| import OAI.Algebra.Finitistic.Dimension.PDCoverRadical | |
| import OAI.Algebra.Finitistic.Dimension.PDRadical | |
| import Mathlib.Algebra.Category.ModuleCat.Projective | |
| namespace OAI | |
| noncomputable section | |
| namespace LittleFinitistic.PDMinimal | |
| open Module LinearMap Submodule | |
| universe u | |
| variable (k A : Type u) [Field k] [Ring A] [Algebra k A] | |
| /-- Finite-dimensional actual left modules, retaining the scalar compatibility. -/ | |
| structure FiniteModule where | |
| carrier : Type u | |
| [add : AddCommGroup carrier] | |
| [moduleA : Module A carrier] | |
| [moduleK : Module k carrier] | |
| [tower : IsScalarTower k A carrier] | |
| [finite : FiniteDimensional k carrier] | |
| attribute [instance] FiniteModule.add FiniteModule.moduleA FiniteModule.moduleK | |
| FiniteModule.tower FiniteModule.finite | |
| instance : CoeSort (FiniteModule k A) (Type u) := ⟨FiniteModule.carrier⟩ | |
| namespace FiniteModule | |
| variable {k A} | |
| abbrev of (M : Type u) [AddCommGroup M] [Module A M] [Module k M] | |
| [IsScalarTower k A M] [FiniteDimensional k M] : FiniteModule k A := ⟨M⟩ | |
| def cat (M : FiniteModule k A) : ModuleCat A := ModuleCat.of A M | |
| instance (M : FiniteModule k A) : Module.Finite A M := | |
| Module.Finite.of_restrictScalars_finite k A M | |
| end FiniteModule | |
| variable [FiniteDimensional k A] | |
| /-- Projective covers with their actual maps and radical-kernel property. -/ | |
| structure Cover (M : FiniteModule k A) where | |
| source : FiniteModule k A | |
| [projective : Module.Projective A source] | |
| π : source →ₗ[A] M | |
| surjective : Function.Surjective π | |
| minimal : PDCovers.RightMinimal π | |
| radical : LinearMap.ker π ≤ Ring.jacobson A • (⊤ : Submodule A source) | |
| attribute [instance] Cover.projective | |
| variable {k A} | |
| theorem cover_nonempty (M : FiniteModule k A) : Nonempty (Cover k A M) := by | |
| obtain ⟨n, S, p, hP, hp, hm, hr⟩ := PDCovers.exists_finite_projective_cover (k := k) (A := A) (N := M) | |
| have := hP | |
| have : FiniteDimensional k S := Module.Finite.of_injective | |
| (S.subtype.restrictScalars k) S.injective_subtype | |
| have : Module.Finite A S := Module.Finite.of_restrictScalars_finite k A S | |
| refine ⟨⟨FiniteModule.of S, p, hp, hm, ?_⟩⟩ | |
| exact hr.trans (PDRadical.jacobson_eq_smul S).le | |
| def cover (M : FiniteModule k A) : Cover k A M := Classical.choice (cover_nonempty M) | |
| abbrev next (M : FiniteModule k A) : FiniteModule k A := by | |
| have : FiniteDimensional k (LinearMap.ker (cover M).π) := | |
| Module.Finite.of_injective ((LinearMap.ker (cover M).π).subtype.restrictScalars k) | |
| (Submodule.injective_subtype _) | |
| exact FiniteModule.of (LinearMap.ker (cover M).π) | |
| @[reducible] def syzygy (M : FiniteModule k A) : ℕ → FiniteModule k A | |
| | 0 => M | |
| | n+1 => next (syzygy M n) | |
| abbrev term (M : FiniteModule k A) (n : ℕ) := (cover (syzygy M n)).source | |
| abbrev augment (M : FiniteModule k A) (n : ℕ) := (cover (syzygy M n)).π | |
| def differential (M : FiniteModule k A) (n : ℕ) : term M (n+1) →ₗ[A] term M n := | |
| (LinearMap.ker (augment M n)).subtype.comp (augment M (n+1)) | |
| @[simp] theorem augment_differential (M : FiniteModule k A) (n : ℕ) : | |
| (augment M n).comp (differential M n) = 0 := by | |
| ext x | |
| exact (augment M (n+1) x).property | |
| @[simp] theorem differential_comp (M : FiniteModule k A) (n : ℕ) : | |
| (differential M n).comp (differential M (n+1)) = 0 := by | |
| rw [differential, LinearMap.comp_assoc, augment_differential, LinearMap.comp_zero] | |
| theorem range_differential (M : FiniteModule k A) (n : ℕ) : | |
| LinearMap.range (differential M n) = LinearMap.ker (augment M n) := by | |
| rw [differential, LinearMap.range_comp, LinearMap.range_eq_top.mpr (cover _).surjective, | |
| Submodule.map_top, Submodule.range_subtype] | |
| theorem ker_differential (M : FiniteModule k A) (n : ℕ) : | |
| LinearMap.ker (differential M n) = LinearMap.ker (augment M (n+1)) := by | |
| ext x | |
| change ((augment M (n+1) x : LinearMap.ker (augment M n)) : term M n) = 0 ↔ | |
| augment M (n+1) x = 0 | |
| constructor | |
| · intro h; exact Subtype.ext h | |
| · intro h; exact congrArg (fun z : LinearMap.ker (augment M n) => z.val) h | |
| theorem exact_differential (M : FiniteModule k A) (n : ℕ) : | |
| LinearMap.range (differential M (n+1)) = LinearMap.ker (differential M n) := by | |
| rw [range_differential, ker_differential] | |
| theorem differential_radical (M : FiniteModule k A) (n : ℕ) : | |
| LinearMap.range (differential M n) ≤ | |
| Ring.jacobson A • (⊤ : Submodule A (term M n)) := by | |
| rw [range_differential] | |
| exact (cover _).radical | |
| end LittleFinitistic.PDMinimal | |
| end | |
| end OAI | |