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 10.103.10. Let $\varphi : A \to B$ be a map of local rings. Assume that $B$ is Noetherian and Cohen-Macaulay and that $\mathfrak m_ B = \sqrt{\varphi (\mathfrak m_ A) B}$. Then there exists a sequence of elements $f_1, \ldots , f_{\dim (B)}$ in $A$ such that $\varphi (f_1), \ldots , \varphi (f_{\dim (B)})$ is a regular sequence in $B$.

Proof. By induction on $\dim (B)$ it suffices to prove: If $\dim (B) \geq 1$, then we can find an element $f$ of $A$ which maps to a nonzerodivisor in $B$. By Lemma 10.103.2 it suffices to find $f \in A$ whose image in $B$ is not contained in any of the finitely many minimal primes $\mathfrak q_1, \ldots , \mathfrak q_ r$ of $B$. By the assumption that $\mathfrak m_ B = \sqrt{\varphi (\mathfrak m_ A) B}$ we see that $\mathfrak m_ A \not\subset \varphi ^{-1}(\mathfrak q_ i)$. Hence we can find $f$ by Lemma 10.14.2. $\square$


Comments (0)

There are also:

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

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 06LC. Beware of the difference between the letter 'O' and the digit '0'.