Lemma of the day

Let A be a valuation ring. Let A→B be a ring map of finite type. Let M be a finite B-module.

  1. If B is flat over A, then B is a finitely presented A-algebra.
  2. If M is flat as an A-module, then M is finitely presented as a B-module.

See Tag 053E.

PS: Much more is true, see the this chapter in the stacks project. The proof of the lemma above however is quite easy.