Let A —> B be a ring map. Assume

- A ⊂ B is an extension of domains,
- A is Noetherian,
- A —> B is of finite type, and
- the extension f.f.(A) ⊂ f.f.(B) is finite.

Let p ⊂ A be a prime such that dim(A_{p}) = 1. Then there are at most finitely many primes of B lying over p. See Tag 02MA.

I believe you can eliminate the Noetherian hypothesis by replacing (3) by the hypothesis (3′) p is finitely generated and by replacing (4) by (4′) there exists nonzero f in A such that B[1/f] is a finitely generated A[1/f]-module.