Recovering each representation coordinate from the reconstructed graded module

theorem MagnitudeConjecture.Graded.FiniteGradedModule.reconstructedCoordinate_eq_single {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) (m : ℕ) (p : ι × Fin (m + 1)) : idempotentComponent R (intervalReconstructedGrading R ⋯ e he0 he F hneg h1 hsum horth m) (e p.1) ↑↑p.2 = singleCoordinate (fun (q : ι × Fin (m + 1)) => ↑(F.obj (Opposite.op (intervalProjectiveLabel R ⋯ e he0 m q)))) p

The idempotent-fixed homogeneous component is exactly one coordinate subspace.

noncomputable def MagnitudeConjecture.Graded.FiniteGradedModule.reconstructedCoordinateEquiv {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) (m : ℕ) (p : ι × Fin (m + 1)) : ↥(idempotentComponent R (intervalReconstructedGrading R ⋯ e he0 he F hneg h1 hsum horth m) (e p.1) ↑↑p.2) ≃ₗ[k] ↑(F.obj (Opposite.op (intervalProjectiveLabel R ⋯ e he0 m p)))

Evaluation at (i,r) recovers the original vector space from the reconstructed coordinate.