| import Mathlib.RingTheory.Artinian.Module |
| import Mathlib.RingTheory.Finiteness.Projective |
| import Mathlib.LinearAlgebra.Projection |
| import Mathlib.RingTheory.Jacobson.Radical |
|
|
| namespace OAI |
| noncomputable section |
| namespace LittleFinitistic.PDCovers |
| open Module LinearMap Submodule |
| universe u v |
| variable {A : Type u} [Ring A] |
| variable {E N : Type v} [AddCommGroup E] [Module A E] |
| [AddCommGroup N] [Module A N] |
|
|
| /-- An epimorphism is right minimal if every endomorphism over it is surjective. -/ |
| def RightMinimal (p : E →ₗ[A] N) : Prop := |
| ∀ f : Module.End A E, p.comp f = p → Function.Surjective f |
|
|
| lemma comp_pow_eq {p : E →ₗ[A] N} {f : Module.End A E} |
| (h : p.comp f = p) (n : ℕ) : p.comp (f ^ n) = p := by |
| induction n with |
| | zero => ext x; rfl |
| | succ n ih => rw [pow_succ, Module.End.mul_eq_comp, ← LinearMap.comp_assoc, ih, h] |
| |
| /-- Fitting decomposition upgrades minimality among projective submodules to |
| right minimality of an actual epimorphism. -/ |
| theorem rightMinimal_of_minimal [IsArtinian A E] [IsNoetherian A E] |
| (p : E →ₗ[A] N) (S : Submodule A E) |
| (hS : Minimal (fun T : Submodule A E => |
| Module.Projective A T ∧ Function.Surjective (p.comp T.subtype)) S) : |
| RightMinimal (p.comp S.subtype) := by |
| have : Module.Projective A S := hS.1.1 |
| intro f hf |
| obtain ⟨n, hn⟩ := Filter.eventually_atTop.mp f.eventually_isCompl_ker_pow_range_pow |
| let r := f ^ (n+1) |
| have hc : IsCompl (LinearMap.range r) (LinearMap.ker r) := |
| (hn (n+1) (by omega)).symm |
| let R := LinearMap.range r |
| have : Module.Projective A R := Module.Projective.of_split R.subtype |
| (R.projectionOnto (LinearMap.ker r) hc) (Submodule.projectionOnto_comp_subtype hc) |
| let T := R.map S.subtype |
| let i : R →ₗ[A] E := S.subtype.comp R.subtype |
| have hi : Function.Injective i := S.injective_subtype.comp R.injective_subtype |
| have he : LinearMap.range i = T := by simp [i, T, LinearMap.range_comp] |
| let e : R ≃ₗ[A] T := (LinearEquiv.ofInjective i hi).trans (LinearEquiv.ofEq _ _ he) |
| have : Module.Projective A T := Module.Projective.of_equiv e |
| have hpr : (p.comp S.subtype).comp r = p.comp S.subtype := comp_pow_eq hf _ |
| have hTsurj : Function.Surjective (p.comp T.subtype) := by |
| intro y |
| obtain ⟨x, hx⟩ := hS.1.2 y |
| refine ⟨⟨S.subtype (r x), ⟨r x, mem_range_self r x, rfl⟩⟩, ?_⟩ |
| exact (DFunLike.congr_fun hpr x).trans hx |
| have hTS : T ≤ S := by |
| rintro x ⟨z, hz, rfl⟩ |
| exact z.property |
| have hST : S ≤ T := hS.2 ⟨inferInstance, hTsurj⟩ hTS |
| have hr : Function.Surjective r := by |
| intro x |
| obtain ⟨z, ⟨w, hw⟩, hz⟩ := hST x.property |
| refine ⟨w, ?_⟩ |
| exact hw.trans (Subtype.ext hz) |
| intro x |
| obtain ⟨y, hy⟩ := hr x |
| refine ⟨(f ^ n) y, ?_⟩ |
| simpa [r, pow_succ', Module.End.mul_eq_comp] using hy |
|
|
| /-- Every projective epimorphism from a finite-length module contains a |
| projective cover. This proves existence rather than assuming covers. -/ |
| theorem exists_rightMinimal [IsArtinian A E] [IsNoetherian A E] |
| [Module.Projective A E] (p : E →ₗ[A] N) (hp : Function.Surjective p) : |
| ∃ S : Submodule A E, Module.Projective A S ∧ |
| Function.Surjective (p.comp S.subtype) ∧ RightMinimal (p.comp S.subtype) := by |
| have : Module.Projective A (⊤ : Submodule A E) := |
| Module.Projective.of_equiv (Submodule.topEquiv : (⊤ : Submodule A E) ≃ₗ[A] E).symm |
| obtain ⟨S, hS⟩ := WellFoundedLT.exists_minimal (inferInstance : WellFoundedLT (Submodule A E)) |
| {T : Submodule A E | Module.Projective A T ∧ Function.Surjective (p.comp T.subtype)} |
| ⟨⊤, inferInstance, fun y => by obtain ⟨x, hx⟩ := hp y; exact ⟨⟨x, mem_top⟩, hx⟩⟩ |
| exact ⟨S, hS.1.1, hS.1.2, rightMinimal_of_minimal p S hS⟩ |
| end LittleFinitistic.PDCovers |
| |
| end |
| |
| end OAI |
| |
