| import OAI.Algebra.AuslanderReiten.HomComplex.FullHomProducts |
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| namespace OAI |
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| noncomputable section |
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| namespace ArExplicit |
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| open CategoryTheory |
| open CochainComplex CochainComplex.HomComplex |
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| variable {C D : Type*} [Category* C] [Category* D] [Preadditive C] [Preadditive D] |
| variable (Φ : C ⥤ D) [Φ.Additive] [Φ.Full] [Φ.Faithful] |
| variable (K L : CochainComplex C ℤ) (n : ℤ) |
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| def fullHomCochainEquiv : Cochain K L n ≃+ |
| Cochain ((Φ.mapHomologicalComplex _).obj K) ((Φ.mapHomologicalComplex _).obj L) n where |
| toFun x := x.map Φ |
| invFun y := Cochain.mk fun p q hpq => Φ.preimage (y.v p q hpq) |
| left_inv x := by |
| apply Cochain.ext |
| intro p q hpq |
| exact Φ.preimage_map _ |
| right_inv y := by |
| apply Cochain.ext |
| intro p q hpq |
| exact Φ.map_preimage _ |
| map_add' := fun x y => Cochain.map_add x y Φ |
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| theorem fullHomCochainEquiv_differential (m : ℤ) (x : Cochain K L n) : |
| δ n m (fullHomCochainEquiv Φ K L n x) = fullHomCochainEquiv Φ K L m (δ n m x) := |
| δ_map n m x Φ |
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| def fullHomCocycleMap : Cocycle K L n →+ |
| Cocycle ((Φ.mapHomologicalComplex _).obj K) ((Φ.mapHomologicalComplex _).obj L) n where |
| toFun x := Cocycle.mk (x.1.map Φ) (n + 1) rfl (by |
| rw [δ_map, Cocycle.δ_eq_zero, Cochain.map_zero]) |
| map_zero' := by apply Cocycle.ext; exact Cochain.map_zero K L n Φ |
| map_add' x y := by apply Cocycle.ext; exact Cochain.map_add x.1 y.1 Φ |
| |
| theorem fullHomCocycleMap_bijective : Function.Bijective (fullHomCocycleMap Φ K L n) := by |
| constructor |
| · intro x y h |
| apply Cocycle.ext |
| apply (fullHomCochainEquiv Φ K L n).injective |
| exact congrArg Subtype.val h |
| · intro y |
| let x := (fullHomCochainEquiv Φ K L n).symm y.1 |
| have hx : δ n (n + 1) x = 0 := by |
| apply (fullHomCochainEquiv Φ K L (n + 1)).injective |
| rw [← fullHomCochainEquiv_differential, map_zero] |
| change δ n (n + 1) (fullHomCochainEquiv Φ K L n |
| ((fullHomCochainEquiv Φ K L n).symm y.1)) = 0 |
| rw [AddEquiv.apply_symm_apply, Cocycle.δ_eq_zero] |
| refine ⟨Cocycle.mk x (n + 1) rfl hx, ?_⟩ |
| apply Cocycle.ext |
| exact (fullHomCochainEquiv Φ K L n).apply_symm_apply y.1 |
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| theorem fullHomCocycleMap_boundary (x : Cocycle K L n) : |
| fullHomCocycleMap Φ K L n x ∈ |
| coboundaries ((Φ.mapHomologicalComplex _).obj K) ((Φ.mapHomologicalComplex _).obj L) n ↔ |
| x ∈ coboundaries K L n := by |
| rw [mem_coboundaries_iff _ (n - 1) (by omega), mem_coboundaries_iff _ (n - 1) (by omega)] |
| constructor |
| · rintro ⟨y, hy⟩ |
| refine ⟨(fullHomCochainEquiv Φ K L (n - 1)).symm y, ?_⟩ |
| apply (fullHomCochainEquiv Φ K L n).injective |
| rw [← fullHomCochainEquiv_differential, AddEquiv.apply_symm_apply] |
| exact hy |
| · rintro ⟨y, hy⟩ |
| refine ⟨y.map Φ, ?_⟩ |
| rw [δ_map, hy] |
| rfl |
|
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| def fullHomCohomologyMap : CohomologyClass K L n →+ |
| CohomologyClass ((Φ.mapHomologicalComplex _).obj K) ((Φ.mapHomologicalComplex _).obj L) n := |
| CohomologyClass.descAddMonoidHom |
| ((CohomologyClass.mkAddMonoidHom _ _ _).comp (fullHomCocycleMap Φ K L n)) (by |
| intro x hx |
| exact (CohomologyClass.mk_eq_zero_iff _).mpr |
| ((fullHomCocycleMap_boundary Φ K L n x).mpr hx)) |
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| @[simp] theorem fullHomCohomologyMap_mk (x : Cocycle K L n) : |
| fullHomCohomologyMap Φ K L n (CohomologyClass.mk x) = |
| CohomologyClass.mk (fullHomCocycleMap Φ K L n x) := rfl |
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| theorem fullHomCohomologyMap_bijective : Function.Bijective (fullHomCohomologyMap Φ K L n) := by |
| constructor |
| · apply (injective_iff_map_eq_zero _).mpr |
| intro x hx |
| obtain ⟨y, rfl⟩ := CohomologyClass.mk_surjective x |
| exact (CohomologyClass.mk_eq_zero_iff y).mpr |
| ((fullHomCocycleMap_boundary Φ K L n y).mp |
| ((CohomologyClass.mk_eq_zero_iff _).mp hx)) |
| · intro y |
| obtain ⟨y, rfl⟩ := CohomologyClass.mk_surjective y |
| obtain ⟨x, rfl⟩ := (fullHomCocycleMap_bijective Φ K L n).surjective y |
| exact ⟨CohomologyClass.mk x, rfl⟩ |
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| def fullHomCohomologyEquiv : CohomologyClass K L n ≃+ |
| CohomologyClass ((Φ.mapHomologicalComplex _).obj K) ((Φ.mapHomologicalComplex _).obj L) n := |
| AddEquiv.ofBijective (fullHomCohomologyMap Φ K L n) (fullHomCohomologyMap_bijective Φ K L n) |
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| variable {K L n} |
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| theorem fullHomCohomologyMap_product {M : CochainComplex C ℤ} {m : ℤ} |
| (x : CohomologyClass K L n) (y : CohomologyClass L M m) : |
| fullHomCohomologyMap Φ K M (n + m) (CompleteComparison.fullHomProduct x y) = |
| CompleteComparison.fullHomProduct (fullHomCohomologyMap Φ K L n x) |
| (fullHomCohomologyMap Φ L M m y) := by |
| obtain ⟨x, rfl⟩ := CohomologyClass.mk_surjective x |
| obtain ⟨y, rfl⟩ := CohomologyClass.mk_surjective y |
| simp only [CompleteComparison.fullHomProduct_mk, fullHomCohomologyMap_mk] |
| congr 1 |
| apply Cocycle.ext |
| apply Cochain.map_comp |
| rfl |
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| end ArExplicit |
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| end |
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| end OAI |
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