By Grégory Berhuy
This booklet is the 1st common advent to Galois cohomology and its purposes. the 1st half is self contained and gives the elemental result of the idea, together with an in depth building of the Galois cohomology functor, in addition to an exposition of the final concept 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 e-book offers an perception of ways Galois cohomology might be precious to unravel a few algebraic difficulties in numerous lively study issues, akin to inverse Galois concept, rationality questions or crucial size of algebraic teams. the writer assumes just a minimum history in algebra (Galois idea, tensor items of vectors areas and algebras).
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Extra resources for Introduction to Galois cohomology and its applications
Conversely, assume that δ 0 (c) is trivial, that is ασ = a σ·a−1 for all σ ∈ Γ, for some a ∈ A. Let b ∈ B be a preimage of c under g. We then have f (a σ·a−1 ) = b−1 σ·b for all σ ∈ Γ, so f (a)σ·f (a)−1 = b−1 σ·b for all σ ∈ Γ. Hence bf (a) ∈ B Γ , and we have c = g(b) = g(bf (a)) ∈ im(g∗ ). Hence ker(δ 0 ) = im(g∗ ), which is what we wanted to prove. Exactness at H 1 (Γ, A): We need to prove that im(δ 0 ) = ker(f∗ ). Let c ∈ C G and let b ∈ B satisfying c = g(b). Then by deﬁnition of f∗ and δ 0 (c), f∗ (δ 0 (c)) is the class of the 1-cocycle Γ −→ B σ −→ b−1 σ·b, which is cohomologous to the trivial cocycle.
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 )). But the fact that M is conjugate or not to M0 by an element of SLn (k) is an intrinsic property of M and of the ﬁeld k, and should certainly not depend on the chosen Galois extension L/k. Therefore, we need to ﬁnd a way to patch these local obstructions together. 3 Cohomology sets: basic properties 27 ﬁrst an appropriate action of GΩ on Mn (Ω) and SLn (Ω). Since we want to patch together the local obstructions, we need this action to coincide with the local actions on the various sets Mn (L) and SLn (L).
1 (σn ) ). Therefore, (f2∗ ◦ f1∗ )([α]) is represented by the cocycle Γn3 −→ A3 (σ1 , . . ,ϕ1 (ϕ2 (σn )) )). Similarly, (f4∗ ◦ f3∗ )([α]) is represented by the cocycle Γn3 −→ A3 (σ1 , . . ,ϕ3 (ϕ4 (σn )) )). Since f2 ◦ f1 = f4 ◦ f3 and ϕ1 ◦ ϕ2 = ϕ3 ◦ ϕ4 by assumption, we get the desired result. 20, unless speciﬁed otherwise. 3 Cohomology sets as a direct limit In this paragraph, we would like to relate the cohomology of proﬁnite groups to the cohomology of its ﬁnite quotients. 6, which says more or less that an n-cocycle α : Γn −→ A is locally deﬁned by a family of n-cocycles α(U ) : (Γ/U )n −→ AU , where U runs through the set of open normal subgroups of Γ.