| import OAI.Algebra.Finitistic.Dimension.PDCovers | |
| import Mathlib.LinearAlgebra.FiniteDimensional.Basic | |
| import Mathlib.RingTheory.Artinian.Instances | |
| namespace OAI | |
| noncomputable section | |
| namespace LittleFinitistic.PDCovers | |
| open Module LinearMap Submodule | |
| universe u | |
| variable {A : Type u} [Ring A] | |
| variable {E N : Type u} [AddCommGroup E] [Module A E] | |
| [AddCommGroup N] [Module A N] | |
| /-- A projective right-minimal epimorphism has a superfluous kernel. -/ | |
| theorem eq_top_of_sup_ker [Module.Projective A E] | |
| (p : E →ₗ[A] N) (hp : Function.Surjective p) (hm : RightMinimal p) | |
| (L : Submodule A E) (hL : L ⊔ LinearMap.ker p = ⊤) : L = ⊤ := by | |
| have hsurj : Function.Surjective (p.comp L.subtype) := by | |
| intro y | |
| obtain ⟨x, hx⟩ := hp y | |
| have hm : x ∈ L ⊔ LinearMap.ker p := hL.symm ▸ mem_top | |
| obtain ⟨z, hz, w, hw, hzw⟩ := Submodule.mem_sup.mp hm | |
| refine ⟨⟨z, hz⟩, ?_⟩ | |
| have hw0 : p w = 0 := hw | |
| change p z = y | |
| rw [← hx, ← hzw, map_add, hw0, add_zero] | |
| obtain ⟨g, hg⟩ := Module.projective_lifting_property (p.comp L.subtype) p hsurj | |
| have hfg : p.comp (L.subtype.comp g) = p := by rwa [← LinearMap.comp_assoc] | |
| have hfgsurj := hm (L.subtype.comp g) hfg | |
| apply top_unique | |
| intro x _ | |
| obtain ⟨z, hz⟩ := hfgsurj x | |
| rw [← hz] | |
| exact (g z).property | |
| /-- Projective-cover kernels lie in the actual module Jacobson radical. -/ | |
| theorem ker_le_jacobson [Module.Projective A E] | |
| (p : E →ₗ[A] N) (hp : Function.Surjective p) (hm : RightMinimal p) : | |
| LinearMap.ker p ≤ Module.jacobson A E := by | |
| apply le_sInf | |
| intro L hL | |
| by_contra h | |
| have hlt : L < L ⊔ LinearMap.ker p := lt_of_le_of_ne le_sup_left (fun he => h (he ▸ le_sup_right)) | |
| have htop : L ⊔ LinearMap.ker p = ⊤ := hL.2 _ hlt | |
| exact hL.1 (eq_top_of_sup_ker p hp hm L htop) | |
| /-- Over a finite-dimensional algebra every finite-dimensional module has an | |
| actual finitely generated projective cover, constructed inside a finite free module. -/ | |
| theorem exists_finite_projective_cover | |
| {k : Type u} [Field k] [Algebra k A] [FiniteDimensional k A] | |
| [Module k N] [IsScalarTower k A N] [FiniteDimensional k N] : | |
| ∃ (n : ℕ) (S : Submodule A (Fin n → A)) (p : S →ₗ[A] N), | |
| Module.Projective A S ∧ Function.Surjective p ∧ | |
| RightMinimal p ∧ LinearMap.ker p ≤ Module.jacobson A S := by | |
| have : Module.Finite A N := Module.Finite.of_restrictScalars_finite k A N | |
| obtain ⟨n, p, hp⟩ := Module.Finite.exists_fin' A N | |
| have : IsArtinian A (Fin n → A) := isArtinian_of_tower k inferInstance | |
| have : IsNoetherian A (Fin n → A) := isNoetherian_of_tower k inferInstance | |
| obtain ⟨S, hS, hs, hm⟩ := exists_rightMinimal p hp | |
| have := hS | |
| exact ⟨n, S, p.comp S.subtype, hS, hs, hm, ker_le_jacobson _ hs hm⟩ | |
| end LittleFinitistic.PDCovers | |
| end | |
| end OAI | |