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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 OAI.Algebra.Finitistic.Dimension.PDLowerBound
namespace OAI
noncomputable section
namespace LittleFinitistic.PDTensor
open CategoryTheory CategoryTheory.Limits
universe u
variable {k A : Type u} [Field k] [Ring A] [Algebra k A] [FiniteDimensional k A]
variable (M : PDMinimal.FiniteModule k A)
theorem bounded_residueTor_of_finite_pd
(h : ∃ b : ℕ, HasProjectiveDimensionLE M.cat b) :
∃ b : ℕ, ∀ n, b < n → IsZero (Tor (k := k) (Residue (A := A)) M.cat n) := by
obtain ⟨b, hb⟩ := h
have : HasProjectiveDimensionLT M.cat (b+1) := hb
refine ⟨b, fun n hn => tor_zero_of_pdLT (Residue (A := A)) M n ?_⟩
exact hasProjectiveDimensionLT_of_ge M.cat (b+1) n (by omega)
/-- Finite projective dimension is equivalent to bounded actual residue Tor. -/
theorem finite_pd_iff_bounded_residueTor :
projectiveDimension M.cat ≠ ⊤ ↔
∃ b : ℕ, ∀ n, b < n → IsZero (Tor (k := k) (Residue (A := A)) M.cat n) := by
rw [projectiveDimension_ne_top_iff]
exact ⟨bounded_residueTor_of_finite_pd M, finite_pd_of_bounded_residueTor M⟩
/-- The upper detector accepts any actual projective resolution, including a
bar resolution once its exactness and projectivity have been proved. -/
theorem finite_pd_of_resolution
(P : ProjectiveResolution M.cat)
(h : ∃ b : ℕ, ∀ n, b < n → IsZero
(((tensorFunctor k A (Residue (A := A))).mapHomologicalComplex _).obj P.complex |>.homology n)) :
projectiveDimension M.cat < ⊤ := by
apply projectiveDimension_lt_top_of_bounded_residueTor M
obtain ⟨b, hb⟩ := h
exact ⟨b, fun n hn => (hb n hn).of_iso
(P.isoLeftDerivedObj (tensorFunctor k A (Residue (A := A))) n)⟩
variable (U : Type u) [AddCommGroup U] [Module k U] [Module Aᵐᵒᵖ U]
/-- The lower detector also accepts any actual projective resolution.
The nonzero class can be supplied by a summand of the bar decomposition. -/
theorem lower_bound_of_resolution (P : ProjectiveResolution M.cat) (r n : ℕ) (hrn : r ≤ n)
(h : ¬ IsZero (((tensorFunctor k A U).mapHomologicalComplex _).obj P.complex |>.homology n)) :
r ≤ projectiveDimension M.cat := by
apply lower_bound_of_nonzero_Tor U M r n hrn
intro hz
exact h (hz.of_iso (P.isoLeftDerivedObj (tensorFunctor k A U) n).symm)
end LittleFinitistic.PDTensor
end
end OAI
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