) := g.(φ(g−1.a)).
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AÕqÛ*Ü.ÕqͳÙ�ðAÈvï@å±×@Ù3Ô�˱Ñ*ÛYÍ
Ô"ÇÎ×a.g := g−1.a
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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
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←−−− F (G)0 := Z[G]∂1←−−− F (G)1 := Z[G2]
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(g0, g1, g2, . . . , gn) = g0.(1, h1, h1h2, . . . , h1h2 . . . hn) =: g0.[h1|h2| . . . hn].
Æ/dzÈ��B�±ÔWÖ!ËNͳÈ[h1|h2| . . . |hn]
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[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).
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
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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
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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$& �/JRJ+���
f 7→ ([g1| . . . |gn] 7→ θ(f([φ(g1)| . . . |φ(gn)]))).
á�ÇÎÌë�ËNÍÎͳÈ�ÐàÕqÛ.ï@åÒÑ*dzÈvÙyòvÛYÌIÆ=È��.ÐYÇÎ×3dzËNÐÊÈ�ÇÎÐYÇÎÏ�È�ÌAëydzï@å¢×3ÇÎÏ�È�Ì=ÞAÖYÖYÇÎͳÑ*ÛYÐYÏ�È�Ð É�È�Ì3ëlÈ�Ð.ÑYÈ�Ð��•ø?äYÌ�È�ÇÎÐ.È3|Iб×@È�Ì3Ï�Ì3ÛYÓYÓ!È
H < GÛYÐ.Ñ�È�ÇÎÐ.È�Ð
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H ↪→ GÛYÐ.Ñ6ÑYÈ�ÌIê�ÑYÈ�Т×3ÇÎ×�ìq×=ÕqÛ*ÜAÑ*dzÈ�« �.I/J=).��&FJ=�RE �
Ì@ÈvÙ: Hn(G,A)→ Hn(H,A).
•ø?äYÌ=È�ÇÎÐ.È�ÐhÁ/ËNÌ3Ô�ÕqÍÎ×@È�ÇÎͳÈ�Ì
U � Gå�ÕqÖ9È�ÐÒëyÇÎÌÝÑ*Ç³È Ð�Õq×3äYÌ3ÍÎdzï�å.È}��Ì@Ëqý3È/{�×3dzËNÐ
φ : G → G/UÛYÐ.ÑÒÑ*dzÈz�ÇÎбÖ!È�×3×3ÛYÐYÏ
Θ : AU ↪→ A~*ëlÈ�ͳï�å.ÈWÈ�ÇÎÐ6É�È�Ì3×3Ì�ìqÏ�ÍÎdzï�å.ÈvÙ³�0Õ�ÕqÌÖYÇÎͳÑYÈ�ÐQé�ÆAdzÈvÙ�ÈvÙyÍÎdzÈdÜoÈ�Ì3×Ñ*dzÈ�p �^À2![J=�=E �ÇÎÐ*Ü
: Hn(G/U,AU )→ Hn(G,A).
•��È�ÇAëydzÈvÑYÈ�Ì3ÛYÔ
H < GÈ�ÇÎÐ.È�|Iб×@È�Ì3Ï�Ì3ÛYÓYÓ!ÈG~
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g ∈ GéláçÇÎÌ6È�Ì3å�ÕqÍÎ×@È�Ð È�ÇÎÐ
É�È�Ì3×3Ì�ìqÏ�ÍÎdzï@å.ÈvÙ_�0Õ�ÕqÌ Ñ*ÛYÌ@ï@åφg : gHg−1 → H, ghg−1 7→ h
ÛYÐ.Ñθg : A → A, a 7→ g.a
é?ÆAdzÈvÙ�ÈvÙÍÎdzÈdÜgÈ�Ì3×æAËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�È�Ð
Hn(gHg−1, A)→ Hn(H,A).ø?äYÌÈ�ÇÎÐ.È�Ð�Á/ËNÌ3Ô�ÕqÍÎ×@È�ÇÎͳÈ�ÌH �G
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Æ/È�ÌWÜgËNÍÎÏ�È�Ð.ÑYÈv�*Õq×3ò�ÍÎdzÈdÜgÈ�Ì3×�ÛQéjÕ*é?È�ÇÎÐ.ÈØÐ�Õq×3äYÌ3ÍÎdzï@å.ÈÚí�È�Ì3×3Ì�ìqÏ�ÍÎdzï@å,{�È�ÇÎ×ØÑYÈ�Ì�Ù@ËÒÑYÈ��.ÐYdzÈ�Ì3×@È�ÐóÞIÖYÖYÇÎͳÑ*ÛYÐYÏ�È�ÐÔ"ÇÎ×Ií�È�Ì3ÖYÇÎÐ.Ñ*ÛYÐYÏ�Ù3å.ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�È�ÐQé�z�ÌdzÙ3×Ñ.ÕNÙY{�ËNå.ËNÔ�ËNͳËNÏ�dzÙ@ï@å.È�ÞIÐ�ÕqͳËNÏ�ËNÐÒÑYÈvÙD�*Õq×3ò{ÈvÙ�ûl��éK��é©�ÿdé���9�=���yô0�KJ�â:�^å,�������
φ : G→ H�����ÂW2).U�*[*,���P#TE 4HE 4HE )+*�#���I.4�U�IªUT�98
Θ���5�gË�UB�,&$JVE )ª8B��)
GZV>XE^8GUB%;�f�5�8G�R�
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GZ+>XE^8 UT%
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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&$E 4�4�UTJ=�V��)/J+Ê
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y
y
f
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Θ�5�^¡^�.&FJ=�5C$�ZYYO=¡n�.&$J+��! U.�����^¡^�.&FJ=�5C$��!�O o ¿ ! �P�vA���%MJ+���0Ê
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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
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(AV )U −→AUÇÎÐ.Ñ*ÛYòvdzÈ�Ì3×{é�Æ/dzÈvÙ@È�ÌIdzÙ3×yÔ"ÇÎ×ÑYÈ�ÌIê�ÑYÈ�Т×3ÇÎ×�ìq×/ÕqÛ*Ü
G/UÉ�È�Ì3×3Ì�ìqÏ�ÍÎdzï�åQé
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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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$�úÑ*dzÈàõ=Ì3ÛYÓYÓ!È�Ð
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Gö�ùÒ˱Ñ*ÛYÍ
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Ñ*ÇÎÌ@È/{�×@ÈvÙs�B�*Ù3×@È�ÔHÖYÇÎͳÑYÈ�Ð�~që�ÕNÙ0ëydzÈvÑYÈ�Ì3ÛYÔ�ÕqÛ*Ü�Æ/ÇÎÔ�È�Ð.Ù3dzËNÐH²Y{�Í ÕqÌ0dzÙ3×�ÛYÐ.Ñ Ñ*ÛYÌ@ï�å Æ/ÇÎÔ�È�Ð.Ù3dzËNÐ.Ù@É�È�Ì@Ù@ï@åYdzÈ�ÖYÛYÐYÏÕqÛ.Ù@Ï�ÈvÑYÈ�åYТ×Ië�È�Ì@ÑYÈ�Ð�{�ÕqÐYÐQéYáçÇÎÌÙ@È�×3ò{È�Ð
Hn(G,A) := lim−→Hn(G/U,AU ).
ûoÆ/dzÈvÙàÙ3ÇÎÐ.Ñ ÐYdzï@å±×àÑ*Ç³È É�ËNÌ3Ð.ÈúÑYÈ��.ÐYdzÈ�Ì3×@È�ÐHn(G,A)
émÿÊÞIÍÎ×@È�Ì3Ð�Õq×3ÇÎÉ È�Ì3Ï�È�Ö9È�Ð>Ù3dzï@åHÑ*dzÈèÐ.È�Û>ÑYÈ��.ÐYdzÈ�Ì3×@È�ÐHn(G,A)
ÕqͳÙx�IËNå.ËNÔ�ËNͳËNÏ�dzÈ=ÑYÈvÙx�AËNÔ"ÓYͳÈ��*ÈvÙ�ÑYÈ�ÌCn(G,A)
ÖYòvëÝéCn(G,A)
~¢ÇÎÐ.ÑYÈ�Ô Ô�ÕqÐØÑYÈ�ÐGö�ùàË�Ñ*ÛYÍ
AÔ"ÇÎ×"ÑYÈ�Ì Ñ*dzÙd{�Ì@È�×@È�Ðj��ËNÓ!ËNͳËNÏ�dzÈ�É�È�Ì@Ù3dzÈ�å±×"ÛYÐ.ÑôòvÛ.Ù�ìq×3òvÍÎdzï@å Ñ*dzÈv�±×@È�×3ÇÎÏG{�È�ÇÎ×�ÕqÍÎͳÈ�Ì ÞAÖYÖYÇÎͳÑ*ÛYÐYÏ�È�Ð(ÜoËNÌ@ÑYÈ�Ì3×{éÞAÛ*Ü�Ñ*dzÈvÙ@È=áúÈ�dzÙ@È�É�È�Ì�ÕqÍÎÍÎÏ�È�Ô�È�ÇÎÐ.È�Ì3×=Ù3dzï@åÒÕqÛ.ï@åÊÑYÈ�Ì�£�È�Ï�Ì3ÇM¦ ÑYÈvÙY{�ËNÇÎÐ.Ñ*ÛYòvdzÈ�Ì3×@È�ÐÒùàË�Ñ*ÛYͳÙyòvÛ
CoindGH(A) := {f : G→ A | f(h.g) = h.f(g) ∀h ∈ H, fÙ3×@È�×3ÇÎÏ
}.
Æ/dzÈyÆ=È��.ÐYÇÎ×3dzËNÐ�ÑYÈvÙ0É�È�Ì3×3Ì�ìqÏ�ÍÎdzï�å.È�Ð �0Õ�ÕqÌ@ÈvÙ�ÛYÐ.Ñ ÑYÈ�Ì�Ô"ÇÎ×�Ù@È�ÇÎÐ.È�Ì�æAÇÎÍ ÜoÈ�È�Ì3å�ÕqÍÎ×@È�Ð.È�Ð�ÞAÖYÖYÇÎͳÑ*ÛYÐYÏ�È�Ð�äYÖ9È�Ì3×3Ì�ìqÏ�×Ù@dzï@åÊÑ*ÇÎÌ@È/{�×{é�xâ:�^�Gë.ñ �9�"�9�¥�9ß5��� Þ �f�T�/;¯ß Þ�÷ �"��vê\$�"ì�ìt�B�êuÐÚÑ*dzÈvÙ@È�Ô ÞIÖ.Ù@ï�åYÐYÇÎ×3×yÙ@È�Ç!Ñ*dzÈ�õ/Ì3ÛYÓYÓ9È
GÙ3×@È�×@Ù�È�Ð.Ñ*ÍÎdzï@å é¢áçÇÎÌ�È�Ì3ÇÎÐYÐ.È�Ì3ÐÊÕqÐØÖYòvëÝé�×3Ì@È�¦-È�Ð�ÜgËNÍÎÏ�È�Ð.ÑYÈ=Æ/È��.ÐYÇ ö
×3dzËNÐ.È�Ð��• NG :=
∑
g∈G g ∈ Z[G]~
• IG := ker(ε : Z[G]→ Z) = (g − 1|g ∈ G) � Z[G]~
• NGA = ker(ANG−−→ A)
ÜoäYÌÈ�ÇÎÐ.È�ÐGö�ùÒ˱Ñ*ÛYÍ
Aé
���9�=���a`�� ö �B�¥�¥â��HÇÆ! È �����X���5�0��!�OQ��%'I/</#:�\W2)�U^*[*P� o ¿ ! �P�X�yI/J
Hom(Z[G], X)→ Z[G]⊗Z X, φ 7→∑
g∈G
g ⊗Z φ(g)
�����æW2Z p InE 4HE )+*�#���I.4�U�I.-�8��.IcI^��� ��4�& �.#B)l!�O�O���%K8 UT�BA`8 UT)d<�#h ⊗Z x 7→ (g 7→ xδg,h)
8��cbt�P�V��)/J7��I�J5-L E�OQ��� δg,h 8[!$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$��ÈvÙAÙ3dzï�åÊ×�Õq×@Ù�ìNï�åYÍÎdzï@å ÛYÔGö�æAËNÔ�ËNÔ�ËNÌ,ö
ÓYåYdzÙ@Ô�È�Ð å�ÕqÐ.ÑYÈ�ÍÎ×^~.ÛYÐ.Ñ6Ì@Èvï�åYÐ.È�×AÑ*dzȣ�Ç ý3È/{±×3ÇÎÉ�ÇÎ×�ìq×Ð�ÕNï@åQé�ûoÑ�ÿlÜoËNÍÎÏ�×Ù�ËqÜoËNÌ3×IÕqÛ.Ù�ûgÕ�ÿ�ö�ûoï&ÿdé��ÛÃûaÖ�ÿ�Ù@È�Ç-òvÛYÐ�ìNï�å.Ù3×a ∈ AG
éYÆ�ÕA ZögÜoÌ@È�Ç-dzÙ3×^~:{[�NÐYÐ.È�Ð ëyÇÎÌIÈ�ÇÎÐ.ÑYÈ�ÛY×3ÇÎÏa =
∑
s∈G
s⊗ xs
Ù�ï@åYÌ@È�ÇÎÖ9È�ÐÊÔ"ÇÎ×xs ∈ X
éYáúÈ�Ï�È�ÐàÑYÈ�ÌGö�ê�бÉNÕqÌ3Ç ÕqÐYòWÏ�ÇÎÍÎ×
a =∑
s∈G
gs⊗ xs =∑
s∈G
s⊗ xg−1s,
ë�ÈvÙ3å�ÕqÍÎÖàÑ*dzÈxsÕqÍÎͳÈ�Ï�ͳÈ�dzï�åàÙ3ÇÎÐ.Ñ-é���ËNÔ"ÇÎ×IdzÙ3×
a = NG(1⊗ xid)È�ÇÎÐSz�ͳÈ�Ô�È�Т×/É�ËNÐ
NGAé
Æ/ÈvÙ@ëlÈ�ÇÎ×@È�Ì@È�ÐàÙ�È�Ç-ТÛYÐa ∈ NGA
Ï�È�Ï�È�Ö!È�Ð�~.Ñ-é åQé0 = NG a = NG
∑
s∈G
s⊗ xs =∑
g∈G
g ⊗∑
s∈G
xs,
Ù�Ë"Ñ.Õ$� ∑
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$�"ÜoäYÌ
n ≥ 1
HomG(F−n, A) = HomG(F ∗n−1, A) ∼= Fn−1 ⊗Z[G] A
Ï�ÇÎÍÎ×{é�ÞIÛ,��È�Ì@ÑYÈ�Ôîå�ÕqÖ9È�Ð ëyÇÎÌIÑ*dzÈ�ê�Ù@ËNÔ�ËNÌ3ÓYåYdzÈZ⊗Z[G] A = AG
NG−−→ AG = HomG(Z, A).
