Theorems · Theorem · commutative algebra
ModuleCat.hasProjectiveDimensionLE_iff_forall_maximalSpectrum
∀ {R : Type u} [inst : CommRing R] (n : ℕ) [inst_1 : Small.{v, u} R] [IsNoetherianRing R] (M : ModuleCat R)
[Module.Finite R ↑M],
CategoryTheory.HasProjectiveDimensionLE M n ↔
∀ (m : MaximalSpectrum R), CategoryTheory.HasProjectiveDimensionLE (M.localizedModule m.asIdeal.primeCompl) n- Cited by
- 2 results in Mathlib
- Foundations
- Depth 121 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites50
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Moduleproof · cited by 20,661
- RingHom.idproof · cited by 18,349
- CommRingstatement and proof · cited by 17,173
- AddCommGroupproof · cited by 12,871
- LinearMapproof · cited by 10,215
- Idealproof · cited by 4,748
- zero_addproof · cited by 2,366
- CategoryTheory.ShortComplexproof · cited by 1,850
- ModuleCatstatement and proof · cited by 1,429
- CategoryTheory.ShortComplex.X₂proof · cited by 1,115
- Module.Finitestatement and proof · cited by 1,032
Cited by2
Results whose statement or proof uses this declaration.
- ModuleCat.hasProjectiveDimensionLE_iff_forall_primeSpectrumproof · cited by 1
- ModuleCat.projectiveDimension_eq_iSup_localizedModule_maximalproof · cited by 0