| import OAI.Algebra.Finitistic.Dimension.PDMinimalResolution | |
| import Mathlib.Algebra.Category.ModuleCat.ProjectiveDimension | |
| namespace OAI | |
| noncomputable section | |
| namespace LittleFinitistic.PDMinimal | |
| open CategoryTheory CategoryTheory.Limits | |
| universe u | |
| variable {k A : Type u} [Field k] [Ring A] [Algebra k A] [FiniteDimensional k A] | |
| lemma cover_subsingleton (M : FiniteModule k A) [Subsingleton M] : | |
| Subsingleton (cover M).source := by | |
| have hp : (cover M).π = 0 := Subsingleton.elim _ _ | |
| have hsurj := (cover M).minimal 0 (by simp [hp]) | |
| refine ⟨fun x y => ?_⟩ | |
| obtain ⟨a, ha⟩ := hsurj x | |
| obtain ⟨b, hb⟩ := hsurj y | |
| exact ha.symm.trans hb | |
| lemma cover_injective (M : FiniteModule k A) [Module.Projective A M] : | |
| Function.Injective (cover M).π := by | |
| obtain ⟨g, hg⟩ := Module.projective_lifting_property (cover M).π LinearMap.id | |
| (cover M).surjective | |
| have he : (cover M).π.comp (g.comp (cover M).π) = (cover M).π := by | |
| rw [← LinearMap.comp_assoc, hg, LinearMap.id_comp] | |
| have hs := (cover M).minimal _ he | |
| have hgs : Function.Surjective g := by | |
| intro x | |
| obtain ⟨y, hy⟩ := hs x | |
| exact ⟨(cover M).π y, hy⟩ | |
| have hgp : g.comp (cover M).π = LinearMap.id := by | |
| ext x | |
| obtain ⟨y, rfl⟩ := hgs x | |
| exact congrArg g (DFunLike.congr_fun hg y) | |
| exact LinearMap.injective_of_comp_eq_id _ _ hgp | |
| lemma next_subsingleton_of_projective (M : FiniteModule k A) [Module.Projective A M] : | |
| Subsingleton (next M) := by | |
| have he := LinearMap.ker_eq_bot.mpr (cover_injective M) | |
| change Subsingleton (LinearMap.ker (cover M).π) | |
| rw [he] | |
| infer_instance | |
| /-- Vanishing of a term of the constructed resolution gives the standard | |
| categorical projective-dimension bound. -/ | |
| theorem pdLT_of_term_zero (M : FiniteModule k A) (n i : ℕ) | |
| (hz : Subsingleton (term M (i+n))) : | |
| HasProjectiveDimensionLT (syzygy M i).cat n := by | |
| induction n generalizing i with | |
| | zero => | |
| have : Subsingleton (term M i) := by simpa using hz | |
| have : Subsingleton (syzygy M i) := | |
| Function.Surjective.subsingleton (cover _).surjective | |
| exact (hasProjectiveDimensionLT_zero_iff_isZero _).mpr | |
| (ModuleCat.isZero_iff_subsingleton.mpr (show Subsingleton (syzygy M i) from inferInstance)) | |
| | succ n ih => | |
| have hnext : HasProjectiveDimensionLT (syzygy M (i+1)).cat n := | |
| ih (i+1) (by simpa [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] using hz) | |
| have hs := LinearMap.shortExact_shortComplexKer (cover (syzygy M i)).surjective | |
| exact hs.hasProjectiveDimensionLT_X₃ n hnext inferInstance | |
| lemma syzygy_projective (M : FiniteModule k A) (n i : ℕ) | |
| (h : HasProjectiveDimensionLT (syzygy M i).cat (n+1)) : | |
| Projective (syzygy M (i+n)).cat := by | |
| induction n generalizing i with | |
| | zero => | |
| simpa using projective_iff_hasProjectiveDimensionLT_one.mpr h | |
| | succ n ih => | |
| have hs := LinearMap.shortExact_shortComplexKer (cover (syzygy M i)).surjective | |
| have hn : HasProjectiveDimensionLT (syzygy M (i+1)).cat (n+1) := | |
| (hs.hasProjectiveDimensionLT_X₃_iff n inferInstance).mp h | |
| simpa [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] using ih (i+1) hn | |
| /-- Finite projective dimension forces the minimal resolution to terminate. -/ | |
| theorem term_zero_of_pdLE (M : FiniteModule k A) (n : ℕ) | |
| (h : HasProjectiveDimensionLE M.cat n) : Subsingleton (term M (n+1)) := by | |
| have hp := syzygy_projective M n 0 h | |
| rw [Nat.zero_add] at hp | |
| have : Module.Projective A (syzygy M n) := | |
| (IsProjective.iff_projective _).mpr (by simpa only [Nat.zero_add, FiniteModule.cat] using hp) | |
| have : Subsingleton (syzygy M (n+1)) := next_subsingleton_of_projective (syzygy M n) | |
| exact cover_subsingleton _ | |
| /-- The actual minimal-resolution term criterion, proved from covers and | |
| short-exact-sequence dimension shifting. -/ | |
| theorem pdLE_iff_term_zero (M : FiniteModule k A) (n : ℕ) : | |
| HasProjectiveDimensionLE M.cat n ↔ Subsingleton (term M (n+1)) := | |
| ⟨term_zero_of_pdLE M n, fun h => pdLT_of_term_zero M (n+1) 0 (by simpa using h)⟩ | |
| end LittleFinitistic.PDMinimal | |
| end | |
| end OAI | |