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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 | import OAI.Algebra.AuslanderReiten.HomComplex.FullHomZeroRepresentative
namespace OAI
noncomputable section
open CategoryTheory CochainComplex.HomComplex
namespace ArExplicit
universe u v w
variable {C : Type w} [Category.{v} C] [Preadditive C]
variable {K L M : CochainComplex C ℤ}
theorem fullHomPostcomp_ofHom (f : K ⟶ L) (g : L ⟶ M) :
fullHomPostcomp g 0 (CohomologyClass.mk (Cocycle.ofHom f)) =
CohomologyClass.mk (Cocycle.ofHom (f ≫ g)) := by
rw [← fullHomPostcomp_identity f, fullHomPostcomp_comp, fullHomPostcomp_identity]
variable (k : Type u) [CommRing k] [Linear k C]
include k in
theorem fullHomZero_projection_inverse (f : K ⟶ L) (g : L ⟶ K)
(e : (HomotopyCategory.quotient C _).obj L ≅ (HomotopyCategory.quotient C _).obj K)
(he : (HomotopyCategory.quotient C _).map g = e.hom)
(h : fullHomPostcomp g 0 (CohomologyClass.mk (Cocycle.ofHom f)) =
CohomologyClass.mk (Cocycle.ofHom (𝟙 K))) :
(HomotopyCategory.quotient C _).map f = e.inv := by
rw [fullHomPostcomp_ofHom] at h
have hh := (fullHomZero_class_eq_iff k (f ≫ g) (𝟙 K)).mp h
rw [CategoryTheory.Functor.map_comp, he, CategoryTheory.Functor.map_id] at hh
apply (cancel_mono e.hom).1
exact hh.trans e.inv_hom_id.symm
end ArExplicit
end
end OAI
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