openAI-math-dataset / lean /OAI /Algebra /AuslanderReiten /Triangular /ConvertedScalarExtension.lean
| 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 | |