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216fb31 | 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 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 | 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)
/-- The exact bounded-projective source category used in Section 3. -/
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))
/-- The actual localization of K^b(proj-B) by the thick subcategory generated by the two cones. -/
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
/-- Tensor evaluation on the manuscript's fixed quotient; no generator-vanishing assumption remains. -/
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)
/-- Evaluation of every bounded-projective representative has bounded finite-dimensional terms. -/
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
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