| import Mathlib.Algebra.Homology.ShortComplex.ModuleCat |
| import OAI.Algebra.Finitistic.Encoding.QuadraticEncoding |
| import OAI.Algebra.Finitistic.Rectification.RectificationAcyclic |
|
|
| namespace OAI |
|
|
| namespace LittleFinitistic.RelationSequence |
| open CategoryTheory CategoryTheory.Limits |
| open QuadraticEncoding |
|
|
| variable {k : Type*} [Field k] {d t : ℕ} (P : Relations k d t) |
|
|
| /-- The actual length-two relation subspace, all paths, and their quotient. -/ |
| noncomputable def sequence : ShortComplex (ModuleCat k) where |
| X₁ := ModuleCat.of k (relationSpace P) |
| X₂ := ModuleCat.of k (Paths k d) |
| X₃ := ModuleCat.of k (Paths k d ⧸ relationSpace P) |
| f := ModuleCat.ofHom |
| { toFun := fun x => x.val |
| map_add' := by intros; rfl |
| map_smul' := by intros; rfl } |
| g := ModuleCat.ofHom |
| { toFun := (relationSpace P).mkQ |
| map_add' := (relationSpace P).mkQ.map_add |
| map_smul' := (relationSpace P).mkQ.map_smul } |
| zero := by ext x; exact (Submodule.Quotient.mk_eq_zero _).2 x.property |
| |
| /-- No additional relation appears: the path quotient sequence is genuinely exact. -/ |
| theorem sequence_shortExact : (sequence P).ShortExact := by |
| apply ModuleCat.shortComplex_shortExact |
| · intro x |
| change (relationSpace P).mkQ x = 0 ↔ ∃ y : relationSpace P, (y : Paths k d) = x |
| constructor |
| · intro hx |
| exact ⟨⟨x,(Submodule.Quotient.mk_eq_zero _).1 hx⟩,rfl⟩ |
| · rintro ⟨y,hy⟩ |
| exact (Submodule.Quotient.mk_eq_zero _).2 (hy ▸ y.property) |
| · exact (relationSpace P).injective_subtype |
| · exact (relationSpace P).mkQ_surjective |
|
|
| /-- Over the base field this exact sequence has an actual splitting. -/ |
| noncomputable def sequenceSplitting : (sequence P).Splitting := |
| (sequence_shortExact P).splittingOfProjective |
|
|
| /-- The finite relation basis used by the manuscript exists for this exact subspace. -/ |
| noncomputable def relationBasis : Module.Basis (Fin (Module.finrank k (relationSpace P))) k |
| (relationSpace P) := by |
| letI : Module.Free k (relationSpace P) := Module.Free.of_divisionRing k (relationSpace P) |
| exact Module.finBasis k (relationSpace P) |
|
|
| /-- Tensoring the split path sequence with a complex, or applying any additive |
| realization functor, gives a short exact sequence of actual complexes. -/ |
| theorem realized_sequence_shortExact {C : Type*} [Category C] [Abelian C] |
| (F : ModuleCat k ⥤ CochainComplex C ℤ) [F.Additive] : |
| ((sequence P).map F).ShortExact := |
| ((sequenceSplitting P).map F).shortExact |
|
|
| /-- The (0,2) associated graded total is acyclic after such a realization. -/ |
| theorem realized_sequence_total_acyclic {C : Type*} [Category C] [Abelian C] |
| (F : ModuleCat k ⥤ CochainComplex C ℤ) [F.Additive] : |
| (RectificationTotal.total ((sequence P).map F).g ((sequence P).map F).f |
| (Homotopy.ofEq ((sequence P).map F).zero)).Acyclic := |
| RectificationAcyclic.short_exact_total_acyclic _ (realized_sequence_shortExact P F) |
| end LittleFinitistic.RelationSequence |
|
|
| end OAI |
|
|
