By Grégory Berhuy

ISBN-10: 0521738660

ISBN-13: 9780521738668

This ebook 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 idea, together with a close development of the Galois cohomology functor, in addition to an exposition of the final idea of Galois descent. the entire thought is stimulated and illustrated utilizing the instance of the descent challenge of conjugacy periods of matrices. the second one a part of the publication provides an perception of ways Galois cohomology might be valuable to unravel a few algebraic difficulties in numerous energetic learn subject matters, reminiscent of inverse Galois idea, 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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**Example text**

For any Γ-set A, we set H 0 (Γ, A) = AΓ . If A is a Γ-group, this is a subgroup of A. The set H 0 (Γ, A) is called the 0th cohomology set of Γ with coeﬃcients in A. 5. We will use this notation only episodically in this book, and will prefer the notation AΓ . We would like now to deﬁne the main object of this chapter, namely the ﬁrst cohomology set H 1 (Γ, A). We ﬁrst need an appropriate deﬁnition of a 1-cocycle. Let us go back to our conjugacy problem. Now that we have a suitable action of GΩ on SLn (Ω), we can mimick the reasoning made in the introduction and obtain a map αQ : GΩ −→ ZSLn (M0 )(Ω) σ −→ Q(σ·Q)−1 which measures the obstruction to the conjugacy problem for the pair of matrices M and M0 .

A topological group Γ is proﬁnite if it is isomorphic as a topological group to an inverse limit of ﬁnite groups (each of them being endowed with the discrete topology). In view of the previous result, the Galois group of an arbitrary Galois extension is proﬁnite. We now list some properties of proﬁnite groups without proof. We refer the reader to [12] for more details. A proﬁnite group Γ is compact and totally disconnected (that is the only non-empty connected subsets are one-point subsets). In particular, one-point subsets are closed, every open subgroup is also closed and has ﬁnite index.

Hence α−1 ({v}) is open, and we are done. 7. Let A be a Γ-group. A 1-cocycle of Γ with values in A is a continuous map α : Γ −→ A such that αστ = ασ σ·ατ for σ, τ ∈ Γ. We denote by Z 1 (Γ, A) the set of all 1-cocycles of Γ with values in A. The constant map Γ −→ A σ −→ 1 is an element of Z 1 (Γ, A), which is called the trivial 1-cocycle. Notice also that for any 1-cocycle α, we have α1 = 1. 8. If Γ acts trivially on A, a 1-cocycle is just a continuous morphism α : Γ −→ A. In order to deﬁne the cohomology set H 1 (Γ, A), we need now an appropriate notion of cohomologous cocycles, which coincides with the one deﬁned in the introduction in a particular case.

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