ÞAТë�È�Ð.Ñ*ÛYÐYÏúÑYÈvÙWø.ÛYÐ,{±×@ËNÌ@ÙHomG(·, A)
È�Ì3Ï�ÇÎÖY× Ð¢ÛYÐ÷Ñ.ÕNÙWÜoËNÍÎÏ�È�Ð.ÑYÈ {�ËNÔ"Ô�ÛY×�Õq×3ÇÎÉ�È6Æ/Ç ÕqÏ�Ì�ÕqÔ"ÔS~0ÑYÈvÙ@Ù@È�ÐËNÖ9È�Ì@Ù3×@È��!È�ÇÎͳÈ�È�ÇÎÐS�IË {�ËNÔ"ÓYͳÈ�� dzÙ3×^�−−−→ F1 ⊗Z[G] A −−−→ F0 ⊗Z[G] A −−−→ HomG(F0, A) −−−→ HomG(F1, A) −−−→
y
x
AGNG−−−→ AG
y
x
0 0.
áçÇÎÌ�ÑYÈ��.ÐYdzÈ�Ì@È�Ð�Ñ*dzÈ}¼ !GJ+��ZR��E #TE 4HE %KEQA �V�RA )�U^*[*P���Hn ÜoäYÌ n ∈ Z
ÕqͳÙ�Ñ*dzÈ2�IËNå.ËNÔ�ËNͳËNÏ�dzÈIÑYÈ�Ì�ËNÖ9È�Ì@Ù3×@È�Ð �!È�ÇÎͳÈ�éÆ/dzÈ7�?Õq×@Èdö+�IËNå.ËNÔ�ËNͳËNÏ�dzÈ
Hn(G, ·)ÖYÇÎͳÑYÈ�×IÈ�ÇÎÐ.È�ÐS{�ËNå.ËNÔ�ËNͳËNÏ�dzÙ�ï@å.È�ÐèøYÛYÐ,{±×@ËNÌ{é�z�ÙÏ�ÇÎÍÎ×IÑYÈ�Ì
���9�=���5ø��KJ�â:�^å,�
Hn(G,A) =
Hn(G,A) n ≥ 1
AG/NGA n = 0
NGA/IGA n = −1
H−n−1(G,A) n ≤ −2A �CB�� Þ ä[� Æ=È�ÐèÈ�Ì@Ù3×@È�ÐàÛYÐ.ÑàͳÈ�×3òv×@È�Ð ø�ÕqÍÎÍ0ÍÎdzÈvÙ3×/Ô�ÕqÐÒÑ*ÇÎÌ@È/{�×=ÕqÛ.Ù=ÑYÈ�Ô Æ/Ç ÕqÏ�Ì�ÕqÔ"Ô ÕqÖQé9ø?äYÌn = 0,−1È�Ì3Ï�ÇÎÖY×WÙ3dzï@åôÑ.ÕNÙz�Ì3Ï�È�ÖYÐYdzÙWÑ.ÕqÌ�ÕqÛ.Ù^~-Ñ.Õ$�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${�×@ËNÌ,ö�ÕqͳÙ�ÕqÛ.ï@å\|AТ×@È�Ì3Ô�Ë�Ñ*ÛYÍ9È�ÇÎÐ.ÈvÙlÇÎÐ.Ñ*ÛYòvdzÈ�Ì3×@È�ÐÚÖYòvëÝé�{�ËNÇÎÐ.Ñ*ÛYòvdzÈ�Ì3×@È�Ð6ùÒ˱Ñ*ÛYͳÙ�~
ÛYÐ.Ñ ëyÇÎÌIÑ*äYÌ,ÜoÈ�Ð Ñ*dzÈ�ùàÈ�×3å.Ë�ÑYÈÝÑYÈ�ÌÆ/ÇÎÔ�È�Ð.Ù3dzËNÐ.Ù3É�È�Ì@Ù@ï�åYdzÈ�ÖYÛYÐYÏÚÇÎÐ6Ö!È�dzÑYÈ�ÐÊðIdzï@å±×3ÛYÐYÏ�È�ÐÊÖ9È�ТÛY×3ò{È�ÐQéÞIÛ*ÜlÑ*dzÈvÙ�È�áúÈ�dzÙ@È�{[�NÐYÐ.È�ÐèëyÇÎÌ=ÛQé!Õ*é!Ñ*dzÈWðAÈvÙ3×3Ì3ÇK{�×3dzËNÐàÛYÐ.ÑÒÑ*dzÈ��IËNÌ@ÈvÙ3×3Ì3ÇK{�×3dzËNÐúÕqÛ*Ü�ÕqÍÎÍ³È ÆAÇÎÔ�È�Ð.Ù@dzËNÐ.È�Ð
n ∈ ZÕqÛ.Ù�ÑYÈ�åYÐ.È�ÐQé
ø�äYÌAÑ*dzÈÝæAËNÔ�ËNͳËNÏ�dzÈÝå�Õq×3×@È�Ð ëyÇÎÌIÑ*dzÈÝÆ=È�Ï�Õq×3dzËNÐ Ö9È�Ì@È�ÇÎ×@ÙyÉ�ËNÌ3Ð.ÈÝÑYÈ��.ÐYdzÈ�Ì3×{é9z�ÙIÙ3×@È�ÍÎÍÎ×Ù@dzï@å å.È�Ì�ÕqÛ.Ù^~�Ñ.Õ$�ÑYÈdÜ
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
Ñ*ÛYÐYÏ�Ù@å.ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�È�ÐQéÞIͳÙ6Ð�ìNï�å.Ù3×@ÈvÙ6ÔH��ï@å±×@È�Ð>ëyÇÎÌÊÕqÛ.ï@åHÈ�ÇÎÐ.ÈúÆ=È�Ï�Õq×3dzËNÐHÕqÛ*Ü Æ/ÇÎÔ�È�Ð.Ù3dzËNÐ
0ÜoäYÌ Ñ*dzÈ@��Õq×@Èdö+�AËNå.ËNÔ�ËNͳËNÏ�dzÈ
ÑYÈ��.ÐYdzÈ�Ì@È�ÐQé�Æ=ÕqòvÛ È�Ì3ÇÎÐYÐ.È�Ì3ÐóëyÇÎÌ"ÛYÐ.Ù"Ñ.ÕqÌ�ÕqÐ�~�Ñ.Õ$� Ñ*dzÈÚí�È�Ì3ÖYÇÎÐ.Ñ*ÛYÐYÏ�Ù3å.ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ@Ô�È�Ð ÕqÛ*Ü=ÆAÇÎÔ�È�Ð.Ù@dzËNÐ−1ÇÎÔ áúÈvÙ@È�Т×3ÍÎdzï�å.È�ÐúÑ*ÛYÌ@ï@å Ñ*dzÈ�Á/ËNÌ3Ô Ï�È�Ï�È�Ö9È�ÐèÙ3ÇÎÐ.Ñ-é�áçÇÎÌ/Ö!È�×3Ì�ÕNï�å¢×@È�ÐÒÜaäYÌ=È�ÇÎÐ.È�È��YÕ${�×@È���Èn ±Û.È�ÐYò
0 →A→ B → C → 0
Ñ.ÕNÙÆ/Ç ÕqÏ�Ì�ÕqÔ"ÔNGC/IGC
δ−1−−−→·NG
AG/NGA
¶/´i^ −1
y
·NU
y
NG/UCU/IG/UC
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= bρστ − bρ + bρσ + bρ + c(ρ, στ) − c(ρ, σ) + ρc(σ, τ)
= bρστ − bρσ + c(ρσ, τ) = (ρσ)bτ
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φ : C(γ) = C ⊕B → IG, c 7→ 0, bσ 7→ σ − 1.
æAdzÈ�Ì�ÕqÛ.ÙIÈ�Ì3å�ÕqÍÎ×@È�ÐÊëyÇÎÌIÑ*dzÈÝÈ��.Õ${±×@È���Èn ±Û.È�ÐYò0 −−−→ C −−−→ C(γ)
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ë�ËNÖ!È�Ç9Ñ*dzÈIÈ�Ì@Ù@×@ÈIÞAÖYÖYÇÎͳÑ*ÛYÐYÏ�Ñ*dzÈAÐ�Õq×3äYÌ3ÍÎdzï@å.Ȫz�ÇÎбÖ!È�×3×3ÛYÐYÏC ↪→ C⊕B
dzÙ3×{é�áúÈ�ÇÎ×@È�Ì�å�ÕqÖ9È�Ð�ëyÇÎÌ�Ñ*dzÈIÈ��.Õ${±×@È��Èn ±Û.È�ÐYò0 −−−→ IG −−−→ Z[G]
ε−−−→ Z −−−→ 0.
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ÛYÐ.Ñ�ý,ÈvÑYÈvÙn ∈ Z
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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)
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,
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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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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).
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
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))
.
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)
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Hi(G(K|k),MK )é�ÞAͳÙ@ËèÖYͳÈ�ÇÎÖY×ØÑ*dzÈSzb�YÕ${�×3å.È�ÇÎ×�Ö9È�ÇÎÔ ÐIÖ9È�Ì3Ï¢ÕqÐYÏúòvÛYÔ ÓYÌ@Ëqý,È/{�×3ÇÎÉ�È�Т�-ÇÎÔ�ÈvÙ
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Hi+1(
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i(
G(K|k),MK
) ÖYòvëÝé
Hi+1(
Gk(p), E(p))
∼= lim←−K|k ´Vµ/¶/·WÊ p ËPÌ�«aÍ/Ê H
i(
Gk(p),MK
) ÛYÐ.Ñ
lim←−K|k ´Vµ/¶/·WÊ H
i(
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lim←−K|k ´+µ�¶/·WÊ p ËaÌC«UÍ�Ê H
i(
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)
∼= Hi(
Gk(p), C(p))
.
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kë�ìqåYͳÈ�ÐØëyÇÎÌ�È�ÇÎÐ.ÈYz�ÇÎбÖ!È�×3×3ÛYÐYÏ
ip : k ↪→ kp
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kp|KP• |kp
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Gp(K|k) = G(KP• |kp)~GK := G(k|K)
ÛYÐ.ÑKp := KP•
éø.È�Ì3Ð.È�ÌAë�ËNÍÎͳÈ�ÐàëyÇÎÌIÛYÐ.ÙAÑ.ÕqÌ�ÕqÛ*Ü�È�ÇÎÐYÇÎÏ�È�Ð�~�ë�ÕNÙAëyÇÎÌÛYТ×@È�Ì=ÑYÈ�Ðfz�ÇÎÐ.Ù�È�ÇÎÐYå.È�ÇÎ×@È�ÐYÏ�Ì3ÛYÓYÓ!È�Ð È�ÇÎÐ.ÈvÙ�9ÕqåYÍK{[�NÌ,ö
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n ≥ 0�
U(n)p :=
Up
ÜoäYÌn = 0, p -∞,
n−×@Èz�ÇÎÐ.Ù@È�ÇÎÐYå.È�ÇÎ×@È�ÐYÏ�Ì3ÛYÓYÓ9È�É�ËNÐ
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n ≥ 1, pÌ@ÈvÈ�ÍÎÍ
,
C× ÜoäYÌp{�ËNÔ"ÓYͳÈ��
.
