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216fb31 | 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 | 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]
/-- The little finitistic dimension: the supremum of the actual projective
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))
/-- Unbounded finite projective dimensions of actual finitely generated modules
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)
/-- The manuscript's concrete lower-bound pattern is sufficient; all finite
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
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