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e3e0a0d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 | 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
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