Theorems · Theorem · commutative algebra

Localization.localRingHom_surjective_of_primesOver_eq_singleton

∀ {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] [inst_6 : q.LiesOver p] [Algebra.IsUnramifiedAt R q]
      [inst_8 : Algebra (Localization.AtPrime p) (Localization.AtPrime q)]
      [inst_9 : Localization.AtPrime.IsLiesOverAlgebra p q],
      Function.Surjective ⇑(algebraMap p.ResidueField q.ResidueField) →
        Function.Surjective ⇑(Localization.localRingHom p q (algebraMap R S) ⋯)

Defined in

Mathlib.RingTheory.Unramified.LocalRing

Cited by

1 results in Mathlib

Foundations

Depth 183 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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