Theorems · Theorem · commutative algebra

conductor_mul_differentIdeal

∀ (A : Type u_1) (K : Type u_2) (L : Type u) {B : Type u_3} [inst : CommRing A] [inst_1 : Field K] [inst_2 : CommRing B]
  [inst_3 : Field L] [inst_4 : Algebra A K] [inst_5 : Algebra B L] [inst_6 : Algebra A B] [inst_7 : Algebra K L]
  [inst_8 : Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [inst_11 : IsDomain A] [IsFractionRing A K]
  [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegralClosure B A L] [IsIntegrallyClosed A]
  [inst_17 : IsDedekindDomain B] [inst_18 : Module.IsTorsionFree A B] (x : B),
  K[(algebraMap B L) x] = ⊤ →
    conductor A x * differentIdeal A B = Ideal.span {(Polynomial.aeval x) (Polynomial.derivative (minpoly A x))}

Defined in

Mathlib.RingTheory.DedekindDomain.Different

Cited by

1 results in Mathlib

Foundations

Depth 197 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingFieldCommRingFieldAlgebraAlgebraAlgebraAlgebraAlgebraIsScalarTowerIsScalarTowerIsDomainIsFractionRingFiniteDimensionalAlgebra.IsSeparableIsIntegralClosureIsIntegrallyClosedIsDedekindDomainModule.IsTorsionFree