| import Mathlib |
| import OAI.RingTheory.Multiplicity.IdealQuotientFunctor |
| import OAI.RingTheory.Multiplicity.ReesRootCech |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace Lech.FiniteModuleCech |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex |
| universe u |
| variable {R : Type u} [CommRing R] {ι : Type} [Fintype ι] [LinearOrder ι] |
| variable (D : Diagram R ι) (F : ModuleCat.{u} R ⥤ ModuleCat.{u} R) |
|
|
| def coefficients : Diagram R ι where |
| obj s := F.obj (D.obj s) |
| res h := F.map (D.res h) |
| res_self s := by rw [D.res_self,F.map_id] |
| res_comp h h' := by rw [←F.map_comp,D.res_comp] |
|
|
| def Map.coefficients {E : Diagram R ι} (φ : Map D E) : |
| Map (coefficients D F) (coefficients E F) where |
| app s := F.map (φ.app s) |
| naturality h := by |
| change F.map (D.res h) ≫ F.map (φ.app _) = F.map (φ.app _) ≫ F.map (E.res h) |
| rw [←F.map_comp,←F.map_comp,φ.naturality] |
|
|
| variable (I : Ideal R) |
| abbrev reduction := coefficients D (TensorIdeal.quotientFunctor I) |
|
|
| lemma quotientPiEquiv_d (q : ℕ) (x : cochains D q) : |
| TensorIdeal.quotientPiEquiv I (fun s : Set.powersetCard ι (q+1) => D.obj s.val) |
| (Submodule.Quotient.mk (d D q x)) = |
| d (reduction D I) q |
| (TensorIdeal.quotientPiEquiv I (fun s : Set.powersetCard ι q => D.obj s.val) |
| (Submodule.Quotient.mk x)) := by |
| rw [TensorIdeal.quotientPiEquiv_mk,TensorIdeal.quotientPiEquiv_mk] |
| funext s |
| change (Submodule.mkQ (I • (⊤ : Submodule R (D.obj s.val)))) |
| (∑ j : Fin (q+1), (-1:ℤ)^j.val • (D.res (AlternatingCech.delete_subset s j)).hom |
| (x (AlternatingCech.delete s j))) = _ |
| rw [map_sum] |
| change (∑ j : Fin (q+1), (Submodule.mkQ (I • (⊤ : Submodule R (D.obj s.val)))) |
| ((-1:ℤ)^j.val • (D.res (AlternatingCech.delete_subset s j)).hom |
| (x (AlternatingCech.delete s j)))) = _ |
| simp only [map_zsmul] |
| rfl |
|
|
| |
|
|
| def reductionComplexIso : |
| ((TensorIdeal.quotientFunctor I).mapHomologicalComplex (.up ℕ)).obj (complex D) ≅ |
| complex (reduction D I) := |
| Hom.isoOfComponents (fun q => (TensorIdeal.quotientPiEquiv I |
| (fun s : Set.powersetCard ι q => D.obj s.val)).toModuleIso) (by |
| intro q r h |
| obtain rfl : q+1=r := h |
| rw [complex_d] |
| change _ ≫ ModuleCat.ofHom (d (reduction D I) q) = |
| (TensorIdeal.quotientFunctor I).map ((complex D).d q (q+1)) ≫ _ |
| rw [complex_d] |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro x |
| induction x using Submodule.Quotient.induction_on with |
| | _ x => exact (quotientPiEquiv_d D I q x).symm) |
|
|
| |
| def reductionPositiveIso : |
| ((TensorIdeal.quotientFunctor I).mapHomologicalComplex (.up ℕ)).obj (positiveComplex D) ≅ |
| positiveComplex (reduction D I) := |
| Hom.isoOfComponents (fun q => (TensorIdeal.quotientPiEquiv I |
| (fun s : Set.powersetCard ι (q+1) => D.obj s.val)).toModuleIso) (by |
| intro q r h |
| obtain rfl : q+1=r := h |
| rw [positiveComplex_d] |
| change _ ≫ ModuleCat.ofHom (d (reduction D I) (q+1)) = |
| (TensorIdeal.quotientFunctor I).map ((positiveComplex D).d q (q+1)) ≫ _ |
| rw [positiveComplex_d] |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro x |
| induction x using Submodule.Quotient.induction_on with |
| | _ x => exact (quotientPiEquiv_d D I (q+1) x).symm) |
|
|
| |
| omit [LinearOrder ι] in |
| lemma reduction_cochains_torsion (q : ℕ) : |
| powerTorsion I (ModuleCat.of R (cochains (reduction D I) q)) := |
| (powerTorsion I).prop_of_iso |
| (TensorIdeal.quotientPiEquiv I (fun s : Set.powersetCard ι q => D.obj s.val)).toModuleIso |
| (TensorIdeal.quotient_torsion I (cochains D q)) |
|
|
| lemma reduction_homology_torsion (q : ℕ) : |
| powerTorsion I ((complex (reduction D I)).homology q) := |
| powerTorsion_epi I ((complex (reduction D I)).homologyπ q) |
| (powerTorsion_mono I ((complex (reduction D I)).iCycles q) |
| (reduction_cochains_torsion D I q)) |
|
|
| lemma reduction_positiveHomology_torsion (q : ℕ) : |
| powerTorsion I ((positiveComplex (reduction D I)).homology q) := |
| powerTorsion_epi I ((positiveComplex (reduction D I)).homologyπ q) |
| (powerTorsion_mono I ((positiveComplex (reduction D I)).iCycles q) |
| (reduction_cochains_torsion D I (q+1))) |
|
|
| namespace Map |
| variable {D} {E : Diagram R ι} (φ : Map D E) |
| variable (P : ObjectProperty (ModuleCat.{u} R)) [P.IsSerreClass] |
|
|
| omit [LinearOrder ι] in |
| lemma cochains_isoModSerre (hφ : ∀ s,P.isoModSerre (φ.app s)) (q : ℕ) : |
| P.isoModSerre (ModuleCat.ofHom (φ.cochains q)) := |
| piLinear_isoModSerre (fun s : Set.powersetCard ι q => (φ.app s.val).hom) P |
| (fun s => hφ s.val) |
|
|
| lemma homology_isoModSerre (hφ : ∀ s,P.isoModSerre (φ.app s)) (q : ℕ) : |
| P.isoModSerre (HomologicalComplex.homologyMap φ.complex q) := |
| homology_isoModSerre_of_termwise P φ.complex (φ.cochains_isoModSerre P hφ) q |
|
|
| lemma positiveHomology_isoModSerre (hφ : ∀ s,P.isoModSerre (φ.app s)) (q : ℕ) : |
| P.isoModSerre (HomologicalComplex.homologyMap φ.positiveComplex q) := |
| homology_isoModSerre_of_termwise P φ.positiveComplex |
| (fun q => φ.cochains_isoModSerre P hφ (q+1)) q |
| end Map |
| end Lech.FiniteModuleCech |
|
|
| end |
|
|
| end OAI |
|
|
