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 10.6.2. The notions finite type and finite presentation have the following permanence properties.

  1. A composition of ring maps of finite type is of finite type.

  2. A composition of ring maps of finite presentation is of finite presentation.

  3. Given $R \to S' \to S$ with $R \to S$ of finite type, then $S' \to S$ is of finite type.

  4. Given $R \to S' \to S$, with $R \to S$ of finite presentation, and $R \to S'$ of finite type, then $S' \to S$ is of finite presentation.

Proof. We only prove the last assertion. Write $S = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m)$ and $S' = R[y_1, \ldots , y_ a]/I$. Say that the class $\bar y_ i$ of $y_ i$ maps to $h_ i \bmod (f_1, \ldots , f_ m)$ in $S$. Then it is clear that $S = S'[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m, h_1 - \bar y_1, \ldots , h_ a - \bar y_ a)$. $\square$


Comments (3)

Comment #584 by Wei Xu on

A in last sentence: "Then it is clear that .", should be and should be .

Comment #586 by Anfang on

There is also an index error. It should be .


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