Theorems · Theorem · commutative algebra

Module.supportDim_le_supportDim_quotSMulTop_succ

∀ {R : Type u_1} [inst : CommRing R] [IsNoetherianRing R] {M : Type u_2} [inst_2 : AddCommGroup M] [inst_3 : Module R M]
  [Module.Finite R M] [inst_5 : IsLocalRing R] {x : R},
  x ∈ IsLocalRing.maximalIdeal R → Module.supportDim R M ≤ Module.supportDim R (QuotSMulTop x M) + 1

If M is a finite module over a Noetherian local ring R, then dim M ≤ dim M/xM + 1 for every x in the maximal ideal of the local ring R.

Defined in

Mathlib.RingTheory.KrullDimension.Regular

Cited by

1 results in Mathlib

Foundations

Depth 143 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingIsNoetherianRingAddCommGroupModuleModule.FiniteIsLocalRing