| import OAI.Algebra.AuslanderReiten.Triangular.TriangularCochain |
| import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorCategory |
| import OAI.Algebra.AuslanderReiten.HomComplex.FullHomTransport |
|
|
| namespace OAI |
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|
| noncomputable section |
|
|
| namespace ArExplicit.TriangularInduction |
|
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| open CategoryTheory CochainComplex.HomComplex |
|
|
| universe u |
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| 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 |
|
|
| def crossHomAdd (M N : ModuleCat.{u} R) : |
| (M ⟶ (BalancedTensor.functor (k := k) (R := R) (S := R) (F := F)).obj N) →+ |
| ((jFunctor k R F).obj M ⟶ (eFunctor k R F).obj N) where |
| toFun f := ModuleCat.ofHom (jHom f.hom) |
| map_zero' := by |
| apply ModuleCat.hom_ext |
| apply jHomEquiv.injective |
| exact map_zero jHomEquiv |>.symm |
| map_add' f g := by |
| apply ModuleCat.hom_ext |
| apply jHomEquiv.injective |
| ext z |
| rfl |
|
|
| theorem crossHomAdd_source {M N L : ModuleCat.{u} R} (f : M ⟶ N) |
| (g : N ⟶ (BalancedTensor.functor (k := k) (R := R) (S := R) (F := F)).obj L) : |
| crossHomAdd M L (f ≫ g) = (jFunctor k R F).map f ≫ crossHomAdd N L g := by |
| apply ModuleCat.hom_ext |
| apply jHomEquiv.injective |
| rfl |
|
|
| theorem crossHomAdd_target {M N L : ModuleCat.{u} R} |
| (f : M ⟶ (BalancedTensor.functor (k := k) (R := R) (S := R) (F := F)).obj N) |
| (g : N ⟶ L) : |
| crossHomAdd M L (f ≫ (BalancedTensor.functor (k := k) (R := R) (S := R) (F := F)).map g) = |
| crossHomAdd M N f ≫ (eFunctor k R F).map g := by |
| apply ModuleCat.hom_ext |
| apply jHomEquiv.injective |
| rfl |
|
|
| abbrev tensorCochain (P : CochainComplex (ModuleCat.{u} R) ℤ) := |
| ((BalancedTensor.functor (k := k) (R := R) (S := R) (F := F)).mapHomologicalComplex _).obj P |
|
|
| variable (P Q : CochainComplex (ModuleCat.{u} R) ℤ) (n : ℤ) |
|
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| def crossCochainEquiv : Cochain P (tensorCochain (k := k) (F := F) Q) n ≃+ |
| Cochain (jCochain (k := k) (F := F) P) (eCochain (k := k) (F := F) Q) n where |
| toFun x := Cochain.mk fun p q hpq => crossHomAdd (P.X p) (Q.X q) (x.v p q hpq) |
| invFun x := Cochain.mk fun p q hpq => ModuleCat.ofHom (TriangularModule.bottomMap (x.v p q hpq).hom) |
| left_inv x := by |
| apply Cochain.ext |
| intro p q hpq |
| rfl |
| right_inv x := by |
| apply Cochain.ext |
| intro p q hpq |
| apply ModuleCat.hom_ext |
| exact jHom_bottomMap _ |
| map_add' x y := by |
| apply Cochain.ext |
| intro p q hpq |
| exact map_add (crossHomAdd (P.X p) (Q.X q)) (x.v p q hpq) (y.v p q hpq) |
|
|
| @[simp] theorem crossCochainEquiv_v |
| (x : Cochain P (tensorCochain (k := k) (F := F) Q) n) (p q : ℤ) (hpq : p+n=q) : |
| (crossCochainEquiv P Q n x).v p q hpq = crossHomAdd (P.X p) (Q.X q) (x.v p q hpq) := rfl |
|
|
| theorem crossCochainEquiv_differential (m : ℤ) |
| (x : Cochain P (tensorCochain (k := k) (F := F) Q) n) : |
| δ n m (crossCochainEquiv P Q n x) = crossCochainEquiv P Q m (δ n m x) := by |
| by_cases hnm : n+1=m |
| · apply Cochain.ext |
| intro p q hpq |
| simp only [δ_v n m hnm _ p q hpq (q-1) (p+1) rfl rfl, crossCochainEquiv_v] |
| let x₁ : P.X p ⟶ (BalancedTensor.functor (k := k) (R := R) (S := R) (F := F)).obj (Q.X (q-1)) := |
| x.v p (q-1) (by omega) |
