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 10.128.1. Let $k$ be a field. Let $S$ be a finite type $k$-algebra. Let $f_1, \ldots , f_ i$ be elements of $S$. Assume that $S$ is Cohen-Macaulay and equidimensional of dimension $d$, and that $\dim V(f_1, \ldots , f_ i) \leq d - i$. Then equality holds and $f_1, \ldots , f_ i$ forms a regular sequence in $S_{\mathfrak q}$ for every prime $\mathfrak q$ of $V(f_1, \ldots , f_ i)$.

Proof. If $S$ is Cohen-Macaulay and equidimensional of dimension $d$, then we have $\dim (S_{\mathfrak m}) = d$ for all maximal ideals $\mathfrak m$ of $S$, see Lemma 10.113.7. By Proposition 10.102.4 we see that for all maximal ideals $\mathfrak m \in V(f_1, \ldots , f_ i)$ the sequence is a regular sequence in $S_{\mathfrak m}$ and the local ring $S_{\mathfrak m}/(f_1, \ldots , f_ i)$ is Cohen-Macaulay of dimension $d - i$. This actually means that $S/(f_1, \ldots , f_ i)$ is Cohen-Macaulay and equidimensional of dimension $d - i$. $\square$


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