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*}

Lemma 10.13.2. Let $R \to S$ be a ring map. Let $M$ be an $S$-module. Let $R \to R'$ be a ring map and let $S' = S \otimes _ R R'$ and $M' = M \otimes _ R R'$ be the base changes.

  1. If $M$ is a finite $S$-module, then the base change $M'$ is a finite $S'$-module.

  2. If $M$ is an $S$-module finite presentation, then the base change $M'$ is an $S'$-module of finite presentation.

  3. If $R \to S$ is of finite type, then the base change $R' \to S'$ is of finite type.

  4. If $R \to S$ is of finite presentation, then the base change $R' \to S'$ is of finite presentation.

Proof. Proof of (1). Take a surjective, $S$-linear map $S^{\oplus n} \to M \to 0$. By Lemma 10.11.3 and 10.11.10 the result after tensoring with $R^\prime $ is a surjection ${S^\prime }^{\oplus n} \to M^\prime \rightarrow 0$, so $M^\prime $ is a finitely generated $S^\prime $-module. Proof of (2). Take a presentation $S^{\oplus m} \to S^{\oplus n} \to M \to 0$. By Lemma 10.11.3 and 10.11.10 the result after tensoring with $R^\prime $ gives a finite presentation ${S^\prime }^{\oplus m} \to {S^\prime }^{\oplus n} \to M^\prime \to 0$, of the $S^\prime $-module $M^\prime $. Proof of (3). This follows by the remark preceding the lemma as we can take $I$ to be finite by assumption. Proof of (4). This follows by the remark preceding the lemma as we can take $I$ and $J$ to be finite by assumption. $\square$


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