á�ÇÎÌIÑYÈ��.ÐYdzÈ�Ì@È�ÐàТÛYÐÊÑ*dzÈ�p 8B��% AG).U�*[*,� ÕqͳÙÑ.ÕNÙÈ�ÇÎÐYÏ�ÈvÙ�ï@åYÌ�ìqÐ,{�×@È���Ì@Ë�Ñ*Û,{�×IK :=
∏
p
(K×p ,U
(0)p ).
Á=ÕNï@å��*Õq×3òØûl��驨*驨�ÿ�dzÙ3×Ñ*dzÈÝê�ÑYÈ�ÍÎÏ�Ì3ÛYÓYÓ9È�Í³Ë {NÕqÍ�{�ËNÔ"Ó�Õ${±×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
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$�ÝÕqÛ*ÜoÏ�Ì3ÛYÐ.Ñ�ÛYÐ.Ù@Èdö
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(0)p
ÕqÐØÌ@ÈvÈ�ÍÎͳÈ�Ðv�±×@È�ÍÎͳÈ�ÐIS(K)
Ï�ͳÈ�dzï@åIT (K)
dzÙ3×^~�ÜgÕqÍÎͳÙlÙ3dzï@åÚÑ*dzÈ���Ì3ÇÎÔ�Ù3×@È�ÍÎͳÈ�ÐYÔ�È�ÐYÏ�È�ÐSÛYÐ.Ñ
TТÛYÌAÕqÐÊÛYÐ.È�Ð.Ñ*ÍÎdzï@å.È�Ðh�±×@È�ÍÎͳÈ�Ð ÛYТ×@È�Ì@Ù@ï�å.È�dzÑYÈ�ÐQé
ùÊÇÎ×OKëlËNÍÎͳÈ�ÐãëyÇÎÌ�ëydzÈÚÏ�È�ë3�NåYÐYÍÎdzï@åçÑYÈ�Ð÷ðIÇÎÐYÏ ÑYÈ�Ì"Ï¢ÕqÐYò{È�ÐÎ�9ÕqåYͳÈ�ÐóÉ�ËNÐ
KÖ9È�ò{È�dzï@åYÐ.È�ÐQé�ø.È�Ì3Ð.È�Ì
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OK,S := {a ∈ K | vp(a) ≥ 0ÜoäYÌIÕqÍÎͳÈ
p 6∈ S, p -∞},
Ñ-é.åQéYÑ.Õ$��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×.
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J SKdzÙ3×^~
ë�ËNÖ!È�ÇSåYdzÈ�ÌIÑ*dzÈÝùÒÈ�ÐYÏ�È�ÕqÍÎͳÈ�Ì
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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
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eiÑYÈ�Ìyí�È�Ì3òvëlÈ�ÇÎÏ�ÛYÐYÏ�Ù3ÇÎÐ.ÑYÈ��àÉ�ËNÐ
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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$�ØÕqͳÙ@Ë
Ik = IG(K|k)K
dzÙ@×^~�ë�ÕNÙ�Ù3dzï�åóÐ�Õq×3äYÌ3ÍÎdzï@å�ÕqÛ.ï�å�ÕqÛ*ÜImK
ÛYÐ.ÑIS(K)
äYÖ9È�Ì3×3Ì�ìqÏ�×{é�Æ=ÕqÔ"ÇÎ×�È�Ì3å�ÕqÍÎ×@È�ÐãëyÇÎÌ ÜaäYÌ"È�ÇÎÐ.ÈÊõ�ÕqͳËNdzÙ,öz�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏM |K
Ñ*dzÙd{±Ì@È�×@ÈG(M |K)
ö�ùàË�Ñ*ÛYÍÎÐÊëydzÈ/ÜgËNÍÎÏ�×^�IM = lim−→
L|K,L⊆M
IL =⋃
L|K,L⊆M
IG(M |L)M
ûaÖYòvëÝé!õ/ͳÈ�dzï@å.ÈvÙIÜaäYÌIS(K)
ÛYÐ.ÑImK
ÿdéá�ÇÎÌ_{[�NÐYÐ.È�ÐóбÛYÐãÕqÛ.ï@å Ñ*dzÈv��Õq×@Èdö+�AËNå.ËNÔ�ËNͳËNÏ�dzÈàÑYÈ�Ì�ê�ÑYÈ�ÍÎÏ�Ì3ÛYÓYÓ9ÈÚÖ!È�×3Ì�ÕNï�å¢×@È�ÐQé�Æ/dzÈvÙ@È�å�Õq× ÜgËNÍÎÏ�È�Ð.ÑYÈ
ÜoÛYÐ.Ñ.ÕqÔ�È�Т×�Õqͳȳz�ÇÎÏ�È�Ð.Ù@ï�å�ÕcÜa×^~�Ñ*dzÈÛYÐ.ÙlÈ�Ì3Í ÕqÛYÖ!È�ÐØëyÇÎÌ@Ñ�~¢ðAÈvÙ3ÛYÍÎ×�Õq×@ÈIÕqÛ.Ù�ÑYÈ�ÔÌ�QË {NÕqͳÈ�ÐØÇÎÐ.Ù�õ/ͳËNÖ�ÕqͳÈIòvÛ�äYÖ9È�Ì,ö×3Ì�ÕqÏ�È�ÐQé���9�y���ÉÅ��KJ�â:�^å,�������
K|k�/���0�����98,%'�R<�#:� WY! %KE ��I.Z=e2) L ���=J+��)�UB�TA o ¿ ! �P�\A ��%MJ9�.NT)�! %5%K� i ∈ Z
Hi(G(K|k), IK) ∼=⊕
p
Hi(Gp(K|k),K×p ).
¿ !�OQ�/�2OQ�d!�<�#BJ+��4H! �@8 �V�D��E �PC ����J=�RE �h8���)³13)��54�I/J+��%5%K���0! U�I L !$#�%"OQ�/�¯e3) L ���=J+��).UT�BA��/� oA �CB�� Þ ä[� Æ=È�Ì?£�È�ëlÈ�dzÙ�Ö9È�Ì3ÛYå¢×IÕqÛ*Ü0ÑYÈ�Ô ��È�Ô"Ô�Õ É�ËNÐ��±å�ÕqÓYÇÎÌ@Ë ÛYÐ.ÑØdzÙ@×�ò�é:£=é�ÇÎÐ ßkÁª�±á�â+~:��Ì@ËNÓ!Ë�Ù@ÇÎ×3dzËNÐ�*éK��驨:~.Ð�ÕNï@åYòvÛYͳÈvÙ@È�ÐQé
2
ÞIͳÙY�AËNÐ.Ù@Èn ±Û.È�ÐYò�É�ËNÐ æAÇÎÍÎÖ!È�Ì3×@ÙD�*Õq×3ò�±G²"È�Ì3å�ÕqÍÎ×@È�Ð ëyÇÎÌIÑ.ÕNÙ���9�y���Kó���ñ ��$�9ß=ß5â,���ËsNT)����98:%;�R<�#:� WY! %KE �yI�Z=e3) L ���=J+��)�UB�BAB��� K|k A ��%MJ+Ê H1(G(K|k), IK) = 1 o���9�y���yô0��ñ ��$�9ß=ß5â,������/�
K|k���5�0��UT�PC$��)Q6 L ���KAGJ+�ª���98,%'�R<�#:��WY! %KE ��I.Z=e2) L ���=J+��)�UB�TA$-"8 �V�?�5�v�/���0�/)x13).�54�I�J+��%;Z%É���P4}���TA��
SCFEG%�%06F��)�%K�RA[Jx��I�J5-s8[!G�P�\AG��%MJP��NB)�! %�%�
i ∈ ZÊHi(G(K|k), IS(K)) = 1 o
A �CB�� Þ ä[� ê�Ù3×p ∈ S
~�Ù@Ë(dzÙ3×6Ñ*dzȥ�!È�Ì3ͳÈ�Ï�ÛYÐYÏ�Ù3Ï�Ì3ÛYÓYÓ9È Ï�ͳÈ�dzï�åG(Kp|kp) = 1
éyÞAÛ,��È�Ì@ÑYÈ�Ô¿Ï�ÇÎÍÎ×ÚÜaäYÌÛYбÉ�È�Ì3òvëlÈ�ÇÎÏ�×@ÈH�±×@È�ÍÎͳÈ�Ð
pÐ�ÕNï�åôßkÁö+���yâ+~Yê@ê�é ®.é©:~YÑ.Õ$�
Hi(G(Kp|kp),Up) = 1dzÙ3×{é
2
à ;��Bßyëèç à ;���âPßyë����/;éJ0�^$âP�"ßcF�ß5â,ä^ä^�T���9$�"ì�ìt�B�ÞAÛ.ÙÝßkÁIöu�0��âQòvÇÎ×3dzÈ�Ì@È�Ð ëyÇÎÌIÑYÈ�ÐÊëydzï@å¢×3ÇÎÏ�È�Ð@�*Õq×3òÝí=ê�éK��éjþ:����9�y���a`��KJ�â:�^å,�
k×��I�J����5�0�_8 ��Ic&F)d�/J+�HUT�98�!�OVA��.I/</#�%ME IcIn�/�0�¥����J+��)RA )�U^*�*,�}CFE �
IkUT��J+��)H8���)_8 �R!^A[E �9!G%K���e2���0Oc�/JRJ=UB�TA
a 7→ (a)p o��ËNÔ"ÇÎ×IdzÙ3×Ñ*dzÈ�Ñ*ÛYÌ@ï�åCK = IK/K
×
ÑYÈ��.ÐYdzÈ�Ì3×@Èp 8���%'&�%K!$IcI^���BA )�U�*[*,� È�ÇÎÐ.ÈyÍ³Ë {�ÕqÍ�{�ËNÔ"Ó�Õ${�×@ÈIæ=ÕqÛ.Ù@ÑYËNÌl¦-Ù@ï@å.ÈIÕqÖ9È�ͳÙ@ï�å.È/õ/Ì3ÛYÓYÓ9È�é¢Æ�Õ�Ñ*dzÈÑ*Ç ÕqÏ�ËNÐ�ÕqͳÈz�ÇÎТÖ9È�×3×3ÛYÐYÏK× ↪→ IK
ÜaäYÌ?È�ÇÎÐ.ÈlÈ�Ð.Ñ*ÍÎdzï�å.È�Ï¢ÕqͳËNdzÙ@Ù@ï@å.È3z�Ì3ë�È�ÇÎ×@È�Ì3ÛYÐYÏK|kÈ�ÇÎÐ
G(K|k)ö�æAËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ@Ô�Û.Ù
dzÙ@×^~�È�Ì3å�ÕqÍÎ×@È�Ð�ëyÇÎÌ�ÕqÛ.ï@å ÕqÛ*ÜCKÈ�ÇÎÐ.È
G(K|k)ö�ùàË�Ñ*ÛYͳÙ3×3Ì3Û,{�×3ÛYÌ^~lÑ*dzÈ6ëydzÈàÑ*Ç³È ê�ÑYÈ�ÍÎÏ�Ì3ÛYÓYÓ!ÈàÈ�ÇÎÐ.È�ÐãÏ�ÛY×@È�Ð
õÝÕqͳËNdzÙ,ö�ÞIÖ.Ù@×3dzÈ�Ï�å�Õq×^~.Ñ-é åQéCk = C
G(K|k)K ,
ë�ÕNÙ ÕqÛ.ÙÚÑYÈ�ÌÚÈ��.Õ${�×@È�Ð ��Èn ±Û.È�ÐYò0 → k× → Ik → C
G(K|k)K → H1(G(K|k),K×)
òvÛ.Ù�ÕqÔ"Ô�È�ÐçÔ"ÇÎ×æAÇÎÍÎÖ9È�Ì3×@ÙD�*Õq×3ò�±G²�ÜgËNÍÎÏ�×{éá�ÇÎÌIÑYÈ��.ÐYdzÈ�Ì@È�ÐÒÑ*Ç³È mO.I/EG%'U:J=�9E )�4 È�ÇÎÐ.ÈvÙê�ÑYÈ�ͳÙ
a ∈ IkÑ*ÛYÌ@ï@å
N (a) :=∏
p
|ap|p.
