File size: 2,846 Bytes
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 | import OAI.Algebra.AuslanderReiten.Resolutions.ResolutionProducts
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Abelian
open CochainComplex CochainComplex.HomComplex
namespace ArExplicit.ResolutionProducts
universe u v
variable {C : Type u} [Category.{v} C] [Abelian C] [HasDerivedCategory C]
theorem conjugate_comp_of_eq {X Y : DerivedCategory C} (e : X ≅ Y)
{n m r : ℤ} (f : ShiftedHom X X n) (g : ShiftedHom X X m) (h : m+n=r) :
conjugate e (f.comp g h) = (conjugate e f).comp (conjugate e g) h := by
obtain rfl : r=n+m := by omega
exact conjugate_comp e f g
omit [HasDerivedCategory C] in
theorem cocycleHom_comp {K L M : CochainComplex C ℤ} {i j l : ℤ} (hl : i+j=l)
(x : Cocycle K L i) (y : Cocycle L M j) (z : Cocycle K M l)
(hz : z.val = x.val.comp y.val hl) :
Cocycle.equivHomShift.symm z =
ShiftedHom.comp (Cocycle.equivHomShift.symm x) (Cocycle.equivHomShift.symm y) (by omega) := by
subst l
have he : z = CompleteComparison.cocycleProduct x y := Subtype.ext hz
subst z
exact CompleteComparison.cocycleHom_product x y
def natCocycleProduct {K L M : CochainComplex C ℤ} {n m : ℕ}
(x : Cocycle K L (n : ℤ)) (y : Cocycle L M (m : ℤ)) :
Cocycle K M ((n+m : ℕ) : ℤ) :=
Cocycle.mk (x.val.comp y.val (by omega)) _ rfl (by
rw [δ_comp x.val y.val (by omega) ((n : ℤ)+1) ((m : ℤ)+1)
(((n+m : ℕ) : ℤ)+1) rfl rfl rfl]
simp)
variable [HasExt.{v} C]
theorem endExt_product {X : C} (P : ProjectiveResolution X) {n m : ℕ}
(x : Cocycle P.cochainComplex P.cochainComplex (n : ℤ))
(y : Cocycle P.cochainComplex P.cochainComplex (m : ℤ)) :
(endExt P x).comp (endExt P y) rfl = endExt P (natCocycleProduct x y) := by
apply Ext.homEquiv.injective
change ((endExt P x).comp (endExt P y) rfl).hom = (endExt P (natCocycleProduct x y)).hom
simp only [Ext.comp_hom, endExt_hom]
rw [cocycleHom_comp (by omega) x y (natCocycleProduct x y) rfl,
ShiftedHom.map_comp]
exact (conjugate_comp_of_eq _ _ _ _).symm
theorem endExt_eq_augmented {X : C} (P : ProjectiveResolution X) {n : ℕ}
(x : Cocycle P.cochainComplex P.cochainComplex (n : ℤ)) :
endExt P x = P.extEquivCohomologyClass.symm
(CohomologyClass.mk (x.postcomp P.π')) := by
apply Ext.homEquiv.injective
change (endExt P x).hom = (P.extEquivCohomologyClass.symm (CohomologyClass.mk (x.postcomp P.π'))).hom
rw [endExt_hom, P.extEquivCohomologyClass_symm_mk_hom]
rw [Cocycle.equivHomShift_symm_postcomp]
rw [← ShiftedHom.comp_mk₀ _ 0 rfl, ShiftedHom.map_comp, ShiftedHom.map_mk₀]
simp only [ShiftedHom.mk₀_comp, ShiftedHom.comp_mk₀]
simp only [conjugate, resolutionIso, Iso.trans_inv, Iso.trans_hom,
asIso_inv, asIso_hom, Functor.map_comp, Category.assoc]
rfl
end ArExplicit.ResolutionProducts
end
end OAI
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