Inflating a cocycle along a compatible pair

A compatible pair for 2-cocycles is a homomorphism f : Aut_K(M) → Aut_K(L) together with an embedding ι : L →ₐ[K] M intertwining it, ι (f g x) = g (ι x). The pair that matters is restriction Gal(M/K) → Gal(L/K) with the inclusion, for finite Galois extensions K ⊆ L ⊆ M. Along such a pair a 2-cocycle c of Aut_K(L) with values in Lˣ inflates to the cocycle c.comap f ι hf : (g, g') ↦ ι (c (f g, f g')) of Aut_K(M); without the intertwining hypothesis this function would not be a cocycle, which is why the hypothesis is an argument of TauCeti.TwoCocycle.comap. The inflation itself is defined in TauCeti.Algebra.CrossedProduct.Basic, and TauCeti.TwoCocycle.Cohomologous.comap (inflation preserves being cohomologous) is in TauCeti.Algebra.CrossedProduct.Cohomologous.

The theorem of the file is that inflation does not change the Brauer class of the crossed product. Write A = (L, Gal(L/K), c), B = (M, Gal(M/K), c.comap f ι hf) and r = [M : L]. Choose an L-basis m_i of M and let m*_i be the dual basis for the trace form of M/L, and let ψ : A → B be the K-linear map x · u_σ ↦ ∑_{f g = σ} ι(x) · u'_g spreading a coefficient over the fibre of f. Then Φ : M_r(A) → B, X ↦ ∑_{i,j} m*_i · ψ(X i j) · m_j is an isomorphism of K-algebras. It is multiplicative because the fibres of f are cosets of the subgroup Gal(M/L), over which the automorphisms sum to the trace, so that ψ(a) · y · ψ(a') = ψ(a · Tr_{M/L}(y) · a') and Tr_{M/L}(m_j m*_k) = δ_{jk}; it is unital by Dedekind's independence of characters, which gives ∑_i m*_i · g(m_i) = δ_{g,1} for g fixing L; and it is bijective because M_r(A) is simple and both sides have dimension [M : K]².

Main results

• TauCeti.CrossedProduct.nonempty_algEquiv_matrix_comap: for a tower K ⊆ L ⊆ M of finite Galois extensions, the crossed product of the inflated cocycle is isomorphic to the algebra of [M : L] × [M : L] matrices over the crossed product of c.
• TauCeti.BrauerGroup.crossedProductClass_comap: inflation does not change the Brauer class, [(M, Gal(M/K), c.comap f ι hf)] = [(L, Gal(L/K), c)].

References

• P. Gille and T. Szamuely, Central Simple Algebras and Galois Cohomology (2006), §4.4.
• J.-P. Serre, Local Fields, GTM 67 (1979), Chapter X.

theorem TauCeti.CrossedProduct.nonempty_algEquiv_matrix_comap {K : Type u} [Field K] {L : Type v} [Field L] [Algebra K L] {M : Type w} [Field M] [Algebra K M] [Algebra L M] [IsScalarTower K L M] [FiniteDimensional K M] (f : Gal(M/K) →* Gal(L/K)) (hf : ∀ (g : Gal(M/K)) (x : L), (IsScalarTower.toAlgHom K L M) ((f g) x) = g ((IsScalarTower.toAlgHom K L M) x)) [IsGalois K M] [IsGalois K L] (c : TwoCocycle K L) : Nonempty (CrossedProduct (TwoCocycle.comap f (IsScalarTower.toAlgHom K L M) hf c) ≃ₐ[K] Matrix (Fin (Module.finrank L M)) (Fin (Module.finrank L M)) (CrossedProduct c))

The crossed product of an inflated cocycle is a matrix algebra over the crossed product. For finite Galois extensions K ⊆ L ⊆ M and f : Gal(M/K) → Gal(L/K) compatible with the inclusion L ⊆ M, the crossed product of c.comap f is isomorphic to the algebra of [M : L] × [M : L] matrices over the crossed product of c.

theorem TauCeti.BrauerGroup.crossedProductClass_comap {K : Type u} [Field K] {L M : Type v} [Field L] [Field M] [Algebra K L] [Algebra K M] [FiniteDimensional K L] [IsGalois K L] [FiniteDimensional K M] [IsGalois K M] (f : Gal(M/K) →* Gal(L/K)) (ι : L →ₐ[K] M) (hf : ∀ (g : Gal(M/K)) (x : L), ι ((f g) x) = g (ι x)) (c : TwoCocycle K L) : crossedProductClass (TwoCocycle.comap f ι hf c) = crossedProductClass c

Inflation does not change the Brauer class. For finite Galois extensions L/K and M/K, an embedding ι : L → M and a homomorphism f : Gal(M/K) → Gal(L/K) intertwined by it, the cocycle c.comap f ι hf of M/K presents the same Brauer class as c.