| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory | |
| open scoped TensorProduct | |
| namespace ArExplicit.Statement | |
| universe u v | |
| abbrev K := FractionRing (MvPolynomial (Fin 3) (ZMod 2)) | |
| def parameters (i : Fin 3) : K := | |
| algebraMap (MvPolynomial (Fin 3) (ZMod 2)) K (MvPolynomial.X i) | |
| def TotallyAcyclicWitness (R : Type u) [Ring R] | |
| (Z : Type u) [AddCommGroup Z] [Module R Z] : Prop := | |
| ∃ P : ChainComplex (ModuleCat.{u} R) ℤ, | |
| (∀ i : ℤ, Module.Finite R (P.X i) ∧ Module.Projective R (P.X i)) ∧ | |
| (∀ i : ℤ, LinearMap.range (P.d (i+1) i).hom = LinearMap.ker (P.d i (i-1)).hom) ∧ | |
| (∀ (Q : Type u) [AddCommGroup Q] [Module R Q] [Module.Projective R Q] | |
| (i : ℤ) (f : P.X i →ₗ[R] Q), f.comp (P.d (i+1) i).hom = 0 → | |
| ∃ g : P.X (i-1) →ₗ[R] Q, f = g.comp (P.d i (i-1)).hom) ∧ | |
| Nonempty (Z ≃ₗ[R] (P.X 0 ⧸ LinearMap.range (P.d 1 0).hom)) | |
| def Conclusions (F R Z : Type u) [Field F] [Ring R] [Algebra F R] | |
| [AddCommGroup Z] [Module R Z] [Module F Z] [IsScalarTower F R Z] : Prop := | |
| FiniteDimensional F R ∧ | |
| FiniteDimensional F Z ∧ | |
| ¬ Module.Projective R Z ∧ | |
| TotallyAcyclicWitness R Z ∧ | |
| (∀ i : ℕ, 0 < i → | |
| Subsingleton (Abelian.Ext (ModuleCat.of R Z) (ModuleCat.of R Z) i)) ∧ | |
| (∀ i : ℕ, 0 < i → | |
| Subsingleton (Abelian.Ext (ModuleCat.of R Z) (ModuleCat.of R R) i)) ∧ | |
| Nonempty ((R ⧸ Ring.jacobson R) ≃ₐ[F] (Fin 8 → F)) ∧ | |
| (Ring.jacobson R)^4 ≠ ⊥ | |
| structure System where | |
| algebraCarrier : Type | |
| [ring : Ring algebraCarrier] | |
| [scalarAlgebra : Algebra K algebraCarrier] | |
| moduleCarrier : Type | |
| [additiveGroup : AddCommGroup moduleCarrier] | |
| [action : Module algebraCarrier moduleCarrier] | |
| [scalarAction : Module K moduleCarrier] | |
| [actionTower : IsScalarTower K algebraCarrier moduleCarrier] | |
| attribute [instance] System.ring System.scalarAlgebra System.additiveGroup | |
| System.action System.scalarAction System.actionTower | |
| def FullStatement : Prop := | |
| AlgebraicIndependent (ZMod 2) parameters ∧ | |
| ∃ s : System, | |
| Conclusions K s.algebraCarrier s.moduleCarrier ∧ | |
| ∀ (E : Type v) [Field E] [Algebra K E], | |
| ∃ action : Module (E ⊗[K] s.algebraCarrier) (E ⊗[K] s.moduleCarrier), | |
| letI := action | |
| ∃ tower : IsScalarTower E (E ⊗[K] s.algebraCarrier) (E ⊗[K] s.moduleCarrier), | |
| letI := tower | |
| (∀ (a b : E) (r : s.algebraCarrier) (z : s.moduleCarrier), | |
| (a ⊗ₜ[K] r) • (b ⊗ₜ[K] z) = (a*b) ⊗ₜ[K] (r • z)) ∧ | |
| Conclusions E (E ⊗[K] s.algebraCarrier) (E ⊗[K] s.moduleCarrier) | |
| end ArExplicit.Statement | |
| namespace ArExplicit.Statement | |
| universe v | |
| theorem main : FullStatement.{v} := by | |
| sorry | |
| end ArExplicit.Statement | |
| end | |
| end OAI | |