| import OAI.Algebra.AuslanderReiten.Evaluation.EvaluatedCoverCoordinates | |
| import OAI.Algebra.AuslanderReiten.Evaluation.EvaluatedBaseShift | |
| import OAI.Algebra.AuslanderReiten.Evaluation.EvaluatedQuotientCubic | |
| import OAI.Algebra.AuslanderReiten.HomComplex.FullHomShiftedMaps | |
| import OAI.Algebra.AuslanderReiten.HomComplex.TensorFullHomZero | |
| namespace OAI | |
| noncomputable section | |
| namespace ArExplicit.T | |
| open CategoryTheory | |
| open CochainComplex CochainComplex.HomComplex CompleteComparison | |
| open scoped TensorProduct ModuleCat.Algebra | |
| universe u | |
| variable {k : Type u} [Field k] [CharP k 2] (q : k) | |
| local notation "R₀" => A k q | |
| local notation "X₀" => A.simpleFiniteModule q | |
| local notation "D₀" => bimoduleTwoConeData q | |
| local notation "hf₀" => bimodule_term_finite_k q | |
| local notation "τ₀" => AFrobeniusTrace k q | |
| local notation "Q₀" => HomotopyCategory.quotient (ModuleCat R₀) (ComplexShape.up ℤ) | |
| abbrev simpleCoverClass (n : ℕ) := | |
| (TwoCone.evaluatedCoverTower D₀ hf₀ X₀ (bimodule_exact q) | |
| (baseTail_right_projective q) (bimodule_term_projective q)).coverClassFrom τ₀ X₀ | |
| (CohomologyClass.mk (Cocycle.ofHom | |
| (FiniteLeftModule.resolutionMap τ₀ (evaluatedBaseZeroEquiv q).symm.toLinearMap))) n | |
| theorem simpleCoverClass_hom (n : ℕ) : | |
| fullHomShiftedHomotopyEquiv k _ _ (n : ℤ) (simpleCoverClass q n) = | |
| (simpleToEvaluatedBaseShiftIso q n).hom := | |
| TwoCone.evaluatedCoverTower_coverClassFrom_hom D₀ hf₀ X₀ (bimodule_exact q) | |
| (baseTail_right_projective q) (bimodule_term_projective q) τ₀ X₀ | |
| (FiniteLeftModule.resolutionMap τ₀ (evaluatedBaseZeroEquiv q).symm.toLinearMap) n | |
| def firstCubicBack : (evaluatedBaseModule q 3).carrier →ₗ[R₀] (X₀).carrier := | |
| (evaluatedBaseZeroEquiv q).toLinearMap.comp | |
| (FiniteLeftModule.evaluationMap X₀ | |
| (M := TwoCone.baseCokernelModule D₀ hf₀ 3) | |
| (N := TwoCone.baseCokernelModule D₀ hf₀ 0) (TwoCone.firstCokernelMap D₀ 0)) | |
| def secondCubicBack : (evaluatedBaseModule q 3).carrier →ₗ[R₀] (X₀).carrier := | |
| (evaluatedBaseZeroEquiv q).toLinearMap.comp | |
| (FiniteLeftModule.evaluationMap X₀ | |
| (M := TwoCone.baseCokernelModule D₀ hf₀ 3) | |
| (N := TwoCone.baseCokernelModule D₀ hf₀ 0) (TwoCone.secondCokernelMap D₀ 0)) | |
| def firstCoverClass : CohomologyClass (A.simpleComplete q) (A.simpleComplete q) 3 := | |
| fullHomPostcomp (FiniteLeftModule.resolutionMap τ₀ (firstCubicBack q)) 3 (simpleCoverClass q 3) | |
| def secondCoverClass : CohomologyClass (A.simpleComplete q) (A.simpleComplete q) 3 := | |
| fullHomPostcomp (FiniteLeftModule.resolutionMap τ₀ (secondCubicBack q)) 3 (simpleCoverClass q 3) | |
| theorem firstCoverClass_hom : | |
| fullHomShiftedHomotopyEquiv k _ _ 3 (firstCoverClass q) = | |
| (simpleToEvaluatedBaseShiftIso q 3).hom ≫ | |
| (TwoCone.evaluatedFirstCokernel D₀ hf₀ X₀ τ₀ 0 ≫ (evaluatedBaseZeroResolutionIso q).hom)⟦(3 : ℤ)⟧' := by | |
| erw [firstCoverClass, fullHomShiftedHomotopyEquiv_postcomp, simpleCoverClass_hom] | |
| congr 1 | |
| apply congrArg (fun f => (CategoryTheory.shiftFunctor | |
| (HomotopyCategory (ModuleCat R₀) (ComplexShape.up ℤ)) (3 : ℤ)).map f) | |
| change (Q₀).map (FiniteLeftModule.resolutionMap τ₀ (firstCubicBack q)) = _ | |
| exact FiniteLeftModule.resolutionMap_comp τ₀ _ _ | |
| theorem secondCoverClass_hom : | |
| fullHomShiftedHomotopyEquiv k _ _ 3 (secondCoverClass q) = | |
| (simpleToEvaluatedBaseShiftIso q 3).hom ≫ | |
| (TwoCone.evaluatedSecondCokernel D₀ hf₀ X₀ τ₀ 0 ≫ (evaluatedBaseZeroResolutionIso q).hom)⟦(3 : ℤ)⟧' := by | |
| erw [secondCoverClass, fullHomShiftedHomotopyEquiv_postcomp, simpleCoverClass_hom] | |
| congr 1 | |
| apply congrArg (fun f => (CategoryTheory.shiftFunctor | |
| (HomotopyCategory (ModuleCat R₀) (ComplexShape.up ℤ)) (3 : ℤ)).map f) | |
| change (Q₀).map (FiniteLeftModule.resolutionMap τ₀ (secondCubicBack q)) = _ | |
| exact FiniteLeftModule.resolutionMap_comp τ₀ _ _ | |
| end ArExplicit.T | |
| end | |
| end OAI | |