áúÈ�Ï�È�ÐÒÑYÈ�Ì=õ=ÈvÙ�ï@åYͳË�Ù@Ù@È�ÐYå.È�ÇÎ×@Ù@Ì@È�Í Õq×3dzËNÐÒdzÙ@×N (a) = 1
ÜaäYÌa ∈ k×
é���ËNÔ"ÇÎ×IëyÇÎÌ@ÑÒÕqÛ.ï@åàÈ�ÇÎÐ.ÈWÞIÖ.Ù�ËNÍÎÛY×3Ð.ËNÌ3ÔÕqÛ*Ü0ÑYÈ�ÌêuÑYÈ�ÍK{�Í ÕNÙ@Ù@È�ÐYÏ�Ì3ÛYÓYÓ9È
CkÇÎÐ.Ñ*ÛYòvdzÈ�Ì3×{éYùÊÇÎ×
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${�×3å.È�ÇÎ×AÑYÈ�Ì/õ=Ì3ÛYÓYÓ!ÈC0k
È�Ì3ëlÈ�dzÙ@È�Ð÷û,ßkÁöu�0��â+~���å.ÈvËNÌ@È�Ôîí/êdéK��é©°�ÿdé
Æ/È�ÐÊÍ³Ë {NÕqÍ�{�ËNÔ"Ó�Õ${�×@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${�×@ÈÊõ/Ì3ÛYÓYÓ9ÈUK,S
Ñ*ÛYÌ@ï@å ∏
p∈S{1} ×∏
p6∈S U(0)p
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UK,S{�ËNå.ËNÔ�ËNͳËNÏ�dzÙ@ï@å
×3Ì3ÇÎÉ�Ç ÕqÍV~9ë�ÕNÙÝÕqÛ.Ù�ÑYÈ�Ô Í³Ë {�ÕqͳÈ�ÐúðIÈvÙ@ÛYÍÎ×�Õq×AÜoäYÌ=ÛYТÉ�È�Ì3òvë�È�ÇÎÏ�×@È}��Ì3ÇÎÔ�Ù3×@È�ÍÎͳÈ�ÐpÉ�ËNÐ
kÜgËNÍÎÏ�×"ûaÉ�Ï�Í�é0ßkÁö+���yâ+~�*Õq×3ò ê@êdé ®.é©�ÿ�~.ÛYÐ.ÑàÑ.ÕqÌ�ÕqÛ.Ù^~�Ñ.Õ$��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!$IcI^���BA )�U�*[*,��4HE^8GUB%KE
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á�ÇÎÌØÖ!È�×3Ì�ÕNï�å¢×@È�Ð ÑYÈ�ÐÎz�ÓYÇÎÔ�ËNÌ3ÓYåYdzÙ@Ô�Û.ÙJK � J SK , a 7→ OK,Sa
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SK = ClS(K)
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ClS(K)È�ÇÎÐàø�Õ${±×@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$��Ñ*dzÈvÙ@È�Ì�Ù@dzï@å6ÕqÛ.Ù
a 7→ (a)ÜaäYÌ
a ∈ K×ÛYÐ.ÑÊËNÖYÇÎÏ�È�ÔχòvÛ.Ù�ÕqÔ"Ô�È�Ð.Ù@È�×3òv×{é
á�ÇÎÌÖ9È�×3Ì�ÕNï@å±×@È�ÐàÑ*dzÈ��±ÛYÌoý3È/{�×3dzËNÐχ : I
(m)K � J m
K/PmK .��È�Ç
a ∈ I(m)K
ÇÎÔ¬��È�Ì3Ð É�ËNÐχéYÆ=ÕqÐYÐÊÏ�ÇÎÖY×AÈvÙÈ�ÇÎÐ
(x) ∈ PmK
~YÑ-é åQéx ∈ U
(np)p
ÜoäYÌp | m
~*Ô"ÇÎ×(x) = χ(a)
éÆ�ÕqÐYРdzÙ3×
apx−1 È�ÇÎÐ@z�ͳÈ�Ô�È�Т×�É�ËNÐ U (0)
p
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ÛYÐ.ÑèÉ�ËNÐU
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~�ÜgÕqÍÎͳÙpÑYÈ�ÐúùÒ˱Ñ*ÛYÍ
m×@È�ÇÎÍÎ×{é-Æ=Õqå.È�Ì
Ï�ÇÎÍÎ×ker(χ) = I
(m)K ∩ Im
KK×.ÞAͳÙ@Ë"È�Ì3å�ÕqÍÎ×@È�ÐàëyÇÎÌ
CK/CmK∼= IK/I
mKK
× ∼= I(m)K K×/Im
KK× ∼= I
(m)K /(I
(m)K ∩ Im
KK×) ∼= J m
K/PmK .
z�Ù�Ù@È�ÇmfÑYÈ�Ì�È�Ð.Ñ*ÍÎdzï@å.ÈÞAТ×@È�ÇÎÍ.ÑYÈvÙ�ùàË�Ñ*ÛYͳÙ
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ðAÈvÙ3×@Ù�Õq×3ò�Ï�ÇÎÍÎ×IÑ*dzÈ�ê�Ù@ËNÔ�ËNÌ3ÓYåYdzÈOK/mf
∼=∏
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OK/pvp(np).
ÞAÛ.ÙÊÑYÈ�Ô êuÙ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�Û.Ù(OK/p
s)× ∼= Up/U(s)p , a + ps 7→ aU
(s)p
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× ∼=∏
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(np)p∼= IS∞(K)/I
mf
K .æAdzÈ�Ì�ÕqÛ.ÙͳÈ�ÇÎ×@È�Ð ëyÇÎÌÑ*dzÈÝê�Ù@ËNÔ�ËNÌ3ÓYåYdzÈ
IS∞(K)/ImK∼=
∏
p|mf
Up/U(np)p ×
∏
p|m, p «�´V´V·k· , np=1
R×/R×+
ÕqÖ�~*ÕqÛ.Ù�ÑYÈ�ÌlëyÇÎÌ�Ñ*dzȳz�Ð.Ñ*ÍÎdzï@å,{�È�ÇÎ×yÉ�ËNÐIS∞(K)/Im
K
ÛYб×@È�Ì2£�È�бÛY×3òvÛYÐYÏ ÑYÈ�Ì?z�Ð.Ñ*ÍÎdzï�å,{�È�ÇÎ×yÉ�ËNÐOK/p
vp(np)È�Ì3å�ÕqÍÎ×@È�ÐQé9£�È�×3Ì�ÕNï@å¢×@È�ÐÒëyÇÎÌyТÛYÐÒÑ*dzÈ�È��YÕ${�×@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
ÞIͳÙÈ�Ì@Ù@×@ÈvÙyëlÈ�Ð.ÑYÈ�ÐàëyÇÎÌÛYÐ.ÙÑYÈ�ÌSZ ¼ #,�dE )��R� òvÛQé���È�Ç
SÕqͳÙ@Ë"È�ÇÎÐ.È�ùàÈ�ÐYÏ�È=É�ËNÐ���Ì3ÇÎÔ�Ù3×@È�ÍÎͳÈ�ÐàÑYÈvÙD�9ÕqåYÍ ö{[�NÌ3Ó9È�Ì@Ù
Ké*áçÇÎÌyå�ÕqÍÎ×@È�Ð òvÛYÐ�ìNï@å.Ù3×IÜoÈvÙ@×^�
���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$��).In�.#:�/��In��� oá�ÇÎÌIÑYÈ��.ÐYdzÈ�Ì@È�ÐIK,S :=
∏
p∈S
(K×p ,U
(0)p ).
Æ/dzÈvÙyå�Õq×IÛYÔ"ÇÎ×3×@È�ÍÎÖ�ÕqÌAÑ*dzÈ7z¯�.Õ${�×3å.È�ÇÎ×IÑYÈ�ÌD��Èn ¢Û.È�ÐYò0→ UK,S → IS(K)
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${�×@ÈG~�æ=ÕqÛ.Ù@ÑYËNÌl¦-Ù@ï@å.ÈG~�ÕqÖ9È�ͳÙ@ï�å.ÈWõ/Ì3ÛYÓYÓ9ÈIK,m :=
∏
p|m
U(np)p .
z�ÙIÈ�Ì3Ï�ÇÎÖY×IÙ3dzï�å ëydzÈvÑYÈ�ÌIÑ*dzÈ�È��.Õ${±×@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
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K|k : H2(G(K|k), IK)→1
[K : k]Z/Z
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[K : k]Z/Z
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G(LabS (p)|k) ∼= ClS(k)(p) oA �Bä ÷ �"$ãP�/F��^�f���$å'B�� Þ �0�"�"�ÁAÛYÐ(ëlÈ�Ð.ÑYÈ�Ð÷ëyÇÎÌWÛYÐ.Ù�ÑYÈ�Ì���å.ÈvËNÌ3dzÈØÑYÈ�Ì�Ö!ÈvÙ�ï@åYÌ�ìqÐ,{�×@È�Ð(í�È�Ì3òvë�È�ÇÎÏ�ÛYÐYÏ òvÛQé�Æ�ÕcÜoäYÌWå�ÕqÖ9È�ÐÃëyÇÎÌ�òvÛYÐ�ìNï@å.Ù@×ÑYÈ�ÐÊëydzï@å¢×3ÇÎÏ�È�Ð���9�y���y�PÅ��KJ�â:�^å,�������
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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
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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$��ÜaäYÌ=ÕqÍÎͳÈWÈ�Ð.Ñ*ÍÎdzï�å.È�ÐàÜgËNÌ3Ô�ÕqͳÈ�Ð����NÌ3Ó9È�Ì@È�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ�È�ÐK|kÑ*dzÈ�ÞAÖ!È�ÍÎdzÙ3dzÈ�Ì3ÛYÐYÏ
GabKÈ�Ð.Ñ*ÍÎdzï@åÊdzÙ@×{é
á�ÇÎÌx�,��dzÈ�Ì@È�Ð6È�ÇÎÐ.È/È�Ð.Ñ*ÍÎdzï@å.È/ÜgËNÌ3Ô�ÕqͳȪ���NÌ3Ó9È�Ì@È�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏK|kÛYÐ.Ñ�Ö9È�ò{È�dzï@åYÐ.È�Ð6Ô"ÇÎ×
K1ÑYÈ�Ðv���NÌ3Ó9È�Ì
Ô"ÇÎ×GabK = G(K1|K)
ÛYÐ.Ñ Ô"ÇÎ×K2ÑYÈ�ÐS�D�NÌ3Ó!È�ÌAÔ"ÇÎ×
GabK1= G(K2|K1).
Á=ÕNï@åàÆ/È��.ÐYÇÎ×3dzËNÐàÈ�Ì3å�ÕqÍÎ×@È�Ð6ëyÇÎÌ^~.Ñ.Õ$�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
SÈ�ÇÎÐ.È���Ì3ÇÎÔ�Ù3×@È�ÍÎͳÈ�ÐYÔ�È�ÐYÏ�È�É�ËNÐ
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${±×@È�Ðàø�ÕqÍÎÍ�È�Ì3å�ÕqÍÎ×@È�ÐÊëyÇÎÌÑ*ÛYÌ@ï�åàÑ*dzÈÝÆ=È��.ÐYÇÎ×3dzËNÐGk := G(LS |k)
ÛYÐ.ÑC := CLS
.
ùÊÇÎ×k1Ö!È�ò{È�dzï�åYÐ.È�Ð6ëyÇÎÌyÑ*dzÈIÔ�ÕF�*ÇÎÔ�ÕqͳÈÝÕqÖ!È�ͳÙ�ï@å.È/ÛYТÉ�È�Ì3òvë�È�ÇÎÏ�×@È�ÛYÐ.Ñ�ÇÎÐ
SÉ�ËNÍÎÍ9ò{È�Ì3ͳÈ�Ï�×@ȳz�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ
É�ËNÐkÛYÐ.ÑèÔ"ÇÎ×
k2Ñ*dzÈ"È�Т×@Ù3ÓYÌ@Èvï�å.È�Ð.ÑYÈ�É�ËNÐ
k1~9ë�È�ͳï@å.È"Ï¢ÕqͳËNdzÙ�Ù@ï@åúäYÖ9È�Ì
kdzÙ3×{é!áúÈ�ÇÎ×@È�Ì�Ù@È�dzÈ�Ð
k1(p)ÛYÐ.Ñ
k2(p)Ñ*dzÈWÔ�ÕF��ÇÎÔ�ÕqͳÈ�ÐÒÇÎÐ
k1ÖYòvëÝé
k2È�Т×3å�ÕqÍÎ×@È�Ð.È�РϢÕqͳËNdzÙ@Ù�ï@å.È�Ð
pö+z�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ�È�ÐQéQÆ�ÕqÐYÐf{�ËNÔ"Ô�ÛY×3dzÈ�Ì@È�Ð
Ñ*dzÈ�ÆAÇ ÕqÏ�Ì�ÕqÔ"Ô�È
ClS(k)χ
←−−−∼
Ck/Nk1|kCk1 ClS(k)(p)χ
←−−−∼
(Ck/Nk1(p)|kCk1(p))(p)
µ�¬ ¥ y
i
y
ÛYÐ.Ñ µ�¬ ¥ y
i
y
ClS(k1)χ
←−−−∼
Ck1/Nk2|k1Ck2 ClS(k1(p))(p)χ
←−−−∼
(Ck1(p)/Nk2(p)|k1(p)Ck2(p))(p)
.
