Theorems · Theorem · commutative algebra

IsIntegralClosure.finite

∀ (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] (C : Type u_4) [inst_5 : CommRing C] [inst_6 : Algebra K L] [inst_7 : Algebra A L]
  [IsScalarTower A K L] [inst_9 : Algebra C L] [IsIntegralClosure C A L] [inst_11 : Algebra A C] [IsScalarTower A C L]
  [FiniteDimensional K L] [IsDomain A] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsNoetherianRing A],
  Module.Finite A C

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

Defined in

Mathlib.RingTheory.DedekindDomain.IntegralClosure

Cited by

3 results in Mathlib

Foundations

Depth 185 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingFieldAlgebraIsFractionRingFieldCommRingAlgebraAlgebraIsScalarTowerAlgebraIsIntegralClosureAlgebraIsScalarTowerFiniteDimensionalIsDomainAlgebra.IsSeparableIsIntegrallyClosedIsNoetherianRing