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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 54 55 56 57 | import OAI.Algebra.Finitistic.Selected.SelectedOrdinaryOrbit
import OAI.Algebra.Finitistic.Selected.SelectedPDConclusion
import OAI.Algebra.Finitistic.Basic
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits
namespace LittleFinitistic.Main
open SelectedOrdinaryAlgebra SelectedOrdinaryModules
/-- The fixed algebra and every actual witness are finite dimensional over ℂ. -/
theorem algebra_finite : FiniteDimensional ℂ A := inferInstance
theorem test_finite (m : ℕ) [NeZero m] : FiniteDimensional ℂ (testModule m) := inferInstance
/-- For the one fixed finite-dimensional complex algebra constructed in the
manuscript, every positive m has the actual finite module N_m with the stated
finite projective-dimension bounds. -/
theorem projective_dimension_bounds (m : ℕ) [NeZero m] :
((2*m-2 : ℕ) : WithBot ℕ∞) ≤ projectiveDimension (testModule m) ∧
projectiveDimension (testModule m) < ⊤ := by
have h := SelectedOrdinaryOrbit.extinction m
exact ⟨SelectedPDConclusion.lower_bound_of_survival (FiniteSelectionFamily.test m) (2*m-2) h.2,
SelectedPDConclusion.finite_pd_of_extinction (FiniteSelectionFamily.test m) (2*m) h.1⟩
/-- The actual constructed algebra has infinite little finitistic dimension. -/
theorem infinite_littleFinitisticDimension : littleFinitisticDimension A = ⊤ := by
apply littleFinitisticDimension_eq_top_of_two_mul_bounds
intro m hm
let : NeZero m := ⟨by omega⟩
exact ⟨testModule m, inferInstance, (projective_dimension_bounds m).2,
(projective_dimension_bounds m).1⟩
/-- Main theorem, with finite dimensionality of the one fixed A explicit and
with actual finitely generated witness modules for every positive m. -/
theorem counterexample : FiniteDimensional ℂ A ∧ littleFinitisticDimension A = ⊤ ∧
∀ m : ℕ, 0 < m → ∃ N : ModuleCat.{0} A, Module.Finite A N ∧
((2*m-2 : ℕ) : WithBot ℕ∞) ≤ projectiveDimension N ∧ projectiveDimension N < ⊤ := by
refine ⟨inferInstance, infinite_littleFinitisticDimension, ?_⟩
intro m hm
let : NeZero m := ⟨by omega⟩
exact ⟨testModule m, inferInstance, projective_dimension_bounds m⟩
theorem exists_counterexample :
∃ (A : Type) (_ : Ring A) (_ : Algebra ℂ A),
FiniteDimensional ℂ A ∧ littleFinitisticDimension A = ⊤ ∧
∀ m : ℕ, 0 < m → ∃ N : ModuleCat.{0} A, Module.Finite A N ∧
((2*m-2 : ℕ) : WithBot ℕ∞) ≤ projectiveDimension N ∧
projectiveDimension N < ⊤ := by
exact ⟨SelectedOrdinaryAlgebra.A, inferInstance, inferInstance, counterexample⟩
end LittleFinitistic.Main
end
end OAI
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