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2c7d975 | 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 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 | import OAI.Algebra.AuslanderReiten.Resolutions.MapResolutionCochain
import OAI.Algebra.AuslanderReiten.Resolutions.ResolutionExtAction
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Abelian CategoryTheory.Localization
open CochainComplex CochainComplex.HomComplex HomologicalComplex
namespace ArExplicit.ResolutionProducts
universe u v u' v'
variable {C : Type u} [Category.{v} C] [Abelian C] [HasExt.{v} C]
variable {D : Type u'} [Category.{v'} D] [Abelian D] [HasExt.{v'} D]
variable (F : C ⥤ D) [F.Additive] [CategoryTheory.Limits.PreservesFiniteLimits F]
[CategoryTheory.Limits.PreservesFiniteColimits F] [F.PreservesProjectiveObjects]
variable {X Y : C} (P : ProjectiveResolution X) {n : ℕ}
def mappedResolutionCocycle
(x : Cocycle P.cochainComplex ((CochainComplex.singleFunctor C 0).obj Y) (n : ℤ)) :
Cocycle (mapResolution F P).cochainComplex
((CochainComplex.singleFunctor D 0).obj (F.obj Y)) (n : ℤ) :=
Cocycle.equivHomShift ((mapResolutionCochainIso F P).hom ≫
ShiftedHom.map (Cocycle.equivHomShift.symm x) (F.mapHomologicalComplex (.up ℤ)) ≫
((F.mapCochainComplexSingleFunctor 0).hom.app Y)⟦(n : ℤ)⟧')
theorem precomp_extOfCocycle
(x : Cocycle P.cochainComplex ((CochainComplex.singleFunctor C 0).obj Y) (n : ℤ)) :
(SmallShiftedHom.mk₀ (quasiIso C (.up ℤ)) 0 rfl P.π').comp
(P.extEquivCohomologyClass.symm (CohomologyClass.mk x)) (add_zero (n : ℤ)) =
SmallShiftedHom.mk (quasiIso C (.up ℤ)) (Cocycle.equivHomShift.symm x) := by
have hπ : quasiIso C (.up ℤ) P.π' := by rw [mem_quasiIso_iff]; infer_instance
change (SmallShiftedHom.precompEquiv P.π' hπ)
((SmallShiftedHom.precompEquiv P.π' hπ).symm
(CohomologyClass.equivOfIsKProjective (CohomologyClass.mk x))) = _
rw [Equiv.apply_symm_apply]
rfl
theorem map_extOfCocycle
(x : Cocycle P.cochainComplex ((CochainComplex.singleFunctor C 0).obj Y) (n : ℤ)) :
(P.extEquivCohomologyClass.symm (CohomologyClass.mk x)).mapExactFunctor F =
(mapResolution F P).extEquivCohomologyClass.symm
(CohomologyClass.mk (mappedResolutionCocycle F P x)) := by
let Φ := F.mapHomologicalComplexUpToQuasiIsoLocalizerMorphism (.up ℤ)
let eP : (F.mapHomologicalComplex (.up ℤ)).obj P.cochainComplex ≅
(mapResolution F P).cochainComplex := (mapResolutionCochainIso F P).symm
let eX : (F.mapHomologicalComplex (.up ℤ)).obj ((CochainComplex.singleFunctor C 0).obj X) ≅
(CochainComplex.singleFunctor D 0).obj (F.obj X) := (F.mapCochainComplexSingleFunctor 0).app X
let eY : (F.mapHomologicalComplex (.up ℤ)).obj ((CochainComplex.singleFunctor C 0).obj Y) ≅
(CochainComplex.singleFunctor D 0).obj (F.obj Y) := (F.mapCochainComplexSingleFunctor 0).app Y
have h := congrArg (fun z => Φ.smallShiftedHomMap eP eY z) (precomp_extOfCocycle P x)
erw [Φ.smallShiftedHomMap_comp eP eX eY,
Φ.smallShiftedHomMap_mk₀ eP eX, Φ.smallShiftedHomMap_mk eP eY] at h
have hπ : eP.inv ≫ Φ.functor.map P.π' ≫ eX.hom = (mapResolution F P).π' :=
mapResolutionCochainIso_π F P
erw [hπ] at h
apply (SmallShiftedHom.precompEquiv (mapResolution F P).π'
(by rw [mem_quasiIso_iff]; infer_instance)).injective
change (SmallShiftedHom.mk₀ (quasiIso D (.up ℤ)) 0 rfl (mapResolution F P).π').comp
_ (add_zero (n : ℤ)) =
(SmallShiftedHom.mk₀ (quasiIso D (.up ℤ)) 0 rfl (mapResolution F P).π').comp
((mapResolution F P).extEquivCohomologyClass.symm
(CohomologyClass.mk (mappedResolutionCocycle F P x))) (add_zero (n : ℤ))
have hpost := precomp_extOfCocycle (mapResolution F P) (mappedResolutionCocycle F P x)
refine h.trans (Eq.trans ?_ hpost.symm)
apply congrArg (SmallShiftedHom.mk (quasiIso D (.up ℤ)))
rw [mappedResolutionCocycle, AddEquiv.symm_apply_apply]
simp only [ShiftedHom.mk₀_comp, ShiftedHom.comp_mk₀]
rfl
end ArExplicit.ResolutionProducts
end
end OAI
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