Theorems · Theorem · commutative algebra

IsLocalRing.finrank_cotangentSpace_le_one_iff

∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsLocalRing R] [IsNoetherianRing R],
  Module.finrank (IsLocalRing.ResidueField R) (IsLocalRing.CotangentSpace R) ≤ 1 ↔
    Submodule.IsPrincipal (IsLocalRing.maximalIdeal R)

Defined in: Mathlib.RingTheory.Ideal.Cotangent
Cited by: 1 results in Mathlib
Foundations: Depth 121 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing IsLocalRing IsNoetherianRing

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Cites33

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
CommRingstatement and proof · cited by 17,173
Semiringproof · cited by 13,802
AddCommMonoidproof · cited by 12,281
Top.topproof · cited by 9,680
Submoduleproof · cited by 7,192
Set.imageproof · cited by 5,609
Idealproof · cited by 4,748
Module.finrankstatement · cited by 1,770
Submodule.spanproof · cited by 1,504
Submodule.mapproof · cited by 614

Cited by1

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tfae_of_isNoetherianRing_of_isLocalRing_of_isDomainproof · cited by 1