Æ=ÕqÔ"ÇÎ×AÈ�Ì3Ï�ÇÎÖY×AÙ3dzï�å Ñ.ÕNÙ���9�y���y�0ø9��ñ}�9F�0ß5ß5â,B� ¿ �V��0!GJ=NB)�%;�=<�#,����mO^O��5%M8 UT�BA��/�
ClS(k)→ ClS(k1)UT�98
ClS(k)(p)→ ClS(k1(p))(p)I����08}J=)���C��=! % o
ÁIÛYÐ(Ù@È�ÇmÈ�ÇÎÐÃùÒË�Ñ*ÛYÍ�ÛYÐ.Ñ
Lm Ñ*dzÈ"Ô�ÕF��ÇÎÔ�ÕqͳÈgz�Ì3ë�È�ÇÎ×@È�Ì3ÛYÐYÏÒÉ�ËNÐ k ÇÎÐYÐ.È�Ì3å�ÕqÍÎÖÃÉ�ËNÐ k ~�Ù@ËÊÑ.Õ$� 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
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kÈ�ÇÎÐÊÕqÍÎÏ�È�ÖYÌ�ÕqdzÙ@ï�å.È�Ì�9ÕqåYÍK{[�NÌ3Ó9È�ÌÛYÐ.Ñ
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ké
• 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
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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)∨.
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A �VO/J2��I&$! �9E ���yIn<�#:��JVE.*:E %KEQA ��I/</#:� p I/EG4_EG)V*�#B�yI�4H�/�Hi
(
G(LS(p)|k), ES(LS(p)))
∼= H2−i(
G(LS(p)|k),Qp/Zp)∨.
A �CB�� Þ ä[� áçÇÎÌWÔWä.Ù@Ù@È�ÐóÑ*dzÈ\£�ÈvÑYÈ�ÛY×3ÛYÐYÏúÑYÈ�Ì�ÇÎÔ ÕqÖ.Ù3×3Ì�Õ${�×@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${�×@È���Èn ¢Û.È�ÐYò
0→ C0 → C → R+ → 0.
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IKÑ*ÛYÌ@ï�å
IS(K)~*ëlÈ�ͳï�å.È�ÜoäYÌ
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ClKë�È�Ì@ÑYÈ�Ì@È�ÓYÌ�ìNÙ@È�Т×3dzÈ�Ì3×�Ñ*ÛYÌ@ï�å
ClS(K)~cë�È�ͳï@å.ÈvÙ�Ð�ÕNï@å �*Õq×3ò�ûl��驨*驨:�&ÿ�dzÙ@ËNÔ�ËNÌ3ÓYå"òvÛYÌ�õ�ÕqͳËNdzÙ,ö�õ=Ì3ÛYÓYÓ!È
ÑYÈ�ÌIÔ�ÕF��ÇÎÔ�ÕqͳÈ�Ð ÛYбÉ�È�Ì3òvëlÈ�ÇÎÏ�×@È�ÐúÕqÖ9È�ͳÙ@ï@å.È�Ð ÛYÐ.Ñ ÇÎÐSÉ�ËNÍÎÍ-ò{È�Ì3ͳÈ�Ï�×@È�Ðfz�Ì3ë�È�ÇÎ×@È�Ì3ÛYÐYÏØÉ�ËNÐ
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lim−→LS⊃K|k
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S ∩ Sp = ∅,
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kÇÎÐk~!Ñ*dzÈ"ÕqÛ,��È�Ì3å�ÕqÍÎÖÒÉ�ËNÐ
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òvÛYÐYÏ"dzÙ3×kS(p)
äYÖ!È�ÌkÇÎÐÊÕqÍÎͳÈ�Ð���Ì3ÇÎÔ�Ù3×@È�ÍÎͳÈ�Ð
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kS ⊃ K|kÙ@ï�åYÌ@È�ÇÎÖ!È�ÐúëyÇÎÌ
ES(K) := O×k,m
ÜoäYÌ�ÑYÈ�ÐùÒ˱Ñ*ÛYÍ
m :=∏
p∈S pé
áçÇÎÌÙ@È�×3ò{È�ÐES(kS) := lim−→
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ES(K)ÛYÐ.Ñ
ES(kS(p)) := lim−→kS(p)⊃K|k
ES(K).
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p = 2ÑYÈ�Ì�õ/Ì3ÛYÐ.Ñ:{[�NÌ3Ó9È�Ì
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ÛYÐ.ÑÚÙ@ËNÔ"ÇÎ×H1(H) = H1(G)
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×3Ì3ÇÎÉ�Ç ÕqÍ�dzÙ3×{é�Æ=ÕqÐYÐÃÔ�Û,�èÕqÖ!È�ÌWÖ!È�Ì@È�ÇÎ×@ÙG/H
Ñ*dzÈ}z�ÇÎÐ.Ù@Ï�Ì3ÛYÓYÓ!ÈØÙ@È�ÇÎÐQé�Æ=È�Ì�*Õq×3òÝÜoËNÍÎÏ�×бÛYÐÊÙ@ËqÜgËNÌ3×{é2z�ÇÎÐ.È2��Ì@Ëqö
pö�õ/Ì3ÛYÓYÓ9È�dzÙ3×0ÕqͳÙ@ËAÏ�È�Ð�ÕqÛ�Ñ.ÕqÐYÐ È�Ð.Ñ*ÍÎdzï@å"È�Ì3ò{È�ÛYÏ�×^~qëlÈ�ÐYÐ
G/Φ(G) = G/Gp[G,G] = H1(G)È�Ð.Ñ*ÍÎdzï�åÊdzÙ3×{éYÆ/dzÈvÙyë�ËNÍÎͳÈ�ÐÊëyÇÎÌͳÈ�dzï�å¢×AÉ�È�Ì�ÕqÍÎÍÎÏ�È�Ô�È�ÇÎÐ.È�Ì3ÐQé|Iб×@È�Ì�£�È�ТÛY×3òvÛYÐYÏÚÑYÈ�ÌÈ��YÕ${�×@È�ÐX��Èn ±Û.È�ÐYò0→ Z/pZ→ Z/prZ→ Z/pr−1Z→ 0
ÜoäYÌr ∈ N
È�Ì3å�ÕqÍÎ×@È�Ð�ëyÇÎÌ�ÇÎÐ.Ñ*Û,{�×3ÇÎÉ0~±Ñ.Õ$�H1(G,Z/prZ)
È�Ð.Ñ*ÍÎdzï@åÚdzÙ3×^~�ÜgÕqÍÎͳÙ�Ñ*dzÈvÙ�ÜaäYÌH1(G,Z/pZ)
ÑYÈ�Ì�ø�ÕqÍÎÍdzÙ@×{éYÞIÛ.ÙÑYÈ�ÌAðIÈvï@åYбÛYÐYÏ
H1(G,Z/prZ) = Hom(G/Gpr [G,G],Z/prZ) = (G/Gpr [G,G])∨
Ù�ï@åYÍÎdzÈ/��È�ÐÊëyÇÎÌ^~�Ñ.Õ$��ÇÎÐ Ñ*dzÈvÙ@È�Ô ø�ÕqÍÎÍ?ÕqÛ.ï@åÊÑ*dzÈWõ=Ì3ÛYÓYÓ!ÈG/Gpr [G,G]
È�Ð.Ñ*ÍÎdzï@åàdzÙ3×{é*áçÇÎÌIÈ�Ì3å�ÕqÍÎ×@È�Ð��®¢þ
�Q���=���y�0� ö �B�¥�¥â��������G���5�0���/�98:%;�=<�#f��)Q6��/U AGJ+�12)lE Z
pZ.W2)�U�*[*,� oÇÆ! È
G/Gpr [G,G]�yI/J3���08:%'�R<�#ª�.NT)�! %5%É�
r ∈ N oÇlO È e3������El�D�/�0�f����J+�/)=A )�U�*[*,�H ≤ G
��I�J2�^OQ���^�n! %5%ÃI_���98:%;�=</#S��)Q6��/U AGJ oA �CB�� Þ ä[� ûgÕ�ÿWå�ÕqÖ9È�ÐóëyÇÎÌ"Ï�È�Ì�ÕNÑYÈÚÏ�È�ò{È�ÇÎÏ�×{éIûaÖ�ÿ�dzÙ3×"ò�鯣=é0ÇÎÐ ß}Æ/Æ=ù@�¢â+~b��Ì@ËNÓ9Ë�Ù3ÇÎ×3dzËNЧ��éûT~0Ö!È�ëydzÈvÙ�È�ÐQé
2��È�ÇN � G
È�ÇÎÐúÕqÖYÏ�ÈvÙ@ï@åYͳË�Ù�Ù@È�Ð.È�Ì�Á/ËNÌ3Ô�ÕqÍÎ×@È�ÇÎͳÈ�Ì�É�ËNÐGé�z�ÇÎÐ e3)d6F��UGA����98��/�,I.w$I�J+�/4 ! %'I®?�E ).4H! %MJ+���5%K�/)
É�ËNÐNdzÙ3×?È�ÇÎÐ.Ès{�ËNбÉ�È�Ì3Ï�È�Т×@ÈY��È�ÇÎÍÎÔ�È�ÐYÏ�È
S ⊆ N~cÙ@Ë/Ñ.Õ$�
NÑYÈ�Ì�{±Í³È�ÇÎÐ.Ù3×@È3Á/ËNÌ3Ô�ÕqÍÎ×@È�ÇÎͳÈ�Ì?dzÙ@×^~cÑYÈ�Ì
SÈ�б×3å�ìqÍÎ×{é
�Q���=���ÉÅ�� ö �B�¥�¥â��������G���5�0�213)dE Z
pZ.W2)�U^*[*P�³UB�08
N�����S!�OVA��.I/</#�%ME IcIn�/�0��)M?7E )�4H! %MJ+����%É��).-¯8��/)D! %'I�?7E )�Z4H! %MJ+���5%K�/)SCFE � 8���)v>@���BA��
S ⊆ N��)Q6���UGAGJ L �5)l8 o Ë"��).����)vIn�/� H � G
���5� L ���=J+��)Q��)X!�OVA��.I/</#�%ME IcIn�/�0��)?�E )�4_!G%;J+���5%É��) o ¿ ! �P� L ��)d8 NH/H ! %'Iõ?�E )�4_!G%;J+���5%É��)_CFE �G/H
�/)d6F��U A[J�CFEG�j8��/)�>@���BA��S8B��) « �.I�J=Z&F%M!$I.In����4HE^8 UT%KE
H8B��)De3%��4}����J+��! UBI
S oA �CB�� Þ ä[� z�ÙWÏ�ÇÎÍÎ×
N =⋂
M~�ëlËNÖ9È�Ç�ÑYÈ�Ì"Æ/ÛYÌ@ï@å.Ù�ï@åYÐYÇÎ×3×"äYÖ9È�Ì ÕqÍÎͳÈÚÕqÖYÏ�ÈvÙ@ï@åYͳË�Ù�Ù@È�Ð.È�Ð�ÁAËNÌ3Ô�ÕqÍÎ×@È�ÇÎͳÈ�Ì
M �GÏ�È�ÖYÇÎͳÑYÈ�×AëyÇÎÌ@Ñ�~.Ñ*dzÈ
SÛYÔ Ü�ÕNÙ@Ù@È�ÐQé.Æ�ÕqÐYРdzÙ3×
NH/H =⋂
S⊆MH/H,M�G
MH/H =⋂
S⊆X,X�G/H
X,
ë�ÕNÙIÑ*dzÈ7£�È�å�ÕqÛYÓY×3ÛYÐYÏØÇÎÔ"ÓYÍÎÇÎòvdzÈ�Ì3×{é2
ÞIÛ.ÙÝßkÁ�±á�âQòvÇÎ×3dzÈ�Ì@È�ÐÊëyÇÎÌ/ü�ËNÌ@ËNÍÎÍ ÕqÌd�S*驱*é©:��Q���=���Kó��KJ�â:�^å,�������
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G o ¿ ! ���v��)Q6��/U AGJb8 �V�>@���BA��S ⊆ N
8B���º?7E )�4H! %MJ+����%É��)N! %'I®?�E )�4_!G%;J+���5%É��)ªAB���9! U@8[! �P�P- L ���P� S ÇVI.�V�.#:� E ��4�4_!�Ç OTo N oañ ÈcÈ
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|S| = dimFpH1(N)G.
