The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

10.7 Finite ring maps

Here is the definition.

Definition 10.7.1. Let $\varphi : R \to S$ be a ring map. We say $\varphi : R \to S$ is finite if $S$ is finite as an $R$-module.

Lemma 10.7.2. Let $R \to S$ be a finite ring map. Let $M$ be an $S$-module. Then $M$ is finite as an $R$-module if and only if $M$ is finite as an $S$-module.

Proof. One of the implications follows from Lemma 10.5.5. To see the other assume that $M$ is finite as an $S$-module. Pick $x_1, \ldots , x_ n \in S$ which generate $S$ as an $R$-module. Pick $y_1, \ldots , y_ m \in M$ which generate $M$ as an $S$-module. Then $x_ i y_ j$ generate $M$ as an $R$-module. $\square$

Lemma 10.7.3. Suppose that $R \to S$ and $S \to T$ are finite ring maps. Then $R \to T$ is finite.

Proof. If $t_ i$ generate $T$ as an $S$-module and $s_ j$ generate $S$ as an $R$-module, then $t_ i s_ j$ generate $T$ as an $R$-module. (Also follows from Lemma 10.7.2.) $\square$

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). $\square$

For more information on finite ring maps, please see Section 10.35.


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