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The Stacks project

Lemma 15.80.9. Let R \to A be a finite type ring map. Let 0 \to M' \to M \to M'' \to 0 be a short exact sequence of A-modules.

  1. If M', M'' are finitely presented relative to R, then so is M.

  2. If M' is a finite type A-module and M is finitely presented relative to R, then M'' is finitely presented relative to R.

Proof. Follows immediately from Algebra, Lemma 10.5.3. \square

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