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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 56 57 58 | 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
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