| import Mathlib.Algebra.Homology.HomotopyCategory.Plus |
| import Mathlib.Algebra.Category.ModuleCat.Projective |
| import Mathlib.Algebra.Category.ModuleCat.Biproducts |
| import Mathlib.RingTheory.Finiteness.Basic |
| import Mathlib.RingTheory.Finiteness.Prod |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits CategoryTheory.Pretriangulated |
| open ZeroObject |
| namespace LittleFinitistic.FiniteProjectiveHomotopy |
| universe u v |
| variable (R : Type u) [Ring R] |
|
|
| /-- Actual bounded complexes of finitely generated projective right/left modules, |
| according to the chosen scalar ring R. -/ |
| def ComplexProperty (K : CochainComplex (ModuleCat.{v} R) ℤ) : Prop := |
| (∃ a b : ℤ, ∀ i, i < a ∨ b < i → IsZero (K.X i)) ∧ |
| ∀ i, Module.Finite R (K.X i) ∧ Projective (K.X i) |
|
|
| def objects : ObjectProperty (HomotopyCategory (ModuleCat.{v} R) (.up ℤ)) := |
| fun T => ComplexProperty R T.as |
|
|
| @[simp] theorem objects_quotient (K : CochainComplex (ModuleCat.{v} R) ℤ) : |
| objects R ((HomotopyCategory.quotient _ _).obj K) ↔ ComplexProperty R K := Iff.rfl |
|
|
| lemma finite_of_iso {M N : ModuleCat.{v} R} (e : M ≅ N) [Module.Finite R M] : Module.Finite R N := |
| Module.Finite.of_surjective e.toLinearEquiv.toLinearMap e.toLinearEquiv.surjective |
|
|
| lemma finite_biprod (M N : ModuleCat.{v} R) [Module.Finite R M] [Module.Finite R N] : |
| Module.Finite R ((M ⊞ N) : ModuleCat R) := |
| finite_of_iso R (ModuleCat.biprodIsoProd M N).symm |
|
|
| lemma mappingCone_property {K L : CochainComplex (ModuleCat.{v} R) ℤ} (f : K ⟶ L) |
| (hK : ComplexProperty R K) (hL : ComplexProperty R L) : |
| ComplexProperty R (CochainComplex.mappingCone f) := by |
| obtain ⟨⟨a,b,hbK⟩,hpK⟩ := hK |
| obtain ⟨⟨c,d,hbL⟩,hpL⟩ := hL |
| constructor |
| · refine ⟨min a c - 1,max b d,fun i hi => ?_⟩ |
| rw [CochainComplex.mappingCone.isZero_X_iff] |
| exact ⟨hbK (i+1) (by omega),hbL i (by omega)⟩ |
| · intro i |
| have := (hpK (i+1)).1 |
| have := (hpK (i+1)).2 |
| have := (hpL i).1 |
| have := (hpL i).2 |
| let e := HomologicalComplex.homotopyCofiber.XIsoBiprod f i (i+1) rfl |
| have := finite_biprod R (K.X (i+1)) (L.X i) |
| exact ⟨finite_of_iso R e.symm, Projective.of_iso e.symm inferInstance⟩ |
|
|
| lemma finite_projective_of_isZero (M : ModuleCat.{v} R) (h : IsZero M) : |
| Module.Finite R M ∧ Projective M := by |
| have := ModuleCat.subsingleton_of_isZero h |
| exact ⟨inferInstance,h.projective⟩ |
|
|
| instance objectsZero : (objects.{u,v} R).ContainsZero where |
| exists_zero := ⟨(HomotopyCategory.quotient _ _).obj 0, |
| (HomotopyCategory.quotient _ _).map_isZero (isZero_zero _), by |
| have hz (i : ℤ) : IsZero ((0 : CochainComplex (ModuleCat.{v} R) ℤ).X i) := |
| (HomologicalComplex.eval _ _ i).map_isZero (isZero_zero _) |
| exact ⟨⟨0,0,fun i _ => hz i⟩,fun i => finite_projective_of_isZero R _ (hz i)⟩⟩ |
|
|
| instance objectsShift : (objects.{u,v} R).IsStableUnderShift ℤ where |
| isStableUnderShiftBy n := |
| { le_shift T hT := by |
| obtain ⟨K,rfl⟩ := T.quotient_obj_surjective |
| rw [ObjectProperty.prop_shift_iff, HomotopyCategory.shift_quotient_obj] |
| change ComplexProperty R (K⟦n⟧) |
| obtain ⟨⟨a,b,hb⟩,hp⟩ := hT |
| constructor |
| · refine ⟨a-n,b-n,fun i hi => ?_⟩ |
| change IsZero (K.X (i+n)) |
| exact hb (i+n) (by omega) |
| · intro i |
| exact hp (i+n) } |
|
|
| instance objectsClosed₃ : (objects.{u,v} R).IsTriangulatedClosed₃ where |
| ext₃' T hT h₁ h₂ := by |
| obtain ⟨f : T.obj₁.as ⟶ T.obj₂.as,hf⟩ := (HomotopyCategory.quotient _ _).map_surjective T.mor₁ |
| refine ⟨(HomotopyCategory.quotient _ _).obj (CochainComplex.mappingCone f), |
| mappingCone_property R f h₁ h₂, ⟨?_⟩⟩ |
| exact Triangle.π₃.mapIso (isoTriangleOfIso₁₂ T _ hT |
| (HomotopyCategory.mappingCone_triangleh_distinguished f) |
| (Iso.refl _) (Iso.refl _) (by |
| change T.mor₁ ≫ 𝟙 _ = 𝟙 _ ≫ (HomotopyCategory.quotient _ _).map f |
| rw [Category.comp_id,Category.id_comp,hf])) |
|
|
| instance objectsTriangulated : (objects.{u,v} R).IsTriangulated where |
| toIsTriangulatedClosed₂ := .of_isTriangulatedClosed₃ |
|
|
| /-- K^b(proj-R), as a full subcategory with actual bounded f.g. projective representatives. -/ |
| abbrev K := (objects.{u,v} R).FullSubcategory |
|
|
| /-- Its full and faithful inclusion into the ambient homotopy category. -/ |
| abbrev inclusion : K.{u,v} R ⥤ HomotopyCategory (ModuleCat.{v} R) (.up ℤ) := (objects R).ι |
| end LittleFinitistic.FiniteProjectiveHomotopy |
|
|
| end |
|
|
| end OAI |
|
|
