Theorems · Theorem · algebraic geometry

RingHom.surjective_localRingHom_iff

∀ {R : Type u_1} [inst : CommRing R] {S : Type u_2} [inst_1 : CommRing S] {f : R →+* S} (P : Ideal S)
  [inst_2 : P.IsPrime],
  Function.Surjective ⇑(Localization.localRingHom (Ideal.comap f P) P f ⋯) ↔
    ∀ (s : S), ∃ x r, ∃ c ∉ P, f r ∉ P ∧ c * f r * s = c * f x

R_p →+* S_q is surjective if and only if every x : S is of the form f x / f r for some f r ∉ q. This is useful when proving SurjectiveOnStalks.

Defined in

Mathlib.RingTheory.SurjectiveOnStalks

Cited by

3 results in Mathlib

Foundations

Depth 66 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingIdeal.IsPrime