Lemma of the day

Let R —> S be a ring map. Let p ⊂ R be a prime. Assume that

  1. there exists a unique prime q ⊂ S lying over p, and
  2. either
    1. going up holds for R —> S, or
    2. going down holds for R —> S and there is at most one prime of S above every prime of R.

Then Sp=Sq. See Tag 00EA.