File size: 4,468 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 | 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
|