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We then have H 1 (GΩ , L ⊗k Ω) H 1 (GΩ , (L1 ⊗k Ω)× ) × · · · × H 1 (GΩ , (Lr ⊗k Ω)× ), and we use the previous case. If E is a finite dimensional algebra over a field F , we denote by NE/F (x) the determinant of left multiplication by x (considered as an endomorphism of the F -vector space E). 2. Let E = k n , n ≥ 1. If x = (x1 , . . , xn ), then we have NE/k (x) = x1 · · · xn , since the representative matrix of x in the canonical basis of E is simply the diagonal matrix whose diagonal entries are x1 , .

We now consider a group-valued functor G such that: (1) For all finite Galois subextension L/k, the map G(L) → G(Ω) is injective and induces a group isomorphism G(L) G(Ω)Gal(Ω/L) (2) For all g ∈ G(Ω), the subgroup {σ ∈ GΩ | σ·g = g} is open. These two conditions say that an element g ∈ G(Ω) ‘comes from’ an element of G(L) for some finite Galois subextension L/k of Ω/k, and that the action of GΩ is in fact the same as the action of GL on g when viewed as an element of G(L). One can show that any functor G defined by a finite set of polynomial equations with coefficients in k satisfy these assumptions.

The Krull topology is the topology generated by the subsets σGal(Ω/K), σ ∈ GΩ , K ⊂ Ω, [K : k] < +∞. We now consider a group-valued functor G such that: (1) For all finite Galois subextension L/k, the map G(L) → G(Ω) is injective and induces a group isomorphism G(L) G(Ω)Gal(Ω/L) (2) For all g ∈ G(Ω), the subgroup {σ ∈ GΩ | σ·g = g} is open. These two conditions say that an element g ∈ G(Ω) ‘comes from’ an element of G(L) for some finite Galois subextension L/k of Ω/k, and that the action of GΩ is in fact the same as the action of GL on g when viewed as an element of G(L).

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An introduction to Galois cohomology and its applications [Lecture notes] by Grégory Berhuy


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