| import OAI.Algebra.AuslanderReiten.Triangular.TriangularCrossHom | |
| import OAI.Algebra.AuslanderReiten.HomComplex.FullHomMap | |
| import OAI.Algebra.AuslanderReiten.Cones.ConeComparison | |
| namespace OAI | |
| noncomputable section | |
| namespace ArExplicit.TriangularInduction | |
| open CategoryTheory CochainComplex.HomComplex | |
| universe u | |
| variable {k R F : Type u} [Field k] [Ring R] [Algebra k R] | |
| variable [AddCommGroup F] [Module k F] [Module R F] [Module Rᵐᵒᵖ F] | |
| variable [SMulCommClass k R F] [SMulCommClass R Rᵐᵒᵖ F] | |
| variable [IsScalarTower k R F] [IsScalarTower k Rᵐᵒᵖ F] [SMulCommClass k Rᵐᵒᵖ F] | |
| attribute [local instance] moduleCatBaseModule moduleCatBaseTower | |
| variable {P : CochainComplex (ModuleCat.{u} R) ℤ} | |
| variable (v : P ⟶ tensorCochain (k := k) (F := F) P) (n : ℤ) | |
| @[simp] theorem crossCohomologyEquiv_mk | |
| (x : Cocycle P (tensorCochain (k := k) (F := F) P) n) : | |
| crossCohomologyEquiv P P n (CohomologyClass.mk x) = | |
| CohomologyClass.mk ((crossFullHomEquivalence P P).cocycle n x) := rfl | |
| theorem crossCohomology_postcomp (x : CohomologyClass P P n) : | |
| crossCohomologyEquiv P P n (fullHomPostcomp v n x) = | |
| fullHomPostcomp (crossChainMap v) n (mapFullHom (jFunctor k R F) P P n x) := by | |
| obtain ⟨x, rfl⟩ := CohomologyClass.mk_surjective x | |
| rw [fullHomPostcomp_mk, crossCohomologyEquiv_mk, mapFullHom_mk, fullHomPostcomp_mk] | |
| congr 1 | |
| apply Cocycle.ext | |
| change crossCochainEquiv P P n (x.1.comp (Cochain.ofHom v) (add_zero n)) = | |
| (x.1.map (jFunctor k R F)).comp (Cochain.ofHom (crossChainMap v)) (add_zero n) | |
| apply Cochain.ext | |
| intro i j hij | |
| rw [crossCochainEquiv_v, | |
| Cochain.comp_v x.1 (Cochain.ofHom v) (add_zero n) i j j hij (add_zero j), | |
| Cochain.comp_v (x.1.map (jFunctor k R F)) (Cochain.ofHom (crossChainMap v)) | |
| (add_zero n) i j j hij (add_zero j)] | |
| simp only [Cochain.ofHom_v, Cochain.map_v] | |
| exact crossHomAdd_source _ _ | |
| theorem crossCohomology_precomp (x : CohomologyClass P P n) : | |
| crossCohomologyEquiv P P n (fullHomPrecomp v n | |
| (mapFullHom (BalancedTensor.functor (k := k) (R := R) (S := R) (F := F)) P P n x)) = | |
| fullHomPrecomp (crossChainMap v) n (mapFullHom (eFunctor k R F) P P n x) := by | |
| obtain ⟨x, rfl⟩ := CohomologyClass.mk_surjective x | |
| rw [mapFullHom_mk, fullHomPrecomp_mk, crossCohomologyEquiv_mk, | |
| mapFullHom_mk, fullHomPrecomp_mk] | |
| congr 1 | |
| apply Cocycle.ext | |
| change crossCochainEquiv P P n ((Cochain.ofHom v).comp | |
| (x.1.map (BalancedTensor.functor (k := k) (R := R) (S := R) (F := F))) (zero_add n)) = | |
| (Cochain.ofHom (crossChainMap v)).comp (x.1.map (eFunctor k R F)) (zero_add n) | |
| apply Cochain.ext | |
| intro i j hij | |
| rw [crossCochainEquiv_v, | |
| Cochain.comp_v (Cochain.ofHom v) | |
| (x.1.map (BalancedTensor.functor (k := k) (R := R) (S := R) (F := F))) | |
| (zero_add n) i i j (add_zero i) hij, | |
| Cochain.comp_v (Cochain.ofHom (crossChainMap v)) (x.1.map (eFunctor k R F)) | |
| (zero_add n) i i j (add_zero i) hij] | |
| simp only [Cochain.ofHom_v, Cochain.map_v] | |
| exact crossHomAdd_target _ _ | |
| def tensorComparison : (CohomologyClass P P n × CohomologyClass P P n) →+ | |
| CohomologyClass P (tensorCochain (k := k) (F := F) P) n where | |
| toFun x := fullHomPostcomp v n x.2 + (n+1).negOnePow • fullHomPrecomp v n | |
| (mapFullHom (BalancedTensor.functor (k := k) (R := R) (S := R) (F := F)) P P n x.1) | |
| map_zero' := by simp | |
| map_add' x y := by simp [smul_add]; abel | |
| theorem cross_tensorComparison (a b : CohomologyClass P P n) : | |
| crossCohomologyEquiv P P n (tensorComparison v n (a,b)) = | |
| coneComparison (crossChainMap v) n | |
| (fullHomCohomologyEquiv (eFunctor k R F) P P n a, | |
| fullHomCohomologyEquiv (jFunctor k R F) P P n b) := by | |
| change crossCohomologyEquiv P P n (fullHomPostcomp v n b + | |
| (n+1).negOnePow • fullHomPrecomp v n | |
| (mapFullHom (BalancedTensor.functor (k := k) (R := R) (S := R) (F := F)) P P n a)) = _ | |
| rw [map_add] | |
| simp only [Units.smul_def, map_zsmul, crossCohomology_postcomp, crossCohomology_precomp] | |
| rfl | |
| end ArExplicit.TriangularInduction | |
| end | |
| end OAI | |