| import OAI.Algebra.Finitistic.Dimension.PDDetection | |
| 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 (U : Type u) [AddCommGroup U] [Module k U] [Module Aᵐᵒᵖ U] | |
| variable (M : PDMinimal.FiniteModule k A) | |
| /-- Any right module detects a lower bound through nonzero Tor; it need not | |
| be flat, finite-dimensional, or annihilated by the radical. -/ | |
| theorem tor_zero_of_pdLT (n : ℕ) (h : HasProjectiveDimensionLT M.cat n) : | |
| IsZero (Tor (k := k) U M.cat n) := by | |
| have ht : Subsingleton (PDMinimal.term M n) := by | |
| cases n with | |
| | zero => | |
| have hm := (hasProjectiveDimensionLT_zero_iff_isZero _).mp h | |
| have : Subsingleton M := ModuleCat.subsingleton_of_isZero hm | |
| exact PDMinimal.cover_subsingleton M | |
| | succ n => exact PDMinimal.term_zero_of_pdLE M n h | |
| have hz : IsZero (PDMinimal.term M n).cat := ModuleCat.isZero_iff_subsingleton.mpr ht | |
| have hm : IsZero ((model U M).X n) := (tensorFunctor k A U).map_isZero hz | |
| exact (HomologicalComplex.ExactAt.of_isZero hm).isZero_homology.of_iso (torIsoModel U M n) | |
| theorem le_projectiveDimension_of_nonzero_Tor (n : ℕ) | |
| (h : ¬ IsZero (Tor (k := k) U M.cat n)) : n ≤ projectiveDimension M.cat := by | |
| by_contra hn | |
| exact h (tor_zero_of_pdLT U M n (projectiveDimension_lt_iff.mp (lt_of_not_ge hn))) | |
| theorem lower_bound_of_nonzero_Tor (r n : ℕ) (hrn : r ≤ n) | |
| (h : ¬ IsZero (Tor (k := k) U M.cat n)) : r ≤ projectiveDimension M.cat := by | |
| exact le_trans (by exact_mod_cast hrn) (le_projectiveDimension_of_nonzero_Tor U M n h) | |
| /-- Bounded radical-quotient Tor gives finite numerical projective dimension. -/ | |
| theorem projectiveDimension_lt_top_of_bounded_residueTor | |
| (h : ∃ b : ℕ, ∀ n, b < n → IsZero (Tor (k := k) (Residue (A := A)) M.cat n)) : | |
| projectiveDimension M.cat < ⊤ := by | |
| obtain ⟨b, hb⟩ := h | |
| exact lt_of_le_of_lt | |
| (projectiveDimension_le_of_residueTor_zero M b (hb (b+1) (by omega))) (WithBot.coe_lt_coe.mpr (ENat.natCast_lt_top b)) | |
| end LittleFinitistic.PDTensor | |
| end | |
| end OAI | |