Theorems · Theorem · commutative algebra

Localization.exists_awayMap_bijective_of_residueField_surjective

∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] {p : Ideal R}
  [inst_3 : p.IsPrime] {q : Ideal S} [inst_4 : q.IsPrime],
  p.primesOver S = {q} →
    ∀ [Module.Finite R S] [FaithfulSMul R S] [inst_7 : q.LiesOver p] [Algebra.IsUnramifiedAt R q]
      [inst_9 : Algebra (Localization.AtPrime p) (Localization.AtPrime q)]
      [inst_10 : Localization.AtPrime.IsLiesOverAlgebra p q],
      Function.Surjective ⇑(algebraMap p.ResidueField q.ResidueField) →
        ∃ r ∉ p, ∀ (r' : R), r ∣ r' → Function.Bijective ⇑(Localization.awayMap (algebraMap R S) r')

Defined in

Mathlib.RingTheory.Unramified.LocalRing

Cited by

0 results in Mathlib

Foundations

Depth 184 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraIdeal.IsPrimeIdeal.IsPrimeModule.FiniteFaithfulSMulIdeal.LiesOverAlgebra.IsUnramifiedAtAlgebraLocalization.AtPrime.IsLiesOverAlgebra