| 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 |
|
|
