Theorems · Theorem · commutative algebra

integralClosure.isNoetherianRing

∀ {A : Type u_1} {K : Type u_2} [inst : CommRing A] [inst_1 : Field K] [inst_2 : Algebra A K] [IsFractionRing A K]
  (L : Type u_3) [inst_4 : Field L] [inst_5 : Algebra K L] [inst_6 : Algebra A L] [IsScalarTower A K L]
  [FiniteDimensional K L] [IsDomain A] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsNoetherianRing A],
  IsNoetherianRing ↥(integralClosure A L)

If L is a finite separable extension of K = Frac(A), where A is integrally closed and Noetherian, the integral closure of A in L is Noetherian.

Defined in

Mathlib.RingTheory.DedekindDomain.IntegralClosure

Cited by

0 results in Mathlib

Foundations

Depth 186 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingFieldAlgebraIsFractionRingFieldAlgebraAlgebraIsScalarTowerFiniteDimensionalIsDomainAlgebra.IsSeparableIsIntegrallyClosedIsNoetherianRing