feat: the derived category of an abelian category (#11806)

The derived category of an abelian category is defined and it is shown that it is a triangulated category.



Co-authored-by: Joël Riou <[email protected]>

Mathlib.lean

@@ -287,6 +287,7 @@ import Mathlib.Algebra.Homology.BifunctorShift
 import Mathlib.Algebra.Homology.ComplexShape
 import Mathlib.Algebra.Homology.ComplexShapeSigns
 import Mathlib.Algebra.Homology.ConcreteCategory
+import Mathlib.Algebra.Homology.DerivedCategory.Basic
 import Mathlib.Algebra.Homology.DifferentialObject
 import Mathlib.Algebra.Homology.Embedding.Basic
 import Mathlib.Algebra.Homology.Exact

Mathlib/Algebra/Homology/DerivedCategory/Basic.lean

@@ -0,0 +1,188 @@
+/-
+Copyright (c) 2024 Joël Riou. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joël Riou
+-/
+import Mathlib.Algebra.Homology.HomotopyCategory.HomologicalFunctor
+import Mathlib.Algebra.Homology.HomotopyCategory.ShiftSequence
+import Mathlib.Algebra.Homology.HomotopyCategory.Triangulated
+import Mathlib.Algebra.Homology.Localization
+
+/-! # The derived category of an abelian category
+
+In this file, we construct the derived category `DerivedCategory C` of an
+abelian category `C`. It is equipped with a triangulated structure.
+
+The derived category is defined here as the localization of cochain complexes
+indexed by `ℤ` with respect to quasi-isomorphisms: it is a type synonym of
+`HomologicalComplexUpToQuasiIso C (ComplexShape.up ℤ)`. Then, we have a
+localization functor `DerivedCategory.Q : CochainComplex C ℤ ⥤ DerivedCategory C`.
+It was already shown in the file `Algebra.Homology.Localization` that the induced
+functor `DerivedCategory.Qh : HomotopyCategory C (ComplexShape.up ℤ) ⥤ DerivedCategory C`
+is a localization functor with respect to the class of morphisms
+`HomotopyCategory.quasiIso C (ComplexShape.up ℤ)`. In the lemma
+`HomotopyCategory.quasiIso_eq_subcategoryAcyclic_W` we obtain that this class of morphisms
+consists of morphisms whose cone belongs to the triangulated subcategory
+`HomotopyCategory.subcategoryAcyclic C` of acyclic complexes. Then, the triangulated
+structure on `DerivedCategory C` is deduced from the triangulated structure
+on the homotopy category (see file `Algebra.Homology.HomotopyCategory.Triangulated`)
+using the localization theorem for triangulated categories which was obtained
+in the file `CategoryTheory.Localization.Triangulated`.
+
+## Implementation notes
+
+If `C : Type u` and `Category.{v} C`, the constructed localized category of cochain
+complexes with respect to quasi-isomorphisms has morphisms in `Type (max u v)`.
+However, in certain circumstances, it shall be possible to prove that they are `v`-small
+(when `C` is a Grothendieck abelian category (e.g. the category of modules over a ring),
+it should be so by a theorem of Hovey.).
+
+Then, when working with derived categories in mathlib, the user should add the variable
+`[HasDerivedCategory.{w} C]` which is the assumption that there is a chosen derived
+category with morphisms in `Type w`. When derived categories are used in order to
+prove statements which do not involve derived categories, the `HasDerivedCategory.{max u v}`
+instance should be obtained at the beginning of the proof, using the term
+`HasDerivedCategory.standard C`.
+
+## TODO (@joelriou)
+
+- define the induced homological functor `DerivedCategory C ⥤ C`.
+- construct the distinguished triangle associated to a short exact sequence
+of cochain complexes, and compare the associated connecting homomorphism
+with the one defined in `Algebra.Homology.HomologySequence`.
+- refactor the definition of Ext groups using morphisms in the derived category
+(which may be shrunk to the universe `v` at least when `C` has enough projectives
+or enough injectives).
+
+## References
+* [Jean-Louis Verdier, *Des catégories dérivées des catégories abéliennes*][verdier1996]
+* [Mark Hovey, *Model category structures on chain complexes of sheaves*][hovey-2001]
+
+-/
+
+universe w v u
+
+open CategoryTheory Limits
+
+variable (C : Type u) [Category.{v} C] [Abelian C]
+
+namespace HomotopyCategory
+
+/-- The triangulated subcategory of `HomotopyCategory C (ComplexShape.up ℤ)` consisting
+of acyclic complexes. -/
+def subcategoryAcyclic : Triangulated.Subcategory (HomotopyCategory C (ComplexShape.up ℤ)) :=
+  (homologyFunctor C (ComplexShape.up ℤ) 0).homologicalKernel
+
+instance : ClosedUnderIsomorphisms (subcategoryAcyclic C).P := by
