By Grégory Berhuy

This e-book is the 1st hassle-free advent to Galois cohomology and its purposes. the 1st half is self contained and offers the elemental result of the speculation, together with a close development of the Galois cohomology functor, in addition to an exposition of the overall concept of Galois descent. the total conception is influenced 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 can be important to resolve a few algebraic difficulties in numerous lively learn themes, akin to inverse Galois concept, rationality questions or crucial size of algebraic teams. the writer assumes just a minimum history in algebra (Galois concept, tensor items of vectors areas and algebras).

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Let ϕ : Γ −→ G be a morphism of proﬁnite groups, and let f = IdA . Then ϕ and f are compatible, and we get a map ϕ∗ : H n (G, A) −→ H n (Γ, A), called the inverse image with respect to ϕ. If [α] ∈ H n (G, A), a cocycle β representing ϕ∗ ([α]) is given by β: Γn −→ A (σ1 , . . ,ϕ(σn ) . 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 .

36 Cohomology of proﬁnite groups (1) In the existing literature, n-cocycles and n-coboundaries are often deﬁned to be elements of ker(dn ) and im(dn−1 ) respectively, and cocycles and coboundaries satisfying the extra condition (2) are called normalized. However, one can show that the two quotient groups obtained with these two diﬀerent deﬁnitions are canonically isomorphic. (2) If Γ is a ﬁnite abstract group, we recover the classical deﬁnition of the cohomology groups associated to a ﬁnite group (modulo the previous remark).

This does not imply that f is injective, unless A and B are groups. Assume that we have an exact sequence 1 /A f /B g /C /1 of pointed Γ-sets. The goal of the next paragraphs is to derive some exact sequences in cohomology, under some reasonable conditions on A, B and C. We will keep this notation throughout. 1 The case of a subgroup Assume that A and B are Γ-groups, that f is a group morphism (hence f is injective), and that g induces a bijection of Γ-sets B/f (A) C, where B/f (A) is the set of left cosets modulo f (A).