Evaluation of reconstruction on all degrees

theorem MagnitudeConjecture.Graded.FiniteGradedModule.principalIntervalInclusion_op_injective {k A : Type u} [Field k] [Ring A] [Algebra k A] [FiniteDimensional k A] (R : VectorGrading k A) (hmul : ∀ {i j : ℤ} {a b : A}, a ∈ R.component i → b ∈ R.component j → a * b ∈ R.component (i + j)) {ι : Type} [Fintype ι] (e : ι → A) (he0 : ∀ (i : ι), e i ∈ R.component 0) (m : ℕ) : Function.Injective (principalIntervalInclusion R ⋯ e he0 m).op.obj

The interval inclusion is injective on objects.

theorem MagnitudeConjecture.Graded.FiniteGradedModule.principalIntervalInclusion_outside {k A : Type u} [Field k] [Ring A] [Algebra k A] [FiniteDimensional k A] (R : VectorGrading k A) (hmul : ∀ {i j : ℤ} {a b : A}, a ∈ R.component i → b ∈ R.component j → a * b ∈ R.component (i + j)) {ι : Type} [Fintype ι] (e : ι → A) (he0 : ∀ (i : ι), e i ∈ R.component 0) (m : ℕ) (p : (PrincipalDegreeCategory R ⋯ e he0)ᵒᵖ) (hp : p ∉ Set.range (principalIntervalInclusion R ⋯ e he0 m).op.obj) : (Opposite.unop p).2 < 0 ∨ ↑m < (Opposite.unop p).2

An object omitted by the interval inclusion has degree outside [0,m].

noncomputable def MagnitudeConjecture.Graded.FiniteGradedModule.supportedEvaluationReconstructionIso {k A : Type u} [Field k] [Ring A] [Algebra k A] [FiniteDimensional k A] (R : VectorGrading k A) (hmul : ∀ {i j : ℤ} {a b : A}, a ∈ R.component i → b ∈ R.component j → a * b ∈ R.component (i + j)) {ι : Type} [Fintype ι] (e : ι → A) (he0 : ∀ (i : ι), e i ∈ R.component 0) (he : ∀ (i : ι), e i * e i = e i) (F : CategoryTheory.Functor (PrincipalDegreeCategory R ⋯ e he0)ᵒᵖ (ModuleCat k)) [F.Additive] [CategoryTheory.Functor.Linear k F] (hneg : ∀ d < 0, R.component d = ⊥) (h1 : 1 ∈ R.component 0) (hsum : ∑ i : ι, e i = 1) (horth : Pairwise fun (i j : ι) => e i * e j = 0) (hfinite : ∀ (p : (PrincipalDegreeCategory R ⋯ e he0)ᵒᵖ), FiniteDimensional k ↑(F.obj p)) (m : ℕ) (hzero : ∀ p ∈ principalOutsideInterval R ⋯ e he0 m, CategoryTheory.Limits.IsZero (F.obj p)) : CoveringHom.restrictedLinearYoneda (principalDegreeInclusion R ⋯ e he0) (intervalReconstructedSupportedObject R ⋯ e he0 he F hneg h1 hsum horth hfinite m).obj ≅ F

Evaluation after reconstruction recovers the whole supported representation.

theorem MagnitudeConjecture.Graded.FiniteGradedModule.supportedEvaluationReconstructionIso_app_interval {k A : Type u} [Field k] [Ring A] [Algebra k A] [FiniteDimensional k A] (R : VectorGrading k A) (hmul : ∀ {i j : ℤ} {a b : A}, a ∈ R.component i → b ∈ R.component j → a * b ∈ R.component (i + j)) {ι : Type} [Fintype ι] (e : ι → A) (he0 : ∀ (i : ι), e i ∈ R.component 0) (he : ∀ (i : ι), e i * e i = e i) (F : CategoryTheory.Functor (PrincipalDegreeCategory R ⋯ e he0)ᵒᵖ (ModuleCat k)) [F.Additive] [CategoryTheory.Functor.Linear k F] (hneg : ∀ d < 0, R.component d = ⊥) (h1 : 1 ∈ R.component 0) (hsum : ∑ i : ι, e i = 1) (horth : Pairwise fun (i j : ι) => e i * e j = 0) (hfinite : ∀ (p : (PrincipalDegreeCategory R ⋯ e he0)ᵒᵖ), FiniteDimensional k ↑(F.obj p)) (m : ℕ) (hzero : ∀ p ∈ principalOutsideInterval R ⋯ e he0 m, CategoryTheory.Limits.IsZero (F.obj p)) (p : ι × Fin (m + 1)) : (supportedEvaluationReconstructionIso R ⋯ e he0 he F hneg h1 hsum horth hfinite m hzero).app (Opposite.op (intervalProjectiveLabel R ⋯ e he0 m p)) = (intervalEvaluationCoordinateEquiv R ⋯ e he0 he F hneg h1 hsum horth hfinite m p).toModuleIso

On an interval object the extended comparison is the original coordinate comparison.