+  dsimp [subcategoryAcyclic]
+  infer_instance
+
+variable {C}
+
+lemma mem_subcategoryAcyclic_iff (X : HomotopyCategory C (ComplexShape.up ℤ)) :
+    (subcategoryAcyclic C).P X ↔ ∀ (n : ℤ), IsZero ((homologyFunctor _ _ n).obj X) :=
+  Functor.mem_homologicalKernel_iff _ X
+
+lemma quotient_obj_mem_subcategoryAcyclic_iff_exactAt (K : CochainComplex C ℤ) :
+    (subcategoryAcyclic C).P ((quotient _ _).obj K) ↔ ∀ (n : ℤ), K.ExactAt n := by
+  rw [mem_subcategoryAcyclic_iff]
+  refine forall_congr' (fun n => ?_)
+  simp only [HomologicalComplex.exactAt_iff_isZero_homology]
+  exact ((homologyFunctorFactors C (ComplexShape.up ℤ) n).app K).isZero_iff
+
+variable (C)
+
+lemma quasiIso_eq_subcategoryAcyclic_W :
+    quasiIso C (ComplexShape.up ℤ) = (subcategoryAcyclic C).W := by
+  ext K L f
+  exact ((homologyFunctor C (ComplexShape.up ℤ) 0).mem_homologicalKernel_W_iff f).symm
+
+end HomotopyCategory
+
+/-- The assumption that a localized category for
+`(HomologicalComplex.quasiIso C (ComplexShape.up ℤ))` has been chosen, and that the morphisms
+in this chosen category are in `Type w`. -/
+abbrev HasDerivedCategory := MorphismProperty.HasLocalization.{w}
+  (HomologicalComplex.quasiIso C (ComplexShape.up ℤ))
+
+/-- The derived category obtained using the constructed localized category of cochain complexes
+with respect to quasi-isomorphisms. This should be used only while proving statements
+which do not involve the derived category. -/
+def HasDerivedCategory.standard : HasDerivedCategory.{max u v} C :=
+  MorphismProperty.HasLocalization.standard _
+
+variable [HasDerivedCategory.{w} C]
+
+/-- The derived category of an abelian category. -/
+def DerivedCategory : Type (max u v) := HomologicalComplexUpToQuasiIso C (ComplexShape.up ℤ)
+
+namespace DerivedCategory
+
+instance : Category.{w} (DerivedCategory C) := by
+  dsimp [DerivedCategory]
+  infer_instance
+
+variable {C}
+
+/-- The localization functor `CochainComplex C ℤ ⥤ DerivedCategory C`. -/
+def Q : CochainComplex C ℤ ⥤ DerivedCategory C := HomologicalComplexUpToQuasiIso.Q
+
+instance : (Q (C := C)).IsLocalization
+    (HomologicalComplex.quasiIso C (ComplexShape.up ℤ)) := by
+  dsimp only [Q, DerivedCategory]
+  infer_instance
+
+/-- The localization functor `HomotopyCategory C (ComplexShape.up ℤ) ⥤ DerivedCategory C`. -/
+def Qh : HomotopyCategory C (ComplexShape.up ℤ) ⥤ DerivedCategory C :=
+  HomologicalComplexUpToQuasiIso.Qh
+
+variable (C)
+
+/-- The natural isomorphism `HomotopyCategory.quotient C (ComplexShape.up ℤ) ⋙ Qh ≅ Q`. -/
+def quotientCompQhIso : HomotopyCategory.quotient C (ComplexShape.up ℤ) ⋙ Qh ≅ Q :=
+  HomologicalComplexUpToQuasiIso.quotientCompQhIso C (ComplexShape.up ℤ)
+
+instance : Qh.IsLocalization (HomotopyCategory.quasiIso C (ComplexShape.up ℤ)) := by
+  dsimp [Qh, DerivedCategory]
+  infer_instance
+
+instance : Qh.IsLocalization (HomotopyCategory.subcategoryAcyclic C).W := by
+  rw [← HomotopyCategory.quasiIso_eq_subcategoryAcyclic_W]
+  infer_instance
+
+noncomputable instance : Preadditive (DerivedCategory C) :=
+  Localization.preadditive Qh (HomotopyCategory.subcategoryAcyclic C).W
+
+instance : (Qh (C := C)).Additive :=
+  Localization.functor_additive Qh (HomotopyCategory.subcategoryAcyclic C).W
+
+instance : (Q (C := C)).Additive :=
+  Functor.additive_of_iso (quotientCompQhIso C)
+
+noncomputable instance : HasZeroObject (DerivedCategory C) :=
+  Q.hasZeroObject_of_additive
+
+noncomputable instance : HasShift (DerivedCategory C) ℤ :=
+  HasShift.localized Qh (HomotopyCategory.subcategoryAcyclic C).W ℤ
+
+noncomputable instance : (Qh (C := C)).CommShift ℤ :=
+  Functor.CommShift.localized Qh (HomotopyCategory.subcategoryAcyclic C).W ℤ
+
+instance (n : ℤ) : (shiftFunctor (DerivedCategory C) n).Additive := by
+  rw [Localization.functor_additive_iff
+    Qh (HomotopyCategory.subcategoryAcyclic C).W]
+  exact Functor.additive_of_iso (Qh.commShiftIso n)
+
+noncomputable instance : Pretriangulated (DerivedCategory C) :=
+  Triangulated.Localization.pretriangulated
+    Qh (HomotopyCategory.subcategoryAcyclic C).W
+
+instance : (Qh (C := C)).IsTriangulated :=
+  Triangulated.Localization.isTriangulated_functor
+    Qh (HomotopyCategory.subcategoryAcyclic C).W
+
+noncomputable instance : IsTriangulated (DerivedCategory C) :=
+  Triangulated.Localization.isTriangulated
+    Qh (HomotopyCategory.subcategoryAcyclic C).W
+
+end DerivedCategory

