| import OAI.Algebra.Finitistic.Localization.FiniteProjectiveHomotopy |
| import Mathlib.Algebra.Homology.DerivedCategory.KProjective |
| import Mathlib.CategoryTheory.EssentialImage |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| attribute [local instance] HasDerivedCategory.standard |
| namespace LittleFinitistic.FiniteProjectiveHomotopy |
| universe u v |
| variable (R : Type u) [Ring R] |
| |
| instance underlyingKProjective (T : K.{u,v} R) : CochainComplex.IsKProjective T.obj.as := by |
| obtain ⟨⟨a,b,hb⟩,hp⟩ := T.property |
| have : ∀ n, Projective (T.obj.as.X n) := fun n => (hp n).2 |
| have : CochainComplex.IsStrictlyLE T.obj.as b := by |
| rw [CochainComplex.isStrictlyLE_iff] |
| intro i hi |
| exact hb i (Or.inr hi) |
| exact CochainComplex.isKProjective_of_projective T.obj.as b |
| |
| / |
| def derivedEmbedding : K.{u,v} R ⥤ DerivedCategory (ModuleCat.{v} R) := |
| inclusion R ⋙ DerivedCategory.Qh |
| |
| instance derivedEmbeddingFaithful : (derivedEmbedding.{u,v} R).Faithful where |
| map_injective {M N} {f g} h := by |
| apply (inclusion R).map_injective |
| exact (CochainComplex.IsKProjective.Qh_map_bijective M.obj.as N.obj).injective h |
| |
| instance derivedEmbeddingFull : (derivedEmbedding.{u,v} R).Full where |
| map_surjective {M N} f := by |
| obtain ⟨g,hg⟩ := (CochainComplex.IsKProjective.Qh_map_bijective M.obj.as N.obj).surjective f |
| refine ⟨(inclusion R).preimage g,?_⟩ |
| change DerivedCategory.Qh.map ((inclusion R).map ((inclusion R).preimage g)) = f |
| erw [Functor.map_preimage] |
| exact hg |
| |
| instance derivedEmbeddingShift : (derivedEmbedding.{u,v} R).CommShift ℤ := |
| inferInstanceAs ((inclusion R ⋙ DerivedCategory.Qh).CommShift ℤ) |
| instance derivedEmbeddingTriangulated : (derivedEmbedding.{u,v} R).IsTriangulated := |
| inferInstanceAs (inclusion R ⋙ DerivedCategory.Qh).IsTriangulated |
| |
| / |
| def boundedDerivedEquivalence : K.{u,v} R ≌ (derivedEmbedding.{u,v} R).EssImageSubcategory := |
| (derivedEmbedding R).toEssImage.asEquivalence |
| end LittleFinitistic.FiniteProjectiveHomotopy |
| |
| end |
| |
| end OAI |
| |
