By Grégory Berhuy
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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Additional info for An Introduction to Galois Cohomology and its Applications
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 ]).
An Introduction to Galois Cohomology and its Applications by Grégory Berhuy