ê�Ô ÜgËNÍÎÏ�È�Ð.ÑYÈ�Ð6ëlËNÍÎͳÈ�Ð ëyÇÎÌÑYÈ�ÐS£�È�Ï�Ì3ÇM¦ ÑYÈ�Ì�ÜaÌ@È�dzÈ�ÐS��Ì@Ëqöpö�õ/Ì3ÛYÓYÓ!È�È�ÇÎÐ*ÜaäYåYÌ@È�Ð6ÛYÐ.ÑÚÇÎåYÐ6Ö9È�ТÛY×3ò{È�Ð�~YÛYÔ
ÑYÈ�Ðf£�È�Ï�Ì3ÇM¦úÑYÈ�ÌðAÈ�Í Õq×3dzËNÐ òvÛÊÑYÈ��.ÐYdzÈ�Ì@È�Ð ÛYÐ.Ñ6òvÛàï@å�ÕqÌ�Õ${�×@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�ÌlÕqÍÎͳÈÈAÛ.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
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Gé
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$� ÈvÙ�òvÛ�É�ËNÌ3Ï�È�Ï�È�Ö!Èdö
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1→ R→ F → G→ 1
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pö�õ/Ì3ÛYÓYÓ9È�ÕqÛ*Ü
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SdzÙ3×�È�ÇÎÐvz�Ì3ò{È�ÛYÏ�È�Ð.ÑYÈ�Ð.ÙQ��Ù3×@È�Ô É�ËNÐ
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�Q���=���a`��KJ�â:�^å,�������G�/���0� ���98:%;�R<�#@��)Q6��/U AGJ+��13)dE Z
pZ.W2)�U^*�*,�_4��=J24����P�54H! %É��4¬e3)d6F��UGA����98��/�,I.w$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
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)∨
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d(G) = dimFp(G/Φ(G)) = rkZp(Gab) + dimFp(pG
ab).
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Ñ*äYÌ,ÜgÈ�ÐôëyÇÎÌGÕqͳÙWÕqÖ9È�ͳÙ@ï�åôÉ�ËNÌ�ÕqÛ.Ù@Ù@È�×3ò{È�ÐQé?Æ�ÕqÐYÐôdzÙ3×
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s = d(G)− rõ/Ì3ÛYÓYÓ9È�Ð6ÑYÈ�Ì�ø�ËNÌ3Ô
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ω : G→ N ∪ {∞},
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ω(xy−1) ≥ min(ω(x), ω(y))
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Gn := {x ∈ G| ω(x) ≥ n},
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ω(x) := max{n| x ∈ Gn}.
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ÜoäYÌIÕqÍÎͳÈx, y ∈ G
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Fp[[G]] = lim←− Fp[G/N ],
ë�ËNÖ!È�Ç�ÑYÈ�Ì=ÓYÌ@Ëqý3È/{±×3ÇÎÉ�È��-ÇÎÔ�ÈvÙ=äYÖ9È�Ì/Ñ*Ç³È Ë$¦-È�Ð.È�Ð@ÁAËNÌ3Ô�ÕqÍÎ×@È�ÇÎͳÈ�Ì=É�ËNÐGÏ�È�ÖYÇÎͳÑYÈ�×/ëyÇÎÌ@Ñ-é9êuÐàÇÎåYÌ=å�ÕqÖ9È�ÐàëyÇÎÌ
Ñ.ÕNÙIêuÑYÈ{ÕqÍI(G) = (g − 1| g ∈ G).ø�È�Ì3Ð.È�ÌAÙ@È�×3ò{È�Ð6ëyÇÎÌIn(G) := (I(G))n.ÁAÛYÐ å�ÕqÖ9È�Ð6ëyÇÎÌIÑYÈ�Ð6ÜgËNÍÎÏ�È�Ð.ÑYÈ�ÐX�*Õq×3òØûaÉ�Ï�Í�é-ßk�/ö�õ/â+~��*Õq×3ò�»�é©��ÿdé
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Fp[[G]] oá�ÇÎÌIÑYÈ��.ÐYdzÈ�Ì@È�ÐÒÑ*Ç³È �"!$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)
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UB�08g!G%�%É�h ∈ G(m)
A �5%MJ+Êgh − 1 ≡ g − 1 + h− 1 (
4HE^8 UT%MEIn+m(G)).
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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${±×@È�Ì3dzÙ3×3ÇK{
p~YÑ.Õqå.È�ÌIdzÙ3×
gpk− 1 = (g − 1)p
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G oA �CB�� Þ ä[� Æ/dzÈ�È�Ì@Ù3×@Èt£�È�å�ÕqÛYÓY×3ÛYÐYÏÜgËNÍÎÏ�×QÛYÐYÔ"ÇÎ×3×@È�ÍÎÖ�ÕqÌ�ÕqÛ.ÙQÑYÈ�Ô>É�ËNÌ�ÕqÛ.Ù@Ï�È�å.È�Ð.ÑYÈ�Ð��QÈ�Ô"Ô�Õ*é ��ÛYÌQòvëlÈ�ÇÎ×@È�ÐÖ9È�×3Ì�ÕNï@å±×@È�ÐàëyÇÎÌÑ*dzÈ�ÞIÖYÖYÇÎͳÑ*ÛYÐYÏ
φ : G(n)/G(n+1) → In(G)/In+1(G), g 7→ g − 1.Æ/dzÈvÙ@ÈdzÙ3×�È�ÇÎÐ�æ/ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�Û.ÙlÐ�ÕNï@åg��È�ÇÎÍ0ûaÖ�ÿ0ÑYÈvÙs�QÈ�Ô"Ô�ÕNÙ&é¢ø.È�Ì3Ð.È�ÌldzÙ3×φÐ�ÕNï@åØÆ=È��.ÐYÇÎ×3dzËNÐØÉ�ËNÐ
G(n+1)ÇÎÐNý3È/{�×3ÇÎÉ!éYÆ�Õqå.È�ÌdzÙ3×G(n)/G(n+1)
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ψU : Fp[[G]]→ Fp[G/U ]Ñ*dzÈWÐ�Õq×3äYÌ3ÍÎdzï@å.È���Ì@Ëqý3È/{�×3dzËNÐQé�Æ�Õ�ÑYÈ�̳��È�Ì3ÐàÉ�ËNÐψUË$¦-È�ÐÊdzÙ3×^~�Ï�ÇÎÖY×AÈvÙ/È�ÇÎÐ.È�ÐÒêuÐ.ÑYÈ��
n~.Ù@Ë�Ñ.Õ$�
In(G)ÇÎÔ��È�Ì3ÐàÈ�Т×3å�ÕqÍÎ×@È�ÐÊdzÙ@×{é.ø?äYÌIÕqÍÎͳÈ
g ∈ G(n)å�ÕqÖ9È�Ð ëyÇÎÌAÕqͳÙ@Ë"Ñ*dzÈWõ=ͳÈ�dzï@åYå.È�ÇÎ×
ψU (g − 1) = 0 = ψU (g) − 1.Æ�Õqå.È�ÌdzÙ@×G(n)ÇÎÐUÈ�Т×3å�ÕqÍÎ×@È�Ð�~�ÛYÐ.Ñ Ñ*dzÈ
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ÑYÈ��.ÐYdzÈ�Ì3×/Ñ*ÛYÌ@ï@åG(1,q) := G
ÛYÐ.ÑG(n+1,q) := (G(n,q))q[G(n,q), G].�Q���=���=�TÅ��KJ�â:�^å,�������
G�/���0�����08:%'�R<�#���)Q6���UGAGJ+�713)dE Z
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K(1) := KUT�98
K(n+1) := (K(n))q[K(n), G].¿ ! ���\A �5%MJP��NB)�!G%�%É�n ∈ N
K(n) ⊆ G(n),L E�OQ��� G(n)
8 �V�nZÆJ+�SW2).U�*[*,�H���¥8B��)��"!$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�
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q = ps���5�0�
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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$�(G(n))
q ⊆ G(nq)ÛYÐ.Ñ
[G(n), G] ⊆ G(n+1)Ï�ÇÎÍÎ×{é�á�ÇÎÌIò{È�ÇÎÏ�È�ÐàÑ*dzÈ7£�È�å�ÕqÛYÓY×3ÛYÐYÏ�ТÛYÐÊÇÎÐ.Ñ*Û,{�×3ÇÎÉ9é�ø?äYÌn = 1
dzÙ3×IÙ3dzÈ{�Í ÕqÌ{é.Æ=È�ÌIêuÐ.Ñ*Û,{±×3dzËNÐ.Ù�Ù@ï@åYÌ3ÇÎ×3×AÜoËNÍÎÏ�×ÕqÛ.Ù
K(n+1) = (K(n))q[K(n), G] ⊆ (G(n))q[G(n), G] ⊆ G(nq)G(n+1) ⊆ G(n+1).Æ/dzÈvÙ�Ï�ÇÎÍÎ×lÜaäYÌq = ps
é�Æ/È�Ì2£�È�ëlÈ�dzÙlÜoäYÌq = 0
È�Ì3Ï�ÇÎÖY×�Ù3dzï@å6ÕqÛ.Ù�Ï�ͳÈ�dzï@å.È�ÌyðIÈvï�åYТÛYÐYÏ"Ñ*ÛYÌ@ï@åÚáúÈ�Ï�Í ÕNÙ@Ù�È�ÐÚÕqÍÎͳÈ�Ì��È�Ì3Ô�ÈG~.ÇÎÐÊÑYÈ�Ð.È�ÐàÈ�ÇÎÐqÕqÛ*Üo×3Ì3ÇÎ×3×{é
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Ù�È�×3ò{È�Ð ëyÇÎÌòvÛYÐ�ìNï@å.Ù@×H := G(n,q) éYÆ�ÕqÐYРdzÙ@× H ÕqͳÙË$¦-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
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G(n,q)/G(n+1,q) = H/Hq[H,G]È�Ð.Ñ*ÍÎdzï�åQé
2
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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
p
}d� �!�h������,!$'&��0©�z&�6 ��.�&¹�0�0�����06 ûÎ,/�C12&Ç�Z�C,�(p(8ä-�C�0�0��&��
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p~Në�È�ͳï@å.ÈyëyÇÎÌ�ÑYÈ�Ìtz�ÇÎÐ*Ü�ÕNï@åYå.È�ÇÎ×lå�ÕqÍÎÖ!È�Ì�ÕqͳÙ0ÛYÐYÏ�È�Ì�ÕNÑYÈAÕqÐYÐ.È�åYÔ�È�Ð�ëlËNÍÎͳÈ�ÐQé��È�Ç
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G = < a1, . . . , ad >�/���0��/�98:%;�=<�#g�/)d6F��U A[J+�x*:EGJ+�/�T6/)Q���R<�#:�}W2).U�*[*,�.-¯UT�98HOQ�Q6F���=</#��0�
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A �5%;JGi+k = Gp
k
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8G�R���)d!GJRJ=������Z����J+��)RA )�U^*�*,��CFE �Gi o
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i−1= {xp
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k ���08 UG6n�R�/)�J¯�������?�9UB)5¡n��&FJ=�=EG� CFEG�Gi/Gi+1
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rk(G) = sup{d(H) | H ≤ GË$¦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,�������
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Gi/Gi+1 � Gi+1/Gi+2, x 7→ xp
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d(A) = d(B)éYÆ/dzÈvÙIÈ�Ì3Í ÕqÛYÖY×AÑ*dzÈÝÆ=È��.ÐYÇÎ×3dzËNÐdim(G) := d(H)
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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
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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%'��&$!GJ=�RE �f4��=J
2���5�\mDU:JVE Z4HE )+*�#���I.4�UBI�yI/J o ¿ �V� p ��CFE %;UTJ=�RE � σ E.*,�/).�V��)Q��! U.� A - B UT�98
C o ¿ ! �P�gAB��%;J+�/�0ÊÇÆ! ÈA+ = {1
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A = A+ ⊕A− oÇu< È¥p I�Jf : A→ B
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Hom(A,B)C ��)cI^�.#:��� L �5)D4��=Jt8���)YA�� L ( #��P%;�=<�#,��� σ Z=m�&$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
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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
p
A �CB�� Þ ä[� Á/ÕNï�åçÑYÈ�ÔÕ�*Õq×3òÊÉ�ËNÐçùÒÕNÙ@ï�å,{�È Ç³Ù3×Fp[< σ >]
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Fp[< σ >]ö�ùàË�Ñ*ÛYÍlå�ÕqÍÎÖ9È�ÇÎÐ*Ü�ÕNï@å�~0ëlÈvÙ@å�ÕqÍÎÖ
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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
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&$!GJ=�RE �f4��=J
2����� p I/EG4_EG)V*�#B�yI�4�UBIª�yI/J5-tUB�98gE.*,�/).�V��)/J28G�R� p �PCFE %;U:J=�=E �
σ! U.�
GÇVUT�98 J=).�5C^�R! %"! U��
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σZÆ���PC$! ).�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$¦-È�Ð.Ù3dzï@å±×3ÍÎdzï@å
R = R− é9Æ�ÕZpÜaÌ@È�Ç-dzÙ3×^~*Ï�ÇÎÍÎ×
H2(Zp,Z/pZ) = 0~YÛYÐ.Ñ ëyÇÎÌÈ�Ì3å�ÕqÍÎ×@È�ÐàÙ@ËNÔ"ÇÎ×IÑYÈ�Ð
σö�ÇÎбÉNÕqÌ3Ç Õqб×@È�ÐÒê�Ù@ËNÔ�ËNÌ3ÓYåYdzÙ3ÔWÛ.Ù
H1(R,Z/pZ)B → H2(B,Z/pZ).Æ�Õ Ð�ÕNï�åÃûaÖ�ÿyÕqÖ9È�ÌH1(R,Z/pZ) = H1(R,Z/pZ)−
Ï�ÇÎÍÎ×^~*ÜgËNÍÎÏ�×Ñ*dzÈÝÞAÛ.Ù@Ù�ÕqÏ�È�é2
p
}d�h}��h�����:�z,�(p(8ä-�����0��&���1264$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$�
dimFpH1(G)ÕqͳÙ0Ô"ÇÎÐYÇÎÔ�ÕqͳÈYz�Ì3ò{È�ÛYÏ�È�Ì3ò&ÕqåYÍ
É�ËNÐGÇÎб×@È�Ì3ÓYÌ@È�×3dzÈ�Ì3×=ëlÈ�Ì@ÑYÈ�Ð�{NÕqÐYÐQé9õ=È�Ð�ÕqÛ.È�Ì/Ï�ÇÎÖY×AÑ*dzÈ�Æ/ÇÎÔ�È�Ð.Ù3dzËNÐèÑYÈvÙ³��ÍÎÛ.Ù,ö�ÖYòvëÝé�ÑYÈvÙ/ùÊÇÎбÛ.Ù3×@È�ÇÎͳÙIÉ�ËNÐ
H1(G)Ñ*dzÈ�Ô"ÇÎÐYÇÎÔ�ÕqͳÈ7z�Ì3ò{È�ÛYÏ�È�Ì3ò&ÕqåYÍ?ÑYÈvÙY��ÍÎÛ.Ù,ö�ÖYòvëÝé.ÑYÈvÙùàÇÎТÛ.Ù@×@È�ÇÎͳÙIÑYÈ�ÌIÞAÖ!È�ÍÎdzÙ@dzÈ�Ì3ÛYÐYÏ"É�ËNÐ
GÕqÐQé
�Q���y���=�nm�� ö �B�Â�hâ������/�G���5�0�����98:%;�R<�#f��)d6F��UGAGJ+�13)dE Z
pZ�W2).U�*[*,� o ¿ ! �P�\A ��%MJ+Ê
d(G)± = dimFp H1(G)± = d((G/G2)
±) = d((Gab)±).
