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The cohomological dimension of G
The global EulerPoincare characteristic
10 other sections not shown
Abelian group Abelian sheaf algebraic number field assumptions automorphism bijective canonically isomorphic Chapter class field theory cohomology groups cohomology sequence commutative diagram compact Corollary cyclic define denote Duality Theorem element etale cohomology exact sequence extension Kjk field extension finite extension finite p-group follows free pro-p-group Frobenius functor G-module G(Kjk Galois Cohomology Galois group group extension Hence Hom(ilf Hom(Jf Hom(M Hom(r homomorphism Hp(U idele induced injective inverse image kernel Kom(M ksjk Lemma Let F Let G Let Kjk logp Math maximal minimal generator system module morphism normal extension obtain open normal subgroup open subgroups p-extension P\ks pairing profinite group Proof Proposition prove ramified rational prime number roots of unity set of places sheaf F sheaves subgroup of G subset sufficiently large extension surjective Tate's Theorem Theorem 4.2 topology totally imaginary trivial universal norms unramified US(K W(ks Z/pZ ZjpZ