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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 | import OAI.Algebra.Finitistic.Selected.SelectedOrdinaryModules
import OAI.Algebra.Finitistic.Dimension.PDDimension
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits
namespace LittleFinitistic.SelectedMinimalResolution
open SelectedOrdinaryAlgebra SelectedOrdinaryModules
abbrev finiteModule (Y : FiniteSelectionFamily.S.{0}) : PDMinimal.FiniteModule ℂ A :=
PDMinimal.FiniteModule.of (Inflated Y)
/-- A genuine minimal projective resolution of each actual constructed A-module. -/
def resolution (Y : FiniteSelectionFamily.S.{0}) : ProjectiveResolution (SelectedOrdinaryModules.module Y) :=
PDMinimal.resolution (finiteModule Y)
/-- The constructed A-module has projective dimension at most n precisely when
its actual minimal projective resolution has zero term n+1. -/
theorem dimension_iff (Y : FiniteSelectionFamily.S.{0}) (n : ℕ) :
HasProjectiveDimensionLE (SelectedOrdinaryModules.module Y) n ↔
Subsingleton (PDMinimal.term (finiteModule Y) (n+1)) :=
PDMinimal.pdLE_iff_term_zero (finiteModule Y) n
/-- The actual minimal-resolution terms are projective. -/
theorem projective (Y : FiniteSelectionFamily.S.{0}) (n : ℕ) :
Projective ((resolution Y).complex.X n) := inferInstance
end LittleFinitistic.SelectedMinimalResolution
end
end OAI
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