| import OAI.Algebra.AuslanderReiten.Triangular.ConvertedStructure |
| import Mathlib.RingTheory.Flat.Basic |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped TensorProduct |
| namespace ArExplicit |
| namespace BaseAlgebra |
| variable {k : Type*} [Field k] (q : k) |
|
|
| theorem e_u_ne_u_e : |
| basisVector q Letter.e * basisVector q Letter.u ≠ |
| basisVector q Letter.u * basisVector q Letter.e := by |
| intro h |
| have hh := congrArg (fun a : BaseAlgebra k q => a.coeff Letter.u) h |
| simp [basis_mul, multiplicationTable] at hh |
|
|
| end BaseAlgebra |
| 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] |
|
|
| def noncommE : ConvertedAlgebra k q F := ⟨T.lower q Letter.e ⊗ₜ[k] 1, 0, 0⟩ |
|
|
| theorem explicit_noncommuting_pair : |
| noncommE q F * radicalU q F ≠ radicalU q F * noncommE q F := by |
| intro h |
| have hh := congrArg (fun a : ConvertedAlgebra k q F => A.originalProjection q a.top) h |
| apply BaseAlgebra.e_u_ne_u_e q |
| have he : A.originalProjection q (T.lower q Letter.e ⊗ₜ[k] 1) = |
| BaseAlgebra.basisVector q Letter.e := by |
| simp [A.originalProjection, T.lower, TrivialExtension.inl] |
| simpa only [Triangular.top_mul, noncommE, radicalU, map_mul, |
| A.originalProjection_radicalU, he] using hh |
|
|
| theorem noncommutative : ¬ ∀ a b : ConvertedAlgebra k q F, a*b = b*a := by |
| intro h |
| exact explicit_noncommuting_pair q F (h _ _) |
|
|
| end ConvertedAlgebra |
|
|
| theorem noncommuting_scalarExtension {k R : Type*} [Field k] [Ring R] [Algebra k R] |
| (E : Type*) [Field E] [Algebra k E] (a b : R) (h : a*b ≠ b*a) : |
| ((1 : E) ⊗ₜ[k] a) * ((1 : E) ⊗ₜ[k] b) ≠ |
| ((1 : E) ⊗ₜ[k] b) * ((1 : E) ⊗ₜ[k] a) := by |
| intro he |
| apply h |
| apply Module.Flat.tensorProduct_mk_injective (R := k) (S := E) (M := R) |
| change (1 : E) ⊗ₜ[k] (a*b) = (1 : E) ⊗ₜ[k] (b*a) |
| simpa only [Algebra.TensorProduct.tmul_mul_tmul, one_mul] using he |
|
|
| end ArExplicit |
|
|
| end |
|
|
| end OAI |
|
|
