Grégory Berhuy's An Introduction to Galois Cohomology and its Applications PDF

By Grégory Berhuy

ISBN-10: 0521738660

ISBN-13: 9780521738668

This booklet is the 1st effortless advent to Galois cohomology and its purposes. the 1st half is self contained and gives the fundamental result of the idea, together with a close development of the Galois cohomology functor, in addition to an exposition of the final concept of Galois descent. the complete concept is prompted and illustrated utilizing the instance of the descent challenge of conjugacy sessions of matrices. the second one a part of the publication supplies an perception of ways Galois cohomology should be precious to resolve a few algebraic difficulties in different lively learn issues, corresponding to inverse Galois thought, rationality questions or crucial size of algebraic teams. the writer assumes just a minimum historical past in algebra (Galois concept, tensor items of vectors areas and algebras).

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The main idea is that the problem locally boils down to the previous case. Let us fix M0 ∈ Mn (k) and let us consider a specific matrix M ∈ Mn (k) such that QM Q−1 = M0 for some Q ∈ SLn (Ω). If L/k is any finite Galois subextension of Ω/k with Galois group GL containing all the entries of Q, then Q ∈ SLn (L) and the equality above may be read in Mn (L). Therefore, for this particular matrix M , the descent problem may be solved by examining the corresponding element in H 1 (GL , ZSLn (M0 )(L)). Now if we take another finite Galois subextension L /k such that M ∈ Mn (L ), we obtain an obstruction living in H 1 (GL , ZSLn (M0 )(L )).

Since L /k is a Galois extension, σ|L is a k-automorphism of L . Hence, there exists x ∈ L ⊂ ks such that σ|L (x) = x . Thus we have x = σ|L (x) = σ(x), and therefore, σ(ks ) = ks . 2 The Galois correspondence We would like now to understand better the structure of the Galois group of a Galois extension Ω/k, not necessarily of finite degree over k. In particular, we would like to have a Galois correspondence between subfields of Ω and subgroups of Gal(Ω/k) as in the case of finite Galois extensions.

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 ]).

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An Introduction to Galois Cohomology and its Applications by Grégory Berhuy


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