File size: 3,230 Bytes
2c7d975 | 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 | 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
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