A �CB�� Þ ä[� Æ/dzÈÒÈ�Ì@Ù3×@Èèõ=ͳÈ�dzï@åYå.È�ÇÎ×6ëlËNÍÎͳÈ�Ð ëyÇÎÌ ÕqͳÙÚÆ=È��.ÐYÇÎ×3dzËNÐçÉ�ËNÐd(G)±
É�È�Ì@Ù@×@È�å.È�ÐQé3z�ÙØÏ�ÇÎÍÎ×ÚÐ�ÕNï�å��È�Ô"Ô�ÕÊûR¨*驨*éK�&ÿHom((G/G2)
±,Z/pZ) = Hom(G/G2,Z/pZ)±,ë�ËNÌ�ÕqÛ.Ù"Ñ*dzÈ�òvëlÈ�ÇÎ×@È õ/ͳÈ�dzï@åYå.È�ÇÎ× ÜgËNÍÎÏ�×{é�áúÈ�Ï�È�ÐHom(Gab,Z/pZ) = Hom(G/G2,Z/pZ)
È�Ì3å�ÕqÍÎ×@È�Ð÷ëyÇÎÌÑ*dzÈ�Ñ*Ì3ÇÎ×3×@È�ÕqÛ.ÙÑYÈ�Ô Æ/Ç ÕqÏ�Ì�ÕqÔ"Ô
Gab = (Gab)+ ⊕ (Gab)−
y
y
y
G/G2 = (G/G2)+ ⊕ (G/G2)
−,ÇÎÐàÑYÈ�Ô ÕqÍÎͳÈ�0ÜgÈ�ÇÎͳÈ��±ÛYÌoý,È/{�×3dzËNÐ.È�ÐàÙ@ÇÎÐ.Ñ-é
2
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pZ.W2)�U^*�*,�.-x! U��³8��/)7�/���0� p �PCFE %;UTJ=�RE �
σE.*,�/).�V��)/J5-InEÂ8�!da
d(G)+ = 0A ��%MJ o ¿ !G�P�¤E.*P��).�V��)/J σ !GU.� G(n)/G(n+1)
!G%ÃI p 8�����J=�=JV�GJ5-b�n! %�%'InA���)d![8����yI/J5-UB�98!G%ÃI p �PC$��).I.�RE �,-¯�n! %�%'I
nUT�BA��/)l![8B�S��I�J o ¿ !�OQ����OQ�c6����R<�#��0� G(n) L �V�d8���)�UT4 8[!$I}e3)Q6���UGA �P��Iv8B��)
nZ�n![<�#:�����E 4�4�U:JV!GJVE )Q��� o
A �CB�� Þ ä[� Æ/È�Ðf£�È�ë�È�dzÙyÜoäYåYÌ@È�Ð6ëyÇÎÌÓ9È�Ìê�Ð.Ñ*Û,{�×3dzËNÐ Ð�ÕNï@ånéTÁ=ÕNï@åÊí�ËNÌ�ÕqÛ.Ù@Ù@È�×3òvÛYÐYÏ�dzÙ@×
G/Gp[G,G] = (G/Gp[G,G])−.
ÁAÛYÐ Ö9È�×3Ì�ÕNï@å±×@È�ÐÊëyÇÎÌIÑ*dzÈ=Ð�Õq×3äYÌ3ÍÎdzï�å.È���Ì@Ëqý3È/{�×3dzËNÐ(G/[G,G])+ � (G/Gp[G,G])+ = (G/Φ(G))+,
ÑYÈ�Ì@È�Ðf£lÇÎͳÑÚ×3Ì3ÇÎÉ�Ç ÕqÍ-dzÙ3×{éYÆ/dzÈ=ÞIÐYò&ÕqåYÍ�ÑYÈ�ÌYz�Ì3ò{È�ÛYÏ�È�Ð.ÑYÈ�ÐÊÉ�ËNÐ(G/[G,G])+
dzÙ3×yÙ@ËNÔ"ÇÎ×0~YÛYÐ.Ñ
G/[G,G]dzÙ3×
Ï�ͳÈ�dzï�å(G/[G,G])−
~Yë�ÕNÙÑYÈ�ÐÊêuÐ.Ñ*Û,{±×3dzËNÐ.Ù�ÕqÐ*ÜgÕqÐYÏ�Ñ.ÕqÌ@Ù@×@È�ÍÎÍÎ×{é��È�Ç�Ñ*dzÈÝÞAÛ.Ù@Ù�ÕqÏ�È=ТÛYÐ ÜaäYÌnÖ9È�Ì@È�ÇÎ×@ÙyÖ!È�ëydzÈvÙ�È�ÐQéYá ìqåYͳÈ
x ∈ G(n) ÛYÐ.Ñ y ∈ G é.Æ�ÕqÐYÐ6dzÙ3×σ[x, y] ≡ [σx, y−1] (
Ô�˱Ñ*ÛYͳËG(3)).
ùÒË�Ñ*ÛYͳËG(n+1) dzÙ3× σx È�Т×uëlÈvÑYÈ�Ì x Ë�ÑYÈ�Ì x−1 ý,È6Ð�ÕNï�åãÑYÈ�ÔS~�ËNÖ n Ï�È�Ì�ÕNÑYÈ Ë�ÑYÈ�Ì�ÛYÐYÏ�È�Ì�ÕNÑYÈÊdzÙ3×{é�ÞIÛ.Ù��È�Ô"Ô�ÕÊûR¨*驨*驱�ÿ�ûoï&ÿ�È�Ì3å�ÕqÍÎ×@È�ÐÊëyÇÎÌ
[x, y−1] ≡ [x, y]−1 (Ô�˱Ñ*ÛYͳË
G(n+2))ÛYÐ.Ñ
[x−1, y−1] ≡ [x, y] (Ô�Ë�Ñ*ÛYͳË
G(n+2)).
Æ�ÕqÌ�ÕqÛ.ÙÜoËNÍÎÏ�×ÑYÈ�ÌIêuÐ.Ñ*Û,{�×3dzËNÐ.Ù@Ù@ï@åYÌ3ÇÎ×3×{é2
�Q���y���=�:��� A �T�h�B8F��������������G���5�0����98:%;�R<�# ��)d6F��UGAGJ+�Y13)dE Z
pZ.W2)�U^*[*P�.-b!GU.�ª8���)ª�/���0� p �PC$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${�×@È�Ð���Èn ±Û.È�ÐYòWòvÛÒÈ�ÇÎÐ.È�ÌI×3Ì3ÇÎÉ�Ç ÕqͳÈ�Ì3ëlÈ�dzÙ@È
σö�ÇÎÐYÉNÕqÌ3Ç ö
ÕqÐY×@È�Ð�{±ÛYÌ3ò{È�ÐÊÈ��.Õ${�×@È�ÐX��Èn ¢Û.È�ÐYò�Ï�È�ëlËNÐYÐ.È�ÐÊëyÛYÌ@ÑYÈG~.dzÙ3×Ù3dzÈ�Ù@È�ÍÎÖ.Ù3×σö�ÇÎбÉNÕqÌ3Ç Õqб×{é
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���
G,H���08:%'�R<�#:��W2)�U^*[*P���\UB�98
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�
σ! U��g�y#������,-YInEX�yI/Jª8G�R�}����)�%É�RA UT�BA
σZÆ���PC$! ).�R! ��J o
A �CB�� Þ ä[� ûgÕ�ÿ�Æ/dzÈvÙ�ÜgËNÍÎÏ�×ÝòvÛYÔ £�È�dzÙ@ÓYdzÈ�ÍlÕqÛ.Ù�ßkÁ�±áãâ+~"z¯�-éQê@êdéK��éû�é�ûaÖ�ÿAÜgËNÍÎÏ�×ÝÇÎÐ.Ñ*Û,{±×3ÇÎÉôÕqÛ.Ù�ûgÕ�ÿ=ÛYб×@È�Ì£�È�бÛY×3òvÛYÐYÏ�É�ËNÐH2((G1 ×G2)×G3, B) = H2(G1 ×G2, B)⊕H2(G3, B)⊕ (H1(G1 ×G2, B)⊗H1(G3, B))
=
3⊕
i=1
H2(Gi, B)⊕⊕
i<j
(H1(Gi, B)⊗H1(Gj , B))
2ÆAdzÈD�=äYÐYÐ.È�×3å*ö�ø.ËNÌ3Ô�È�Í-ëlÈ�Ì@ÑYÈ�Ð�ëyÇÎÌ�Ö!È�бÛY×3ò{È�Ð�~¢ÛYÔ Ñ*dzÈIÆ/ÇÎÔ�È�Ð.Ù3dzËNÐ�É�ËNÐH2(G/G2)
± òvÛØÖ!ÈvÙ@×3ÇÎÔ"Ô�È�ÐQé�Q���y���=�Bó��KJ�â:�^å,�������
G�/���0�����08:%'�R<�#���)Q6���UGAGJ+�713)dE Z
pZ.W2)�U^*�*,� oÇÆ! È �����
d = d(G) o ¿ ! �P�vA ��%MJ+ÊdimFp H
2(G/G2) =d(d+ 1)
2.
