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216fb31 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 | 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
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