File size: 4,172 Bytes
ae00908 | 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 | import OAI.Algebra.Finitistic.Ordinary.OrdinaryRestrictionHomology
import Mathlib.Algebra.Homology.ShortComplex.ExactFunctor
import OAI.Algebra.Finitistic.Selected.SelectedBaseIterates
import OAI.Algebra.Finitistic.Selected.SelectedDerivedFunctor
import Mathlib.Algebra.Homology.DerivedCategory.ExactFunctor
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits
namespace LittleFinitistic.SelectedFirstBlockFamily
attribute [local instance] preservesBinaryBiproducts_of_preservesBiproducts
open SelectedOrdinaryAlgebra SelectedFixedDiagram QuadraticEncoding
local instance : HasDerivedCategory.{1} (ModuleCat.{0} (B P)) :=
HasDerivedCategory.standard (ModuleCat.{0} (B P))
local instance : HasDerivedCategory.{1} (ModuleCat.{0} D) :=
HasDerivedCategory.standard (ModuleCat.{0} D)
/-- The actual exact inflation functor from B-modules to the first D-block. -/
abbrev inflation : ModuleCat.{0} (B P) ⥤ ModuleCat.{0} D := ModuleCat.restrictScalars SelectedOrdinaryModules.firstProjection
instance inflation_preservesFiniteLimits : PreservesFiniteLimits inflation := by
have h : PreservesFiniteLimits inflation ∧ PreservesFiniteColimits inflation :=
((Functor.exact_tfae inflation).out 2 4).mp
(RestrictionHomology.mapsExact SelectedOrdinaryModules.firstProjection)
exact h.1
instance inflation_preservesFiniteColimits : PreservesFiniteColimits inflation := by
have h : PreservesFiniteLimits inflation ∧ PreservesFiniteColimits inflation :=
((Functor.exact_tfae inflation).out 2 4).mp
(RestrictionHomology.mapsExact SelectedOrdinaryModules.firstProjection)
exact h.2
abbrev derivedInflation := inflation.mapDerivedCategory
instance derivedInflation_additive : derivedInflation.Additive := by
rw [Localization.functor_additive_iff DerivedCategory.Qh
(HomotopyCategory.subcategoryAcyclic (ModuleCat.{0} (B P))).trW]
exact Functor.additive_of_iso inflation.mapDerivedCategoryFactorsh.symm
/-- This is the original selected input, with precisely its constructed D-action. -/
def moduleIso (Y : FiniteSelectionFamily.S.{0}) :
inflation.obj (FiniteSelectionFamily.module Y) ≅ SelectedBaseIterates.input Y :=
Iso.refl _
abbrev derived (Y : FiniteSelectionFamily.S.{0}) :=
(DerivedCategory.singleFunctor (ModuleCat D) 0).obj (SelectedBaseIterates.input Y)
/-- Exact inflation on derived categories agrees with the selected first-block stalk. -/
def stalkIso (Y : FiniteSelectionFamily.S.{0}) :
derivedInflation.obj (FiniteSelectionFamily.derived Y) ≅ derived Y :=
(inflation.mapDerivedCategorySingleFunctor 0).app (FiniteSelectionFamily.module Y) ≪≫
(DerivedCategory.singleFunctor (ModuleCat D) 0).mapIso (moduleIso Y)
theorem module_zero (Y : FiniteSelectionFamily.S.{0}) :
IsZero (SelectedBaseIterates.input Y) ↔ Subsingleton Y.val := by
rw [← FiniteSelectionFamily.module_zero Y]
rw [ModuleCat.isZero_iff_subsingleton, ModuleCat.isZero_iff_subsingleton]
rfl
theorem derived_zero (Y : FiniteSelectionFamily.S.{0}) :
IsZero (derived Y) ↔ Subsingleton Y.val := by
rw [← module_zero Y]
constructor
· intro h
exact ((DerivedCategory.singleFunctorCompHomologyFunctorIso (ModuleCat D) 0).app
(SelectedBaseIterates.input Y)).isZero_iff.mp
((DerivedCategory.homologyFunctor (ModuleCat D) 0).map_isZero h)
· exact (DerivedCategory.singleFunctor (ModuleCat D) 0).map_isZero
/-- The already proved Phi realization, transported by the actual first-block
inflation, with no ordinary F² identity assumed or asserted. -/
def inflated_realization (Y : FiniteSelectionFamily.S.{0}) :
derivedInflation.obj (SelectedDerivedFunctor.Phi.obj (FiniteSelectionFamily.derived Y)) ≅
derived (FiniteSelectionFamily.H Y) ⊞ (derived (FiniteSelectionFamily.H Y))⟦(3 : ℤ)⟧ :=
derivedInflation.mapIso (SelectedDerivedFunctor.comparison Y ≪≫ FiniteSelectionFamily.realization Y) ≪≫
derivedInflation.mapBiprod _ _ ≪≫
biprod.mapIso (stalkIso _) ((derivedInflation.commShiftIso (3 : ℤ)).app _ ≪≫
(shiftFunctor _ (3 : ℤ)).mapIso (stalkIso _))
end LittleFinitistic.SelectedFirstBlockFamily
end
end OAI
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