openAI-math-dataset / lean /OAI /Algebra /AuslanderReiten /Resolutions /ResolutionProductCompatibility.lean
| 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 | |