| let x₂ : P.X (p+1) ⟶ (BalancedTensor.functor (k := k) (R := R) (S := R) (F := F)).obj (Q.X q) := |
| x.v (p+1) q (by omega) |
| change crossHomAdd _ _ x₁ ≫ (eFunctor k R F).map (Q.d (q-1) q) + |
| m.negOnePow • ((jFunctor k R F).map (P.d p (p+1)) ≫ crossHomAdd _ _ x₂) = |
| crossHomAdd (P.X p) (Q.X q) (x₁ ≫ |
| (BalancedTensor.functor (k := k) (R := R) (S := R) (F := F)).map (Q.d (q-1) q) + |
| m.negOnePow • (P.d p (p+1) ≫ x₂)) |
| rw [map_add] |
| simp only [Units.smul_def, map_zsmul] |
| rw [crossHomAdd_target, crossHomAdd_source] |
| · rw [δ_shape n m hnm, δ_shape n m hnm, map_zero] |
|
|
| def crossFullHomEquivalence : FullHomEquivalence P (tensorCochain (k := k) (F := F) Q) |
| (jCochain (k := k) (F := F) P) (eCochain (k := k) (F := F) Q) where |
| cochain := crossCochainEquiv P Q |
| differential := crossCochainEquiv_differential P Q |
|
|
| def crossCohomologyEquiv : |
| CohomologyClass P (tensorCochain (k := k) (F := F) Q) n ≃+ |
| CohomologyClass (jCochain (k := k) (F := F) P) (eCochain (k := k) (F := F) Q) n := |
| (crossFullHomEquivalence P Q).cohomology n |
|
|
| variable {P Q n} |
|
|
| def crossChainMap (f : P ⟶ tensorCochain (k := k) (F := F) Q) : |
| jCochain (k := k) (F := F) P ⟶ eCochain (k := k) (F := F) Q where |
| f i := crossHomAdd (P.X i) (Q.X i) (f.f i) |
| comm' i j _ := by |
| change crossHomAdd _ _ (f.f i) ≫ (eFunctor k R F).map (Q.d i j) = |
| (jFunctor k R F).map (P.d i j) ≫ crossHomAdd _ _ (f.f j) |
| exact (crossHomAdd_target (f.f i) (Q.d i j)).symm.trans |
| ((congrArg (crossHomAdd (P.X i) (Q.X j)) (f.comm i j)).trans |
| (crossHomAdd_source (P.d i j) (f.f j))) |
|
|
| theorem crossChainMap_ofHom (f : P ⟶ tensorCochain (k := k) (F := F) Q) : |
| crossCochainEquiv P Q 0 (Cochain.ofHom f) = Cochain.ofHom (crossChainMap f) := by |
| apply Cochain.ext₀ |
| intro p |
| simp only [crossCochainEquiv_v, Cochain.ofHom_v, crossChainMap] |
|
|
| variable {L : CochainComplex (ModuleCat.{u} R) ℤ} {m : ℤ} |
|
|
| theorem crossCochainEquiv_source (a : Cochain P Q n) |
| (b : Cochain Q (tensorCochain (k := k) (F := F) L) m) : |
| crossCochainEquiv P L (n+m) (a.comp b rfl) = |
| (a.map (jFunctor k R F)).comp (crossCochainEquiv Q L m b) rfl := by |
| apply Cochain.ext |
| intro p q hpq |
| simp only [crossCochainEquiv_v, Cochain.comp_v a b rfl p (p+n) q rfl (by omega), |
| Cochain.comp_v (a.map (jFunctor k R F)) (crossCochainEquiv Q L m b) rfl |
| p (p+n) q rfl (by omega), Cochain.map_v] |
| exact crossHomAdd_source _ _ |
|
|
| theorem crossCochainEquiv_target |
| (a : Cochain P (tensorCochain (k := k) (F := F) Q) n) (b : Cochain Q L m) : |
| crossCochainEquiv P L (n+m) |
| (a.comp (b.map (BalancedTensor.functor (k := k) (R := R) (S := R) (F := F))) rfl) = |
| (crossCochainEquiv P Q n a).comp (b.map (eFunctor k R F)) rfl := by |
| apply Cochain.ext |
| intro p q hpq |
| simp only [crossCochainEquiv_v, |
| Cochain.comp_v a (b.map (BalancedTensor.functor (k := k) (R := R) (S := R) (F := F))) rfl |
| p (p+n) q rfl (by omega), |
| Cochain.comp_v (crossCochainEquiv P Q n a) (b.map (eFunctor k R F)) rfl |
| p (p+n) q rfl (by omega), Cochain.map_v] |
| exact crossHomAdd_target _ _ |
|
|
| end ArExplicit.TriangularInduction |
|
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| end |
|
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| end OAI |
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|
