Class field theory : the Bonn Lectures by Jürgen Neukirch; Alexander Schmidt

By Jürgen Neukirch; Alexander Schmidt

Half I Cohomology of Finite teams -- half II neighborhood category box idea -- half III worldwide classification box idea

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15) Proposition. If f : A → B is a G-homomorphism of the G-modules A, B, and g is a subgroup of G, then the following diagram commutes àßÞÝÜÛ A) H q (G, res f¯ cor H q (G, B) res H q (g, A) f¯ cor H q (g, B). This is clear in case of dimension q = 0, and the general case follows easily by dimension shifting. In fact, the homomorphism f : A → B induces a homomorphism f : Aq → B q , and in the following diagram f¯ ðïîíìëêéèçæåäãâá Aq ) H 0 (G, H 0 (G, B q ) cor δq δq res H 0 (g, Aq ) f¯ H q (G, A) cor δ q res H q (g, A) f¯ cor res H 0 (g, B q ) H q (G, B) cor f¯ δq res H q (g, B) 42 Part I.

6) y0 = − a1 (τ ) ⊗ τ (σ − 1) = τ ∈G a1 (τ ) ⊗ τ − τ ∈G a1 (τ ) ⊗ τ σ. τ ∈G The 1-cocycle a1 (τ ) satisfies a1 (τ ) = a1 (τ σ) − τ a1 (σ). Substituting this into the last sum, we find τ a1 (σ) ⊗ τ σ. y0 = τ ∈G Therefore y0 − x0 = shows that x0 = y 0 . τ ∈G τ a1 (σ) ⊗ τ (σ − 1) = NG (a1 (σ) ⊗ (σ − 1)), which § 6. 8) is of particular interest for us. Note that if we take an element a2 in the group H 2 (G, A), it provides us with the homomorphism a2 ∪ : H −2 (G, ZZ) −→ H 0 (G, A), which maps each σ ∈ H −2 (G, ZZ) to the cup product a2 ∪ σ ∈ H 0 (G, A); we thus get a canonical mapping from the abelianization Gab to the norm residue group AG /NG A.

If 0 −→ A −→ B −→ C −→ 0 is an exact sequence of G-modules, we write the corresponding long exact cohomology sequence in the form of an exact hexagon: ✩✪✫✬✭✮ H −1 (G, A) H −1 (G, B) H −1 (G, C) H 0 (G, C) H 0 (G, B) H 0 (G, A). 1) fits into the commutative diagram § 6.

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