| import OAI.Algebra.Finitistic.Localization.FiniteProjectiveHomotopy |
| import OAI.Algebra.Finitistic.Localization.AmbientQuotientEvaluation |
| import OAI.Algebra.Finitistic.Selected.SelectedFiniteTensorEvaluation |
| import Mathlib.Algebra.Homology.HomotopyCategory.Triangulated |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| namespace LittleFinitistic |
| open BalancedTensor ThickEvaluation FiniteProjectiveHomotopy IdempotentTensor |
| universe u v |
|
|
| namespace FiniteProjectiveHomotopy |
| variable {R : Type u} [Ring R] |
|
|
| lemma single_property (M : ModuleCat.{v} R) [Module.Finite R M] [Projective M] (n : ℤ) : |
| ComplexProperty R ((HomologicalComplex.single _ (.up ℤ) n).obj M) := by |
| constructor |
| · exact ⟨n,n,fun i hi => HomologicalComplex.isZero_single_obj_X (.up ℤ) n M i (by omega)⟩ |
| · intro i |
| by_cases h : i=n |
| · subst i |
| let e := HomologicalComplex.singleObjXSelf (.up ℤ) n M |
| exact ⟨finite_of_iso R e.symm,Projective.of_iso e.symm inferInstance⟩ |
| · exact finite_projective_of_isZero R _ |
| (HomologicalComplex.isZero_single_obj_X (.up ℤ) n M i h) |
| end FiniteProjectiveHomotopy |
|
|
| namespace QuadraticEncoding |
| variable {k : Type u} [Field k] {d t : ℕ} (P : Relations k d t) |
| |
| instance vertexModuleFinite (i : Fin 3) : Module.Finite (B P)ᵐᵒᵖ (vertexModule P i) := by |
| let e := Representation.vertexId (P := P) i |
| have he : e*e=e := (ThreeVertexAlgebra.directedSystem (composition P)).idem i |
| apply Module.Finite.of_surjective (splitProjection he) |
| intro m |
| exact ⟨MulOpposite.op m.val,Subtype.ext m.property⟩ |
| |
| instance vertexModuleProjective (i : Fin 3) : Projective (vertexModule P i) := by |
| have : Module.Projective (B P)ᵐᵒᵖ (vertexModule P i) := |
| rightIdeal_projective ((ThreeVertexAlgebra.directedSystem (composition P)).idem i) |
| exact ModuleCat.projective_of_categoryTheory_projective (vertexModule P i) |
| |
| / |
| abbrev BoundedK := K.{u,u} (B P)ᵐᵒᵖ |
| |
| def boundedConeS : BoundedK P := ⟨coneS P, |
| mappingCone_property _ _ (single_property (vertexModule P 0) 0) (single_property (vertexModule P 1) 0)⟩ |
| def boundedConeS' : BoundedK P := ⟨coneS' P, |
| mappingCone_property _ _ (single_property (vertexModule P 1) 0) (single_property (vertexModule P 2) 0)⟩ |
| |
| def boundedConeGenerators : ObjectProperty (BoundedK P) := |
| fun T => T = boundedConeS P ∨ T = boundedConeS' P |
| |
| abbrev boundedDenominators : MorphismProperty (BoundedK P) := |
| ObjectProperty.trW (Thick (boundedConeGenerators P)) |
| |
| / |
| abbrev BoundedQ := (boundedDenominators P).Localization |
| |
| variable {P} {Y : Type v} [AddCommGroup Y] [Module k Y] (X : Representation P Y) |
| |
| def boundedEvaluation : BoundedK P ⥤ HomotopyCategory (ModuleCat.{max u v} k) (.up ℤ) := |
| inclusion (B P)ᵐᵒᵖ ⋙ homotopyEvaluation k (B P) X.Evaluation |
| |
| instance boundedEvaluationAdditive : (boundedEvaluation X).Additive := |
| inferInstanceAs (inclusion (B P)ᵐᵒᵖ ⋙ homotopyEvaluation k (B P) X.Evaluation).Additive |
| instance boundedEvaluationShift : (boundedEvaluation X).CommShift ℤ := |
| inferInstanceAs ((inclusion (B P)ᵐᵒᵖ ⋙ homotopyEvaluation k (B P) X.Evaluation).CommShift ℤ) |
| instance boundedEvaluationTriangulated : (boundedEvaluation X).IsTriangulated := |
| inferInstanceAs (inclusion (B P)ᵐᵒᵖ ⋙ homotopyEvaluation k (B P) X.Evaluation).IsTriangulated |
| |
| lemma boundedEvaluation_kills_generators : ∀ T, boundedConeGenerators P T → |
| IsZero ((boundedEvaluation X).obj T) := by |
| rintro T (rfl | rfl) |
| · exact evaluation_coneS_zero X |
| · exact evaluation_coneS'_zero X |
| |
| / |
| def boundedQuotientEvaluation : BoundedQ P ⥤ HomotopyCategory (ModuleCat.{max u v} k) (.up ℤ) := |
| quotientEvaluation (boundedConeGenerators P) (boundedEvaluation X) |
| (boundedEvaluation_kills_generators X) |
| |
| def boundedQuotientEvaluationFactors : |
| (boundedDenominators P).Q ⋙ boundedQuotientEvaluation X ≅ boundedEvaluation X := |
| quotientEvaluationFactors (boundedConeGenerators P) (boundedEvaluation X) |
| (boundedEvaluation_kills_generators X) |
| |
| / |
| theorem boundedEvaluation_finite [FiniteDimensional k Y] (T : BoundedK P) : |
| (∃ a b : ℤ, ∀ i, i < a ∨ b < i → IsZero (((boundedEvaluation X).obj T).as.X i)) ∧ |
| ∀ i, FiniteDimensional k (((boundedEvaluation X).obj T).as.X i) := by |
| obtain ⟨⟨a,b,hb⟩,hp⟩ := T.property |
| constructor |
| · exact ⟨a,b,bounded_support X.Evaluation T.obj.as a b hb⟩ |
| · intro i |
| have := (hp i).1 |
| have : FiniteDimensional k (B P) := finiteDimensional P |
| exact tensorFunctor_finite X.Evaluation (T.obj.as.X i) |
| end QuadraticEncoding |
| end LittleFinitistic |
| |
| end |
| |
| end OAI |
| |
