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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 | 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
/-- The actual first-block D-module underlying the inflated A-module. -/
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)
/-- Genuine bar-derived iteration of the actual X over the actual 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
/-- A global-dimension bound on D bounds the amplitude of every derived iterate. -/
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
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