| import OAI.Algebra.AuslanderReiten.Modules.TInducedHom |
| import OAI.Algebra.AuslanderReiten.Ext.ExtendShortExact |
| import OAI.Algebra.AuslanderReiten.Homology.DerivedHomVanishing |
| import OAI.Algebra.AuslanderReiten.Homology.TriangleRecurrence |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory |
| namespace ArExplicit.Resolution |
| universe u |
| variable {k : Type u} [Field k] [CharP k 2] (q : k) |
| local notation "C" => BaseAlgebra k q |
| local notation "T" => ArExplicit.T k q |
|
|
| def tExtendedSequence : ShortComplex (CochainComplex (ModuleCat T) ℤ) := |
| (TrivialInduction.sequence (k := k) (complex q)).map |
| (ComplexShape.embeddingDownNat.extendFunctor (ModuleCat T)) |
|
|
| theorem tExtendedSequence_shortExact : (tExtendedSequence q).ShortExact := |
| extend_shortExact _ _ (TrivialInduction.sequence_shortExact (complex q)) |
|
|
| variable [HasDerivedCategory (ModuleCat.{u} (ArExplicit.T k q))] |
|
|
| def tSimpleDerived : DerivedCategory (ModuleCat T) := |
| (DerivedCategory.singleFunctor (ModuleCat T) 0).obj (ModuleCat.of T (Simple q)) |
|
|
| def tInducedDerived : DerivedCategory (ModuleCat T) := |
| DerivedCategory.Q.obj (tInducedCochain q) |
|
|
| def tTriangleA (hq : q ≠ 0) (hq1 : 1 + q ≠ 0) (h : NoPositivePowerOne q) : |
| (tSimpleDerived q)⟦(2 : ℤ)⟧ ⟶ tInducedDerived q := |
| (tDualDerivedIso q hq hq1 h).inv ≫ DerivedCategory.Q.map (tExtendedSequence q).f |
|
|
| def tTriangleB (h : NoPositivePowerOne q) : tInducedDerived q ⟶ tSimpleDerived q := |
| DerivedCategory.Q.map (tExtendedSequence q).g ≫ (tQuotientDerivedIso q h).hom |
|
|
| def tTriangleD (hq : q ≠ 0) (hq1 : 1 + q ≠ 0) (h : NoPositivePowerOne q) : |
| tSimpleDerived q ⟶ ((tSimpleDerived q)⟦(2 : ℤ)⟧)⟦(1 : ℤ)⟧ := |
| (tQuotientDerivedIso q h).inv ≫ |
| DerivedCategory.triangleOfSESδ (tExtendedSequence_shortExact q) ≫ |
| (tDualDerivedIso q hq hq1 h).hom⟦(1 : ℤ)⟧' |
|
|
| def tInductionTriangle (hq : q ≠ 0) (hq1 : 1 + q ≠ 0) (h : NoPositivePowerOne q) := |
| Pretriangulated.Triangle.mk (tTriangleA q hq hq1 h) (tTriangleB q h) (tTriangleD q hq hq1 h) |
|
|
| def tInductionTriangleIso (hq : q ≠ 0) (hq1 : 1 + q ≠ 0) (h : NoPositivePowerOne q) : |
| DerivedCategory.triangleOfSES (tExtendedSequence_shortExact q) ≅ tInductionTriangle q hq hq1 h := |
| Pretriangulated.Triangle.isoMk _ _ (tDualDerivedIso q hq hq1 h) (Iso.refl _) |
| (tQuotientDerivedIso q h) |
| (by exact (Category.comp_id _).trans (Iso.hom_inv_id_assoc _ _).symm) |
| (by exact (Category.id_comp _).symm) |
| (by exact (Iso.hom_inv_id_assoc _ _).symm) |
|
|
| theorem tInductionTriangle_distinguished (hq : q ≠ 0) (hq1 : 1 + q ≠ 0) |
| (h : NoPositivePowerOne q) : |
| tInductionTriangle q hq hq1 h ∈ distTriang (DerivedCategory (ModuleCat T)) := |
| Pretriangulated.isomorphic_distinguished _ |
| (DerivedCategory.triangleOfSES_distinguished (tExtendedSequence_shortExact q)) |
| _ (tInductionTriangleIso q hq hq1 h).symm |
|
|
| theorem tInducedDerived_hom_zero (hq1 : 1 + q ≠ 0) (n : ℤ) (hn : n ≠ 0) |
| (f : tInducedDerived q ⟶ (tSimpleDerived q)⟦n⟧) : f = 0 := |
| derivedHom_zero_of_cochain_zero (tInducedCochain q) |
| ((CochainComplex.singleFunctor (ModuleCat T) 0).obj (ModuleCat.of T (Simple q))) n |
| (tInduced_fullCochain_zero q hq1 n hn) f |
|
|
| end ArExplicit.Resolution |
|
|
| end |
|
|
| end OAI |
|
|
