| import OAI.Algebra.Finitistic.Selected.SelectedOrdinaryModules |
| import OAI.Algebra.Finitistic.Iteration.AdditiveIterateAmplitude |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| namespace LittleFinitistic.SelectedBaseIterates |
| open SelectedOrdinaryAlgebra SelectedOrdinaryModules |
|
|
| / |
| abbrev input (Y : FiniteSelectionFamily.S.{0}) := ModuleCat.of D (Inflated Y) |
|
|
| local instance (N : ModuleCat.{0} D) : Module ℂ N := DerivedBar.BimoduleTensor.scalarModule ℂ D N |
| local instance (N : ModuleCat.{0} D) : IsScalarTower ℂ D N where |
| smul_assoc c d n := by |
| change (c • d) • n = (algebraMap ℂ D c) • (d • n) |
| have hd : c • d = (algebraMap ℂ D c) * d := by |
| exact Algebra.smul_def c d |
| rw [hd, mul_smul] |
| |
| instance input_finiteD (Y : FiniteSelectionFamily.S.{0}) : Module.Finite D (input Y) := |
| Module.Finite.of_restrictScalars_finite ℂ D (Inflated Y) |
| |
| instance input_finite (Y : FiniteSelectionFamily.S.{0}) : FiniteDimensional ℂ (input Y) := |
| Module.Finite.trans D (input Y) |
| |
| local instance : HasDerivedCategory.{1} (ModuleCat.{0} D) := |
| HasDerivedCategory.standard (ModuleCat.{0} D) |
| |
| / |
| abbrev object (Y : FiniteSelectionFamily.S.{0}) (r : ℕ) := |
| DerivedBar.IterateSupport.object ℂ D X (input Y) r |
| |
| theorem cohomology_finite (Y : FiniteSelectionFamily.S.{0}) (r : ℕ) (i : ℤ) : |
| FiniteDimensional ℂ ((ModuleCat.restrictScalars (algebraMap ℂ D)).obj |
| ((DerivedCategory.homologyFunctor (ModuleCat D) i).obj (object Y r))) := by |
| let : Module ℂ (input Y) := DerivedBar.BimoduleTensor.scalarModule ℂ D (input Y) |
| have : IsScalarTower ℂ D (input Y) := by |
| refine ⟨fun c d n => ?_⟩ |
| change (c • d) • n = (algebraMap ℂ D c) • (d • n) |
| have hd : c • d = (algebraMap ℂ D c) * d := by exact Algebra.smul_def c d |
| rw [hd, mul_smul] |
| have : FiniteDimensional ℂ (input Y) := Module.Finite.trans D (input Y) |
| exact DerivedBar.IterateSupport.cohomology_finite ℂ D X (input Y) r i |
| |
| / |
| theorem bounded (g : ℕ) (hg : ∀ M : ModuleCat.{0} D, HasProjectiveDimensionLE M g) |
| (Y : FiniteSelectionFamily.S.{0}) (r : ℕ) : |
| (object Y r).IsGE (-((r*(g+2) : ℕ) : ℤ)) ∧ (object Y r).IsLE 0 := |
| DerivedBar.IterateSupport.bounded_amplitude ℂ D X g hg (input Y) r |
| end LittleFinitistic.SelectedBaseIterates |
| |
| end |
| |
| end OAI |
| |
