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2c7d975 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 | import OAI.Algebra.AuslanderReiten.Triangular.ConvertedStructure
import OAI.Algebra.AuslanderReiten.BaseChange.ScalarExtensionStructure
namespace OAI
namespace ArExplicit.ConvertedAlgebra
open TensorProduct
variable {k : Type*} [Field k] (q : k) (F : Type*) [AddCommGroup F]
variable [Module (A k q) F] [Module (A k q)ᵐᵒᵖ F]
variable [SMulCommClass (A k q) (A k q)ᵐᵒᵖ F]
variable [Module k F]
variable [IsScalarTower k (A k q) F] [IsScalarTower k (A k q)ᵐᵒᵖ F]
variable [SMulCommClass k (A k q) F] [SMulCommClass k (A k q)ᵐᵒᵖ F]
noncomputable def diagonalAlg : ConvertedAlgebra k q F →ₐ[k] (Fin 8 → k) :=
(eightCoordinates k).toAlgHom.comp (Triangular.quotientMap (F := F) k (A.diagonalAlg q))
theorem diagonalAlg_surjective : Function.Surjective (diagonalAlg q F) :=
(eightCoordinates k).surjective.comp
(Triangular.quotientMap_surjective k (A.diagonalAlg q) (A.diagonalAlg_surjective q))
theorem radical_eq_ker_diagonal :
Ring.jacobson (ConvertedAlgebra k q F) = RingHom.ker (diagonalAlg q F) := by
rw [Triangular.radical_eq_ker_quotientMap k (A.diagonalAlg q) (A.radical_eq_ker_diagonal q)]
ext a
change Triangular.quotientMap k (A.diagonalAlg q) a = 0 ↔
eightCoordinates k (Triangular.quotientMap k (A.diagonalAlg q) a) = 0
exact ⟨fun h => by rw [h, map_zero], fun h => (eightCoordinates k).injective (by simpa using h)⟩
variable [Module.Finite k F] (E : Type*) [Field E] [Algebra k E]
noncomputable def scalarExtensionQuotientEquiv :
(E ⊗[k] ConvertedAlgebra k q F ⧸ Ring.jacobson (E ⊗[k] ConvertedAlgebra k q F)) ≃ₐ[E]
(Fin 8 → E) :=
ArExplicit.scalarExtension_semisimpleQuotientEquiv E (diagonalAlg q F)
(diagonalAlg_surjective q F) (radical_eq_ker_diagonal q F)
theorem scalarExtension_jacobson_fourth_ne_bot (hq : q ≠ 0) (hq1 : 1 + q ≠ 0) :
(Ring.jacobson (E ⊗[k] ConvertedAlgebra k q F)) ^ 4 ≠ ⊥ :=
scalarExtension_radical_pow_ne_bot E 4 (jacobson_fourth_ne_bot q F hq hq1)
end ArExplicit.ConvertedAlgebra
end OAI
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