An Introduction to Galois Cohomology and its Applications by Grégory Berhuy

By Grégory Berhuy

This booklet is the 1st user-friendly advent to Galois cohomology and its functions. the 1st half is self contained and gives the fundamental result of the speculation, together with an in depth development of the Galois cohomology functor, in addition to an exposition of the final concept of Galois descent. the entire thought is encouraged and illustrated utilizing the instance of the descent challenge of conjugacy periods of matrices. the second one a part of the e-book supplies an perception of ways Galois cohomology might be priceless to unravel a few algebraic difficulties in different lively learn subject matters, similar to inverse Galois concept, rationality questions or crucial measurement of algebraic teams. the writer assumes just a minimum heritage in algebra (Galois idea, tensor items of vectors areas and algebras).

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32. The sets H n (Γ/U, AU ) together with the maps inf U,U form a directed system of pointed sets (resp. of groups if A is abelian). Moreover, we have fU = fU ◦ inf U,U . We now come to the main result of this section. 33. Let Γ be a profinite group, and let A be a Γ-group. Then we have an isomorphism of pointed sets (resp. an isomorphism of groups if A is abelian) n U lim −→ H (Γ/U, A ) H n (Γ, A). U ∈N If [ξU ] ∈ H n (Γ/U, AU ), this isomorphism maps [ξU ]/∼ onto fU ([ξU ]). Proof. We first prove that there exists a well-defined map n U n f: − lim → H (Γ/U, A ) −→ H (Γ, A), U ∈N which sends the equivalence class of [ξU ] ∈ H n (Γ/U, AU ) onto fU ([ξU ]).

We would like to observe now that the map ϕ∗ depends on ϕ only up to conjugation. For, let ρ ∈ G and set ψ = Int(ρ) ◦ ϕ. Then ψ ∗ ([α]) is represented by the cocycle γ defined by γ: Γn −→ A (σ1 , . . ,ρϕ(σn )ρ−1 . 21. Assume now that Γ also acts trivially on G, so that H 1 (Γ, G) is nothing but the set of conjugacy classes of continuous morphisms ϕ : Γ −→ G. The previous observations then imply that for each class [α] ∈ H n (G, A), we have a well-defined map H 1 (Γ, G) −→ H n (Γ, A) [ϕ] −→ ϕ∗ ([α]).

5. There is a natural bijection of pointed sets between the orbit set C Γ /B Γ and ker(H 1 (Γ, A) −→ H 1 (Γ, B)). More precisely, the bijection sends the orbit of c ∈ C Γ onto δ 0 (c). Proof. 4, we have ker(H 1 (Γ, A) −→ H 1 (Γ, B)) = im(δ 0 ). Hence we have to construct a bijection between C Γ /B Γ and im(δ 0 ). Let c, c ∈ C Γ lying in the same orbit, that is c = β·c for some β ∈ B Γ . Then c = g(βb), for some preimage b ∈ B of c, and βb is a preimage of c . Since we have (βb)−1 σ·(βb) = b−1 β −1 (σ·β)(σ·b) = b−1 σ·b, it turns out that δ 0 (c ) = δ 0 (c).

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