Mathlib/CategoryTheory/Triangulated/HomologicalFunctor.lean

@@ -230,7 +230,7 @@ lemma mem_homologicalKernel_W_iff {X Y : C} (f : X ⟶ Y) :
   simp only [mem_homologicalKernel_iff, h₁, ← h₂, ← h₃]
   constructor
   · intro h n
-    obtain ⟨m, rfl⟩ : ∃ (m : ℤ), n = m + 1 := ⟨n - 1, by omega⟩
+    obtain ⟨m, rfl⟩ : ∃ (m : ℤ), n = m + 1 := ⟨n - 1, by simp⟩
     have := (h (m + 1)).1
     have := (h m).2
     apply isIso_of_mono_of_epi

Mathlib/CategoryTheory/Triangulated/Subcategory.lean

@@ -5,7 +5,8 @@ Authors: Joël Riou
 -/
 import Mathlib.CategoryTheory.ClosedUnderIsomorphisms
 import Mathlib.CategoryTheory.Localization.CalculusOfFractions
-import Mathlib.CategoryTheory.Triangulated.Triangulated
+import Mathlib.CategoryTheory.Localization.Triangulated
+import Mathlib.CategoryTheory.Shift.Localization
 
 /-! # Triangulated subcategories
 
@@ -185,6 +186,11 @@ lemma W.unshift {X₁ X₂ : C} {f : X₁ ⟶ X₂} {n : ℤ} (hf : S.W (f⟦n
   (S.respectsIso_W.arrow_mk_iso_iff
      (Arrow.isoOfNatIso (shiftEquiv C n).unitIso (Arrow.mk f))).2 (hf.shift (-n))
 
+instance : S.W.IsCompatibleWithShift ℤ where
+  condition n := by
+    ext K L f
+    exact ⟨fun hf => hf.unshift, fun hf => hf.shift n⟩
+
 instance [IsTriangulated C] : S.W.IsMultiplicative where
   comp_mem := by
     rw [← isoClosure_W]
@@ -241,6 +247,15 @@ instance [IsTriangulated C] : S.W.HasRightCalculusOfFractions where
       dsimp at eq
       rw [← sub_eq_zero, ← comp_sub, hq, reassoc_of% eq, zero_comp]
 
+instance [IsTriangulated C] : S.W.IsCompatibleWithTriangulation := ⟨by
+  rintro T₁ T₃ mem₁ mem₃ a b ⟨Z₅, g₅, h₅, mem₅, mem₅'⟩ ⟨Z₄, g₄, h₄, mem₄, mem₄'⟩ comm
+  obtain ⟨Z₂, g₂, h₂, mem₂⟩ := distinguished_cocone_triangle (T₁.mor₁ ≫ b)
+  have H := someOctahedron rfl mem₁ mem₄ mem₂
+  have H' := someOctahedron comm.symm mem₅ mem₃ mem₂
+  let φ : T₁ ⟶ T₃ := H.triangleMorphism₁ ≫ H'.triangleMorphism₂
+  exact ⟨φ.hom₃, S.W.comp_mem _ _ (W.mk S H.mem mem₄') (W.mk' S H'.mem mem₅'),
+    by simpa [φ] using φ.comm₂, by simpa [φ] using φ.comm₃⟩⟩
+
 section
 
 variable (T : Triangle C) (hT : T ∈ distTriang C)

docs/references.bib

@@ -1746,6 +1746,22 @@ @Book{            Hofstadter-1979
   year          = "1979"
 }
 
+@Article{         hovey-2001,
+  author        = {Hovey, Mark},
+  title         = {Model category structures on chain complexes of sheaves},
+  journal       = {Trans. Amer. Math. Soc.},
+  fjournal      = {Transactions of the American Mathematical Society},
+  volume        = {353},
+  year          = {2001},
+  number        = {6},
+  pages         = {2441--2457},
+  issn          = {0002-9947,1088-6850},
+  mrclass       = {18F20 (14F05 18E15 18E30 55U35)},
+  mrnumber      = {1814077},
+  doi           = {10.1090/S0002-9947-01-02721-0},
+  url           = {https://doi.org/10.1090/S0002-9947-01-02721-0}
+}
+
 @Misc{            howard,
   title         = {Second Order Elliptic PDE: The Lax-Milgram Theorem},
   url           = {https://www.math.tamu.edu/~phoward/m612/s20/elliptic2.pdf},