ÇlO È Y�*P��).�V��)/JY������� p �PC$E %'U:J=�RE �σ! U��
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
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
K(1) = K(2) ÔWÛ,�ÚÑ.Õqå.È�Ì K(1) ~�ÕqͳÙ@Ë [G,G]~
×3Ì3ÇÎÉ�Ç ÕqÍ�Ù@È�ÇÎÐQé2
% �Bß5â:� Þ �9�"�B� Þ �`ìt���^�B�"å[$� Þ5÷ �"�B� ´ $�,ëpënêg �"ì"ìt�T�
ÞAÛ.ÙÑYÈ�ÌWûσö�ÇÎбÉNÕqÌ3Ç Õqб×@È�Ð�ÿÈ��.Õ${±×@È�Ðf��Èn ±Û.È�ÐYò
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$��Ö9È�Ç9È�ÇÎÐ.È�Ì�È�Ð.Ñ*ÍÎdzï�åÊÈ�Ì3ò{È�ÛYÏ�×@È�Ð6Ó9ËN×@È�ÐYòvÌ@È�Ç ö
ï�å.È�Ðf��Ì@Ëqöpö�õ=Ì3ÛYÓYÓ!È
GÙ3×@È�×@Ù
r(G) ≥
(
d(G)
2
)
Ï�ÇÎÍÎ×^~�Ñ.Õd(G2) ≤ d(G)
dzÙ3×{é9�±×�ìqÌd{�È�ÌIÐ.˱ï�å Ï�ÇÎÍÎ×IÑ.ÕNÙ
�Q���y���=�n`������"����$�B�`�������G���������/�98:%;�=<�#���)Q6���UGAGJ+��*,EGJ+���T6n)d���R<�#:�12)lE Z
pZ.W2)�U�*[*,� o ¿ !G�P�gA �5%MJ+Ê
(
d(G)
2
)
≤ r(G) ≤
(
d(G)
2
)
+ d(G) − dim(G).
p �PInOQ�.InE �98��/)d�ªA �5%MJ38[! #:��)r(G) =
(
d(G)
2
)
,
�n! %5%ÃIGUB��� �nE )�4 �yI/J o
A �CB�� Þ ä[� áçÇÎÌ�ÜaäYåYÌ@È�Ð ÑYÈ�Ð6Ï�Ì3ÛYÓYÓ9È�Т×3å.ÈvËNÌ@È�×3dzÙ�ï@å.È�Ð�£�È�ë�È�dzÙ^~*ëydzÈ�È�ÌyÙ3dzï@å6ÇÎÐúß}Æ=Æ=ù@�¢â+~T��å.ÈvËNÌ@È�Ô¬®.é©�þ:~�.Ð.ÑYÈ�×{éá�ÇÎÌIÙ�È�×3ò{È�Ð
di := (Gi : Gi+1)é.Æ�Õ
GÓ9ËN×@È�ÐYòvÌ@È�dzï@åàdzÙ3×^~YÈ�Ì3å�ÕqÍÎ×@È�Ð ëyÇÎÌIÈ�ÇÎÐ.ÈWÕqÖ.Ù3×@È�ÇÎÏ�È�Ð.ÑYÈ���È�×3×@È
l := d(G) = d1 ≥ d2 ≥ · · · ≥ dk = dim(G),
Ù�Ë"Ñ.Õ$�GkÛYÐYÇ ÜoËNÌ3ÔîdzÙ@×{éÁ/ÕNï�å ß}Æ=Æ=ù@�¢â+~"��å.ÈvËNÌ@È�Ô *é©°:~�Ï�ÇÎÖY× ÈvÙ�È�ÇÎÐjz�Ì3ò{È�ÛYÏ�È�Ð.ÑYÈ�Ð.Ùd�*Ù3×@È�Ô
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G~0Ù@ËàÑ.Õ$�ÒÑ*dzÈ
ùÒÈ�ÐYÏ�È{xp
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ab)± ≤ d(G)±,
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ÕqÛ*Üa×3Ì@È�×@È�ÐQéNÆ�Õ3 = r(G) = d(G) =
dimFppGab = dimFpH
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(pGab)∨ ∼= H2(G)
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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
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f1 · · ·Qfss Qs
fs Ô"ÇÎ×ei, fj ∈ Z,
ë�ËNÖ!È�ÇPi = Pi
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pÈ�ÇÎÐôÇÎÐ
k|k+ ÛYТÉ�È�Ì3òvë�È�ÇÎÏ�×@ÈvÙH��Ì3ÇÎÔ"dzÑYÈ{ÕqÍ�É�ËNÐ k+ ~�Ù@Ë Ç³Ù3×�È�Т×uëlÈvÑYÈ�Ì ϕ(p) = PÔ"ÇÎ×
P = P˱ÑYÈ�Ì
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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$�
(PSk )+ = PSk+
Ï�ÇÎÍÎ×{é�Æ/ÛYÌ@ï�å�ÐAÖ9È�Ì3Ï¢ÕqÐYÏ�òvÛèÑYÈ�Ð ÈAÛ.ËN×3Ç öÈ�б×@È�ÐÊÔ�Ë�Ñ*ÛYͳË
PSk+
È�Ì3å�ÕqÍÎ×@È�ÐÊëyÇÎÌAÕqÛ.ÙϕÑYÈ�Ð æAËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ@Ô�Û.Ù
φ : ClS(k+) = J Sk+/PSk+ → (J Sk )+/PSk+ = (ClS(k))+,
ë�È�ͳï@å.È�Ì=Ñ.Õqå.È�ÌAÕqÛ.ï@åàÈ�ÇÎÐ.È�Ð��IË {�È�Ì3ÐàÉ�ËNÔ¬z¯�*Ó!ËNÐ.È�б×@È�Ð2Ö!ÈvÙ@ÇÎ×3òv×{é��È�Ç
aÈ�ÇÎÐSz�ͳÈ�Ô�È�Т×=ÑYÈvÙY��È�Ì3Ð.ÈvÙIÉ�ËNÐ
φé�Æ=ÕqÐYÐÊdzÙ3×
aOk,S = αOk,SÔ"ÇÎ×È�ÇÎÐ.È�Ô
α ∈ PSkéYÆ�Õqå.È�ÌÏ�ÇÎÍÎ×
(1) = Ok,S =aOk,S
aOk,S=α
αOk,S .
αα
dzÙ@×ÝÈ�ÇÎÐhz�ͳÈ�Ô�È�Т×WÉ�ËNÐ(ES(k))−
ÛYÐ.ÑúÙ�ËNÔ"ÇÎ×�Ð�ÕNï@år�*Õq×3òÊûR¨*é©*éK�G�&ÿ=È�ÇÎÐ.È}z�ÇÎÐYå.È�ÇÎ×@Ù3ëyÛYÌ3ò{È�Í�é?õ=ÇÎÖY×ÝÈvÙ�È�ÇÎÐ.Èz�ÇÎÐYå.È�ÇÎ×@Ù3ëyÛYÌ3ò{È�Íζ ∈ µ(k)
ÑYÈ�Ì�ÕqÌ3×^~¢Ñ.Õ$� 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
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dzÙ@×{éûaÖ�ÿ�Æ=È�Ì�£�È�ëlÈ�dzÙÉ�È�Ì3Í ìqÛ*Üo×É�ËNÍÎÍK{�ËNÔ"Ô�È�ÐèÕqÐ�ÕqͳËNÏ.é
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k+ o ¿ ! �P�fI.�5�98g8 �R�}W2)�U^*[*P���Clmk+(p) ∼= (Clmk (p))+�yInE 4HE )V*9# o
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SA���%É�RA�������13)���4�I/J+��%�%É� o ËsNT) E := ES(k)
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k+ o ËsNT) E := O×k,m
AB��%;J+�/�¥8[! �P�E− ⊆ µk
UT�98(E : E+E−) ≤ 2.
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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
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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,�������
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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�
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m =∏
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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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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$��Ù3dzï�åÊÈ�ÇÎÐÊæ/ËNÔ�ËNÔ�ËNÌ3ÓYåYdzÙ3Ô�Û.Ùψ : B− → (pA)−
È�Ì3Ï�ÇÎÖY×{éê�Ô ø�ÕqÍÎÍ�ûgÕ�ÿ�dzÙ3×�È�ÇÎÐ\z�ͳÈ�Ô�È�Т×
b ∈ BÇÎÔ ��È�Ì3ÐØÉ�ËNÐ
φ~�Ü�ÕqÍÎͳÙ
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a ∈ ES(k)Ï�ÇÎÍÎ×{é±Æ�Õqå.È�Ì
dzÙ@×ker(ψ) ≤ (ES(k)/(ES(k))p)−.
êuÔîø�ÕqÍÎÍlûaÖ�ÿyÈ�Ì3Ï�È�Ö!È�ÐàÕqÐ�ÕqͳËNÏ�È�ÐIÖ9È�Ì3ͳÈ�Ï�ÛYÐYÏ�È�Ðker(ψ) ≤ (O×
k,m/(O×k,m)p)−.
Æ�ÕqÖ9È�ÇQëyÛYÌ@ÑYÈ��*Õq×3òØûR¨*é©*éK�G�&ÿ�Ö!È�бÛY×3òv×{éÁIÛYÐ6ÜgËNÍÎÏ�×IÕqÛ.ÙD�*Õq×3òØûR¨*é©*éK�/®±ÿyÔ"ÇÎ×3×@È�ͳÙIêuÐ.ÑYÈ���Ö9È�×3Ì�ÕNï@å±×3ÛYÐYÏ�È�Ð�~�Ñ.Õ$�(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${�×@È���Èn ¢Û.È�ÐYò1→ ker(ψ)→ B− ψ
−→ (pA)−,
ÛYÐ.ÑÊÑ.Õqå.È�ÌIÈ�Ì3å�ÕqÍÎ×@È�ÐÊëyÇÎÌ
d(B−) ≤ d(ker(ψ)) + d(pA)− ≤ d(Z/pZ) + d(A−) = 1 + d(A)−.
áúÈ�Ï�È�ÐàÑYÈ�ÌIËNÖ!È�ÐÊÏ�È�ò{È�ÇÎÏ�×@È�ÐÒõ/ͳÈ�dzï@åYå.È�ÇÎ×d(B−) = d(A)+
dzÙ@×ÑYÈ�ÌD�*Õq×3òÝÖ!È�ëydzÈvÙ�È�ÐQé2
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á�ÇÎÌÖ9È�×3Ì�ÕNï@å±×@È�ÐàÑ*dzÈ�Ö9È�dzÑYÈ�Ð6ÜgËNÍÎÏ�È�Ð.ÑYÈ�ÐX�±ÇÎ×3Û�Õq×3dzËNÐ.È�Ð��ûoê3ÿ
• SÙ@È�ÇQÈ�ÇÎÐ.È�ûaÔH�NÏ�ÍÎdzï�å.È�Ì3ëlÈ�dzÙ@ÈWͳÈvÈ�Ì@È&ÿ�ùÒÈ�ÐYÏ�È�É�ËNÐ���Ì3ÇÎÔ�Ù3×@È�ÍÎͳÈ�ÐÊÉ�ËNÐ
k+ é•ê�Ðk|k+ ò{È�Ì,Ü�ÕqÍÎͳÈ{�È�ÇÎÐ.È�ÇÎÐ S Ï�È�ͳÈ�Ï�È�Ð.È���Ì3ÇÎÔ�Ù3×@È�ÍÎͳÈ�é
• LS(p)|kÙ@È�Ç!Ñ*dzÈ/Ô�ÕF�*ÇÎÔ�ÕqͳÈ/ÛYТÉ�È�Ì3òvëlÈ�ÇÎÏ�×@ÈÝÏ¢ÕqͳËNdzÙ@Ù@ï@å.È
pö+z�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ�É�ËNÐ
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SÉ�ËNÍÎÍQò{È�Ì3ͳÈ�Ï�×AÙ3ÇÎÐ.Ñ-é
• G := G(LS(p)|k)Ù@È�ÇQÑ*dzÈWõÝÕqͳËNdzÙ,ö�õ/Ì3ÛYÓYÓ9ÈÝÑ*dzÈvÙ@È�Ì�z�Ì3ë�È�ÇÎ×@È�Ì3ÛYÐYÏ.é
• E := ES(LS(p)) = lim−→LS(p)⊃K⊃k, K|k ´Vµ/¶/·k¸k¹=º E(K)
~�ëlËNÖ9È�ÇE(K) := ES(K)
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Sp ∩ S = ∅é
• kS(p)|kÙ�È�Ç�Ñ*dzÈÝÔ�ÕF�*ÇÎÔ�ÕqͳÈWÕqÛ,��È�Ì3å�ÕqÍÎÖÒÉ�ËNÐ
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pö+z�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ�É�ËNÐ
ké
• G := G(kS(p)|k)Ù@È�ÇQÑ*dzÈ�õ�ÕqͳËNdzÙ,ö�õ/Ì3ÛYÓYÓ9È�Ñ*dzÈvÙ@È�Ì�z�Ì3ë�È�ÇÎ×@È�Ì3ÛYÐYÏ.é
• E(K) := O×K,m
~.ëlËNÖ9È�ÇkS(p) ⊃ K
È�ÇÎÐ.È�È�Ð.Ñ*ÍÎdzï@å.ÈWÏ¢ÕqͳËNdzÙ@Ù@ï@å.È����NÌ3Ó9È�Ì@È�Ì3ëlÈ�ÇÎ×@È�Ì3ÛYÐYÏ�È�Ð É�ËNÐkÙ@È�Ç�éYÆ�ÕqÖ!È�Ç-dzÙ@×
mÑYÈ�ÌAùàË�Ñ*ÛYÍ ∏
p∈S pé
• E := O×kS(p),m = lim−→
K
E(K)~?ë�ËNÖ!È�ÇyÑYÈ�Ì"Ñ*ÇÎÌ@È/{±×@Èg�-ÇÎÔ�ÈvÙ äYÖ9È�Ì"Ñ*dzÈ�ÇÎÐ
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á�ÇÎÌ�å�ÕqÍÎ×@È�ÐÚÜoÈvÙ@×^~¢Ñ.Õ$� 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
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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)−)∨.
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dimFp (pH2(G,Qp/Zp)
−) ≤
{
0Ü�ÕqÍÎͳÙ
µp(k) 6⊆ k
1Ü�ÕqÍÎͳÙ
µp(k) ⊆ k.
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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
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éê�Ôîø�ÕqÍÎÍ
d(G)+ = 0dzÙ3×
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d(P ) ≤ d(G) ≤ 3Ï�ÇÎÍÎ×{éYá ìqÌ@È
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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 )+
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