| import OAI.Algebra.Finitistic.Localization.BoundedHomotopy |
| import Mathlib.Algebra.Category.ModuleCat.Projective |
| import Mathlib.Algebra.Category.ModuleCat.Biproducts |
| import Mathlib.RingTheory.Finiteness.Prod |
| import Mathlib.RingTheory.Finiteness.Basic |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory Limits |
| namespace LittleFinitistic.ProjectiveComplexes |
| universe u |
| variable (B : Type u) [Ring B] |
|
|
| /-- Right B-modules are left modules over the opposite ring. -/ |
| abbrev RightModule := ModuleCat.{u} Bᵐᵒᵖ |
|
|
| def finiteProjective : ObjectProperty (RightModule B) := |
| fun M => Module.Finite Bᵐᵒᵖ M ∧ Module.Projective Bᵐᵒᵖ M |
| |
| instance : (finiteProjective B).IsClosedUnderIsomorphisms where |
| of_iso e h := by |
| let := h.1 |
| let := h.2 |
| exact ⟨Module.Finite.equiv e.toLinearEquiv, |
| Module.Projective.of_equiv e.toLinearEquiv⟩ |
| |
| instance : (finiteProjective B).ContainsZero where |
| exists_zero := ⟨ModuleCat.of Bᵐᵒᵖ PUnit, ModuleCat.isZero_of_subsingleton _, |
| by constructor <;> infer_instance⟩ |
| |
| instance : (finiteProjective B).IsClosedUnderBinaryProducts where |
| limitsOfShape_le := by |
| rintro X ⟨p⟩ |
| let M := p.diag.obj (.mk .left) |
| let N := p.diag.obj (.mk .right) |
| have hM := p.prop_diag_obj (.mk .left) |
| have hN := p.prop_diag_obj (.mk .right) |
| let := hM.1 |
| let := hM.2 |
| let := hN.1 |
| let := hN.2 |
| have hp : finiteProjective B (ModuleCat.of Bᵐᵒᵖ (M × N)) := |
| ⟨inferInstance, inferInstance⟩ |
| apply (finiteProjective B).prop_of_iso _ hp |
| exact IsLimit.conePointUniqueUpToIso (ModuleCat.binaryProductLimitCone M N).isLimit |
| ((IsLimit.postcomposeHomEquiv (diagramIsoPair p.diag) _).2 p.isLimit) |
| |
| instance : (finiteProjective B).IsClosedUnderFiniteProducts := .mk' |
|
|
| /-- The additive category of finitely generated projective right B-modules. -/ |
| abbrev Proj := (finiteProjective B).FullSubcategory |
| |
| instance : HasBinaryBiproducts (Proj B) := .of_hasBinaryProducts |
|
|
| /-- The actual category of bounded complexes of finitely generated projectives. -/ |
| abbrev Complex := BoundedHomotopy.Complex (Proj B) |
|
|
| /-- Section 3's K = Kᵇ(proj-B), with homotopy classes of chain maps. -/ |
| abbrev K := BoundedHomotopy.K (Proj B) |
|
|
| example : HasBinaryBiproducts (Proj B) := inferInstance |
| example : IsTriangulated (K B) := inferInstance |
|
|
| /-- The passage from bounded projective complexes to their homotopy category. -/ |
| abbrev quotient : Complex B ⥤ K B := BoundedHomotopy.quotient (Proj B) |
|
|
| end LittleFinitistic.ProjectiveComplexes |
|
|
| end |
|
|
| end OAI |
|
|
