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*}

Regular sequences in Cohen-Macaulay local rings are characterized by cutting out something of the correct dimension.

Lemma 10.103.2. Let $R$ be a Noetherian local Cohen-Macaulay ring with maximal ideal $\mathfrak m $. Let $x_1, \ldots , x_ c \in \mathfrak m$ be elements. Then

\[ x_1, \ldots , x_ c \text{ is a regular sequence } \Leftrightarrow \dim (R/(x_1, \ldots , x_ c)) = \dim (R) - c \]

If so $x_1, \ldots , x_ c$ can be extended to a regular sequence of length $\dim (R)$ and each quotient $R/(x_1, \ldots , x_ i)$ is a Cohen-Macaulay ring of dimension $\dim (R) - i$.

Proof. Special case of Proposition 10.102.4. $\square$


Comments (1)

Comment #916 by Matthieu Romagny on

Suggested slogan: Regular sequences of Cohen-Macaulay local rings

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  • 7 comment(s) on Section 10.103: Cohen-Macaulay rings

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