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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 75 76 77 78 79 80 81 82 83 84 85 86 | import OAI.Algebra.AuslanderReiten.HomComplex.FullHomMap
import OAI.Algebra.AuslanderReiten.HomComplex.FullHomProductLinear
import OAI.Algebra.AuslanderReiten.HomComplex.FullHomComposition
namespace OAI
noncomputable section
open CategoryTheory
open CochainComplex CochainComplex.HomComplex
namespace ArExplicit
variable {k : Type*} [CommRing k]
variable {C D : Type*} [Category* C] [Category* D] [Preadditive C] [Preadditive D]
[Linear k C] [Linear k D]
variable (Φ : C ⥤ D) [Φ.Additive] [Φ.Linear k]
variable (K L : CochainComplex C ℤ) (n : ℤ)
theorem mapFullHom_smul (a : k) (x : CohomologyClass K L n) :
mapFullHom Φ K L n (a • x) = a • mapFullHom Φ K L n x := by
obtain ⟨x,rfl⟩ := CohomologyClass.mk_surjective x
rw [← CompleteComparison.fullHom_mk_smul, mapFullHom_mk, mapFullHom_mk,
← CompleteComparison.fullHom_mk_smul]
congr 1
apply Subtype.ext
apply Cochain.ext
intro p q hpq
exact Φ.map_smul a (x.val.v p q hpq)
def mapFullHomLinear : CohomologyClass K L n →ₗ[k]
CohomologyClass ((Φ.mapHomologicalComplex _).obj K)
((Φ.mapHomologicalComplex _).obj L) n where
__ := mapFullHom Φ K L n
map_smul' := mapFullHom_smul Φ K L n
theorem mapFullHom_identity :
mapFullHom Φ K K 0 (CohomologyClass.mk (Cocycle.ofHom (𝟙 K))) =
CohomologyClass.mk (Cocycle.ofHom (𝟙 ((Φ.mapHomologicalComplex _).obj K))) := by
rw [mapFullHom_mk]
congr 1
apply Subtype.ext
change (Cochain.ofHom (𝟙 K)).map Φ =
Cochain.ofHom (𝟙 ((Φ.mapHomologicalComplex _).obj K))
rw [Cochain.map_ofHom, CategoryTheory.Functor.map_id]
variable {K L n}
theorem fullHomPostcomp_smul (f : K ⟶ L) (n : ℤ) (a : k)
(x : CohomologyClass K K n) :
fullHomPostcomp f n (a • x) = a • fullHomPostcomp f n x := by
obtain ⟨x,rfl⟩ := CohomologyClass.mk_surjective x
rw [← CompleteComparison.fullHom_mk_smul, fullHomPostcomp_mk, fullHomPostcomp_mk,
← CompleteComparison.fullHom_mk_smul]
congr 1
apply Subtype.ext
exact Cochain.smul_comp a x.val (Cochain.ofHom f) (add_zero n)
theorem fullHomPrecomp_smul (f : K ⟶ L) (n : ℤ) (a : k)
(x : CohomologyClass L L n) :
fullHomPrecomp f n (a • x) = a • fullHomPrecomp f n x := by
obtain ⟨x,rfl⟩ := CohomologyClass.mk_surjective x
rw [← CompleteComparison.fullHom_mk_smul, fullHomPrecomp_mk, fullHomPrecomp_mk,
← CompleteComparison.fullHom_mk_smul]
congr 1
apply Subtype.ext
exact Cochain.comp_smul (Cochain.ofHom f) a x.val (zero_add n)
theorem fullHomPostcomp_identity (f : K ⟶ L) :
fullHomPostcomp f 0 (CohomologyClass.mk (Cocycle.ofHom (𝟙 K))) =
CohomologyClass.mk (Cocycle.ofHom f) := by
rw [fullHomPostcomp_mk]
congr 1
apply Subtype.ext
exact Cochain.id_comp _
theorem fullHomPrecomp_identity (f : K ⟶ L) :
fullHomPrecomp f 0 (CohomologyClass.mk (Cocycle.ofHom (𝟙 L))) =
CohomologyClass.mk (Cocycle.ofHom f) := by
rw [fullHomPrecomp_mk]
congr 1
apply Subtype.ext
exact Cochain.comp_id _
end ArExplicit
end
end OAI
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