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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 | 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
/-- The canonical derived-category functor on the actual bounded projective source. -/
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
/-- Bounded complexes of f.g. projectives form a full subcategory of the actual derived category. -/
def boundedDerivedEquivalence : K.{u,v} R ≌ (derivedEmbedding.{u,v} R).EssImageSubcategory :=
(derivedEmbedding R).toEssImage.asEquivalence
end LittleFinitistic.FiniteProjectiveHomotopy
end
end OAI
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