The Stacks project

Lemma 10.7.4. Let $\varphi : R \to S$ be a ring map.

  1. If $\varphi $ is finite, then $\varphi $ is of finite type.

  2. If $S$ is of finite presentation as an $R$-module, then $\varphi $ is of finite presentation.

Proof. For (1) if $x_1, \ldots , x_ n \in S$ generate $S$ as an $R$-module, then $x_1, \ldots , x_ n$ generate $S$ as an $R$-algebra. For (2), suppose that $\sum r_ j^ ix_ i = 0$, $j = 1, \ldots , m$ is a set of generators of the relations among the $x_ i$ when viewed as $R$-module generators of $S$. Furthermore, write $1 = \sum r_ ix_ i$ for some $r_ i \in R$ and $x_ ix_ j = \sum r_{ij}^ k x_ k$ for some $r_{ij}^ k \in R$. Then

\[ S = R[t_1, \ldots , t_ n]/ (\sum r_ j^ it_ i,\ 1 - \sum r_ it_ i,\ t_ it_ j - \sum r_{ij}^ k t_ k) \]

as an $R$-algebra which proves (2). $\square$

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