Theorems · Theorem · commutative algebra

IsIntegralClosure.rank

∀ (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] [IsPrincipalIdealRing A] [Module.IsTorsionFree A L],
  Module.finrank A C = Module.finrank K L

If L is a finite separable extension of K = Frac(A), where A is a principal ring and L has no zero smul divisors by A, the A-rank of the integral closure C of A in L is equal to the K-rank of L.

Defined in

Mathlib.RingTheory.DedekindDomain.IntegralClosure

Cited by

1 results in Mathlib

Foundations

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

Assumes

CommRingFieldAlgebraIsFractionRingFieldCommRingAlgebraAlgebraIsScalarTowerAlgebraIsIntegralClosureAlgebraIsScalarTowerFiniteDimensionalIsDomainAlgebra.IsSeparableIsPrincipalIdealRingModule.IsTorsionFree