| import OAI.Algebra.AuslanderReiten.Algebras.TensorStructure | |
| import OAI.Algebra.AuslanderReiten.Triangular.TriangularRadical | |
| import OAI.Algebra.AuslanderReiten.Homology.SplitQuotient | |
| namespace OAI | |
| namespace ArExplicit | |
| abbrev ConvertedAlgebra (k : Type*) [Field k] (q : k) (F : Type*) := Triangular (A k q) F | |
| namespace ConvertedAlgebra | |
| 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] | |
| def radicalU : ConvertedAlgebra k q F := ⟨A.radicalU q, 0, 0⟩ | |
| def radicalT : ConvertedAlgebra k q F := ⟨A.radicalT q, 0, 0⟩ | |
| theorem radicalU_mem : radicalU q F ∈ Ring.jacobson (ConvertedAlgebra k q F) := by | |
| apply Triangular.mem_jacobson_iff _ |>.mpr | |
| exact ⟨A.radicalU_mem q, Submodule.zero_mem _⟩ | |
| theorem radicalT_mem : radicalT q F ∈ Ring.jacobson (ConvertedAlgebra k q F) := by | |
| apply Triangular.mem_jacobson_iff _ |>.mpr | |
| exact ⟨A.radicalT_mem q, Submodule.zero_mem _⟩ | |
| theorem jacobson_fourth_ne_bot (hq : q ≠ 0) (hq1 : 1 + q ≠ 0) : | |
| (Ring.jacobson (ConvertedAlgebra k q F)) ^ 4 ≠ ⊥ := by | |
| have h4 := Ideal.mul_mem_mul | |
| (Ideal.mul_mem_mul (Ideal.mul_mem_mul (radicalU_mem q F) (radicalT_mem q F)) | |
| (radicalU_mem q F)) (radicalT_mem q F) | |
| have hmem : ((radicalU q F * radicalT q F) * radicalU q F) * radicalT q F ∈ | |
| (Ring.jacobson (ConvertedAlgebra k q F)) ^ 4 := by | |
| simpa only [Submodule.pow_succ, Submodule.pow_zero, Submodule.one_mul] using h4 | |
| intro h | |
| rw [h] at hmem | |
| have ht := congrArg Triangular.top (show _ = 0 from hmem) | |
| exact A.fourfold_product_ne_zero q hq hq1 ht | |
| 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 semisimpleQuotientEquiv : | |
| (ConvertedAlgebra k q F ⧸ Ring.jacobson (ConvertedAlgebra k q F)) ≃ₐ[k] (Fin 8 → k) := | |
| (Triangular.semisimpleQuotientEquiv (F := F) k (A.diagonalAlg q) | |
| (A.diagonalAlg_surjective q) (A.radical_eq_ker_diagonal q)).trans (eightCoordinates k) | |
| end ConvertedAlgebra | |
| end ArExplicit | |
| end OAI | |