The Stacks project

Lemma 10.31.4. Let $R$ be a Noetherian ring.

  1. Any finite $R$-module is of finite presentation.

  2. Any submodule of a finite $R$-module is finite.

  3. Any finite type $R$-algebra is of finite presentation over $R$.

Proof. Let $M$ be a finite $R$-module. By Lemma 10.5.4 we can find a finite filtration of $M$ whose successive quotients are of the form $R/I$. Since any ideal is finitely generated, each of the quotients $R/I$ is finitely presented. Hence $M$ is finitely presented by Lemma 10.5.3. This proves (1).

Let $N \subset M$ be a submodule. As $M$ is finite, the quotient $M/N$ is finite. Thus $M/N$ is of finite presentation by part (1). Thus we see that $N$ is finite by Lemma 10.5.3 part (5). This proves part (2).

To see (3) note that any ideal of $R[x_1, \ldots , x_ n]$ is finitely generated by Lemma 10.31.1. $\square$

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