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

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

Then S_{p}=S_{q}. See Tag 00EA.