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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 48 49 50 51 52 53 54 55 | 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
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