| import OAI.Algebra.AuslanderReiten.Complete.CompleteNullhomotopy |
| import Mathlib.Algebra.Homology.HomotopyCategory |
| import Mathlib.GroupTheory.QuotientGroup.Basic |
|
|
| namespace OAI |
|
|
| open CategoryTheory |
|
|
| namespace ArExplicit.CompleteComparison |
|
|
| universe u |
|
|
| variable {R : Type u} [Ring R] |
| variable {P Q : ChainComplex (ModuleCat.{u} R) ℤ} |
|
|
| def stableCokernel : (P ⟶ Q) →+ StableHom R (Cokernel P 0) (Cokernel Q 0) where |
| toFun F := StableHom.classOf R _ _ (cokernelMap F 0) |
| map_zero' := by simp |
| map_add' F G := by simp |
|
|
| theorem stableCokernel_surjective (hP : TotallyAcyclicComplex P) |
| (hQ : ExactProjectiveComplex Q) : Function.Surjective (stableCokernel (P := P) (Q := Q)) := by |
| intro x |
| obtain ⟨f, rfl⟩ := StableHom.classOf_surjective x |
| obtain ⟨F, rfl⟩ := cokernelMap_surjective hP hQ f |
| exact ⟨F, rfl⟩ |
|
|
| private theorem comparison_kernels (hP : TotallyAcyclicComplex P) |
| (hQ : ExactProjectiveComplex Q) : |
| (Functor.mapAddHom (HomotopyCategory.quotient (ModuleCat.{u} R) (ComplexShape.down ℤ)) |
| (X := P) (Y := Q)).ker = (stableCokernel (P := P) (Q := Q)).ker := by |
| ext F |
| change (HomotopyCategory.quotient _ _).map F = 0 ↔ stableCokernel F = 0 |
| rw [HomotopyCategory.quotient_map_eq_zero_iff] |
| exact nullhomotopy_iff_stable_zero hP hQ F |
|
|
| noncomputable def homotopyCokernelEquiv (hP : TotallyAcyclicComplex P) |
| (hQ : ExactProjectiveComplex Q) : |
| ((HomotopyCategory.quotient (ModuleCat.{u} R) (ComplexShape.down ℤ)).obj P ⟶ |
| (HomotopyCategory.quotient (ModuleCat.{u} R) (ComplexShape.down ℤ)).obj Q) ≃+ |
| StableHom R (Cokernel P 0) (Cokernel Q 0) := |
| (QuotientAddGroup.quotientKerEquivOfSurjective |
| (Functor.mapAddHom (HomotopyCategory.quotient (ModuleCat.{u} R) (ComplexShape.down ℤ))) |
| (Functor.map_surjective _)).symm.trans |
| ((QuotientAddGroup.quotientAddEquivOfEq (comparison_kernels hP hQ)).trans |
| (QuotientAddGroup.quotientKerEquivOfSurjective stableCokernel |
| (stableCokernel_surjective hP hQ))) |
|
|
| @[simp] |
| theorem homotopyCokernelEquiv_map (hP : TotallyAcyclicComplex P) |
| (hQ : ExactProjectiveComplex Q) (F : P ⟶ Q) : |
| homotopyCokernelEquiv hP hQ ((HomotopyCategory.quotient _ _).map F) = |
| StableHom.classOf R _ _ (cokernelMap F 0) := by |
| let f := Functor.mapAddHom |
| (HomotopyCategory.quotient (ModuleCat.{u} R) (ComplexShape.down ℤ)) (X := P) (Y := Q) |
| let e := QuotientAddGroup.quotientKerEquivOfSurjective f (Functor.map_surjective _) |
| have he : e (QuotientAddGroup.mk F) = (HomotopyCategory.quotient _ _).map F := rfl |
| change (QuotientAddGroup.quotientKerEquivOfSurjective stableCokernel |
| (stableCokernel_surjective hP hQ)) |
| (QuotientAddGroup.quotientAddEquivOfEq (comparison_kernels hP hQ) |
| (e.symm ((HomotopyCategory.quotient _ _).map F))) = _ |
| rw [← he, e.symm_apply_apply, QuotientAddGroup.quotientAddEquivOfEq_mk] |
| rfl |
|
|
| theorem homotopyCokernelEquiv_comp {S : ChainComplex (ModuleCat.{u} R) ℤ} |
| (hP : TotallyAcyclicComplex P) (hQ : TotallyAcyclicComplex Q) |
| (hS : ExactProjectiveComplex S) |
| (f : (HomotopyCategory.quotient _ _).obj P ⟶ (HomotopyCategory.quotient _ _).obj Q) |
| (g : (HomotopyCategory.quotient _ _).obj Q ⟶ (HomotopyCategory.quotient _ _).obj S) : |
| homotopyCokernelEquiv hP hS (f ≫ g) = |
| StableHom.comp (homotopyCokernelEquiv hQ hS g) |
| (homotopyCokernelEquiv hP hQ.toExactProjectiveComplex f) := by |
| obtain ⟨F, rfl⟩ := Functor.map_surjective (HomotopyCategory.quotient _ _) f |
| obtain ⟨G, rfl⟩ := Functor.map_surjective (HomotopyCategory.quotient _ _) g |
| rw [← Functor.map_comp, homotopyCokernelEquiv_map, homotopyCokernelEquiv_map, |
| homotopyCokernelEquiv_map, StableHom.comp_classOf, cokernelMap_comp] |
|
|
| end ArExplicit.CompleteComparison |
|
|
| end OAI |
|
|
