| import Mathlib.Algebra.Category.ModuleCat.Abelian |
| import Mathlib.CategoryTheory.Abelian.Projective.Dimension |
| import Mathlib.RingTheory.Finiteness.Defs |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory |
| namespace LittleFinitistic |
| universe u |
| variable (A : Type u) [Ring A] |
|
|
| / |
| dimensions of finitely generated modules whose projective dimension is finite. |
| The value of the zero module follows mathlib's -∞ convention. -/ |
| def littleFinitisticDimension : WithBot ℕ∞ := |
| ⨆ (M : ModuleCat.{u} A) (_ : Module.Finite A M) (_ : projectiveDimension M < ⊤), |
| projectiveDimension M |
|
|
| theorem le_littleFinitisticDimension (M : ModuleCat.{u} A) [Module.Finite A M] |
| (h : projectiveDimension M < ⊤) : |
| projectiveDimension M ≤ littleFinitisticDimension A := by |
| unfold littleFinitisticDimension |
| exact le_iSup_of_le M (le_iSup_of_le inferInstance (le_iSup_of_le h le_rfl)) |
|
|
| / |
| force the little finitistic dimension to be infinite. -/ |
| theorem littleFinitisticDimension_eq_top_of_unbounded |
| (h : ∀ n : ℕ, ∃ M : ModuleCat.{u} A, Module.Finite A M ∧ |
| projectiveDimension M < ⊤ ∧ (n : WithBot ℕ∞) ≤ projectiveDimension M) : |
| littleFinitisticDimension A = ⊤ := by |
| apply ENat.WithBot.eq_top_iff_forall_ge.mpr |
| intro n |
| obtain ⟨M, hM, hfinite, hn⟩ := h n |
| let := hM |
| exact hn.trans (le_littleFinitisticDimension A M hfinite) |
|
|
| / |
| projective-dimension premises must still be supplied by the constructed modules. -/ |
| theorem littleFinitisticDimension_eq_top_of_two_mul_bounds |
| (h : ∀ m : ℕ, 0 < m → ∃ M : ModuleCat.{u} A, Module.Finite A M ∧ |
| projectiveDimension M < ⊤ ∧ ((2*m-2 : ℕ) : WithBot ℕ∞) ≤ projectiveDimension M) : |
| littleFinitisticDimension A = ⊤ := by |
| apply littleFinitisticDimension_eq_top_of_unbounded |
| intro n |
| obtain ⟨M,hM,hfinite,hm⟩ := h (n+1) (by omega) |
| refine ⟨M,hM,hfinite,le_trans ?_ hm⟩ |
| exact_mod_cast (show n ≤ 2*(n+1)-2 by omega) |
| end LittleFinitistic |
|
|
| end |
|
|
| end OAI |
|
|
