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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 | import OAI.Algebra.Finitistic.Selected.SelectedOrdinaryModules
import OAI.Algebra.Finitistic.Bar.FiniteBarResolution
namespace OAI
noncomputable section
open CategoryTheory
namespace LittleFinitistic.SelectedBarResolution
open SelectedOrdinaryAlgebra SelectedOrdinaryModules
/-- The genuine absolute bar projective resolution of each actual inflated
module. It can have infinitely many nonzero degrees. -/
def resolution (Y : FiniteSelectionFamily.S.{0}) : ProjectiveResolution (module Y) :=
DerivedBar.projectiveResolution ℂ A (Inflated Y)
theorem finite_free (Y : FiniteSelectionFamily.S.{0}) (n : ℕ) :
Module.Finite A ((resolution Y).complex.X n) ∧
Module.Free A ((resolution Y).complex.X n) :=
⟨DerivedBar.projectiveResolution_finite ℂ A (Inflated Y) n,
DerivedBar.projectiveResolution_free ℂ A (Inflated Y) n⟩
end LittleFinitistic.SelectedBarResolution
end
end OAI
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