Theorems · Theorem · commutative algebra

Localization.finite_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] [inst_7 : Algebra (Localization.AtPrime p) (Localization.AtPrime q)]
      [Localization.AtPrime.IsLiesOverAlgebra p q], Module.Finite (Localization.AtPrime p) (Localization.AtPrime q)

Defined in: Mathlib.RingTheory.Unramified.LocalRing
Cited by: 1 results in Mathlib
Foundations: Depth 132 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing CommRing Algebra Ideal.IsPrime Ideal.IsPrime Module.Finite Ideal.LiesOver Algebra Localization.AtPrime.IsLiesOverAlgebra

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Idealstatement and proof · cited by 4,748
Algebra.algebraMapproof · cited by 4,706
Set.rangeproof · cited by 4,705

Cited by1

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Localization.localRingHom_surjective_of_primesOver_eq_singletonproof · cited by 1