Lemma of the day

Let A —> B be a ring map. Assume

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

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

One thought on “Lemma of the day

  1. 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.

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