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 10.118.3. Let $R$ be a local ring with maximal ideal $\mathfrak m$. Assume $R$ is Noetherian, has dimension $1$, and that $\dim (\mathfrak m/\mathfrak m^2) > 1$. Then there exists a ring map $R \to R'$ such that

  1. $R \to R'$ is finite,

  2. $R \to R'$ is not an isomorphism,

  3. the kernel and cokernel of $R \to R'$ are annihilated by a power of $\mathfrak m$, and

  4. $\mathfrak m$ is not an associated prime of $R'$.

Proof. This follows from Lemma 10.118.2 and the fact that $R$ is not Artinian, not regular, and does not have depth $\geq 2$ (the last part because the depth does not exceed the dimension by Lemma 10.71.3). $\square$


Comments (2)

Comment #1504 by Kollar on

Dear Johan,

2 comments on Lemma tag 00P9.

  1. About R' you want to also claim that m is not an associated prime of R'. (To avoid eg R'=R+(R/m) with R/m being nilpotent.)

  2. You may like the following variant of Serre's criterion of normality, which implies the Lemma (and other things as well).

Let be a local Noetherian ring. Then exactly one of the following holds.

  1. is Artinian,
  2. is regular of dimension ,
  3. or
  4. is a non-normal pair (= in your terminology: there is an R' as you want)

\end{lem}

Proof.
is not Artinian iff is nowhere dense; we assume this from now on.

Let be the largest ideal killed by a power of . If then shows that is a non-normal pair.

Otherwise and there is an that is not a zero-divisor. If is not an associated prime of then . Thus we are left with the case when there is an such that .

If then, by the determinantal trick proposition 2.4 of Atiyah--Macdonald, satisfies a monic polynomial hence shows that is a non-normal pair.

Otherwise there is a such that ; in particular is not a zero-divisor. For any we have for some . Thus .
Since is not a zero-divisor this implies that and so . Thus is regular of dimension 1.

Comment #1508 by on

OK, thank you very much. I added your lemma and I used it to prove this one. See this commit. Of course this lemma can be used to give a quick proof of Serre's criterion, but I haven't made those changes yet.


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 00P9. Beware of the difference between the letter 'O' and the digit '0'.