| 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 | |