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*}

Noetherian property is stable by passage to finite type extension and localization.

Lemma 10.30.1. Any finitely generated ring over a Noetherian ring is Noetherian. Any localization of a Noetherian ring is Noetherian.

Proof. The statement on localizations follows from the fact that any ideal $J \subset S^{-1}R$ is of the form $I \cdot S^{-1}R$. Any quotient $R/I$ of a Noetherian ring $R$ is Noetherian because any ideal $\overline{J} \subset R/I$ is of the form $J/I$ for some ideal $I \subset J \subset R$. Thus it suffices to show that if $R$ is Noetherian so is $R[X]$. Suppose $J_1 \subset J_2 \subset \ldots $ is an ascending chain of ideals in $R[X]$. Consider the ideals $I_{i, d}$ defined as the ideal of elements of $R$ which occur as leading coefficients of degree $d$ polynomials in $J_ i$. Clearly $I_{i, d} \subset I_{i', d'}$ whenever $i \leq i'$ and $d \leq d'$. By the ascending chain condition in $R$ there are at most finitely many distinct ideals among all of the $I_{i, d}$. (Hint: Any infinite set of elements of $\mathbf{N} \times \mathbf{N}$ contains an increasing infinite sequence.) Take $i_0$ so large that $I_{i, d} = I_{i_0, d}$ for all $i \geq i_0$ and all $d$. Suppose $f \in J_ i$ for some $i \geq i_0$. By induction on the degree $d = \deg (f)$ we show that $f \in J_{i_0}$. Namely, there exists a $g\in J_{i_0}$ whose degree is $d$ and which has the same leading coefficient as $f$. By induction $f - g \in J_{i_0}$ and we win. $\square$


Comments (1)

Comment #911 by Matthieu Romagny on

Suggested slogan: Noetherian property is stable by passage to finite type extension and localization.


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