Theorems · Theorem · commutative algebra

Algebra.trace_quotient_eq_of_isDedekindDomain

∀ {R : Type u_1} (S : Type u_2) [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal R)
  [p.IsMaximal] (x : S) [inst_4 : IsDedekindDomain R] [inst_5 : IsDomain S] [inst_6 : Module.IsTorsionFree R S]
  [inst_7 : Module.Finite R S] [inst_8 : IsIntegrallyClosed S],
  (Algebra.trace (R ⧸ p) (S ⧸ Ideal.map (algebraMap R S) p)) ((Ideal.Quotient.mk (Ideal.map (algebraMap R S) p)) x) =
    (Ideal.Quotient.mk p) ((Algebra.intTrace R S) x)

The trace map on B → A coincides with the trace map on B⧸pB → A⧸p.

Defined in

Mathlib.RingTheory.Trace.Quotient

Cited by

3 results in Mathlib

Foundations

Depth 187 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraIdeal.IsMaximalIsDedekindDomainIsDomainModule.IsTorsionFreeModule.FiniteIsIntegrallyClosed