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 10.156.13. Let $(R, \mathfrak m)$ be a local ring. If $R$ is Noetherian, a domain, and Nagata, then $R$ is analytically unramified.

Proof. By induction on $\dim (R)$. The case $\dim (R) = 0$ is trivial. Hence we assume $\dim (R) = d$ and that the lemma holds for all Noetherian Nagata domains of dimension $< d$.

Let $R \subset S$ be the integral closure of $R$ in the field of fractions of $R$. By assumption $S$ is a finite $R$-module. By Lemma 10.156.5 we see that $S$ is Nagata. By Lemma 10.111.4 we see $\dim (R) = \dim (S)$. Let $\mathfrak m_1, \ldots , \mathfrak m_ t$ be the maximal ideals of $S$. Each of these lies over the maximal ideal $\mathfrak m$ of $R$. Moreover

\[ (\mathfrak m_1 \cap \ldots \cap \mathfrak m_ t)^ n \subset \mathfrak mS \]

for sufficiently large $n$ as $S/\mathfrak mS$ is Artinian. By Lemma 10.96.2 $R^\wedge \to S^\wedge $ is an injective map, and by the Chinese Remainder Lemma 10.14.4 combined with Lemma 10.95.9 we have $S^\wedge = \prod S^\wedge _ i$ where $S^\wedge _ i$ is the completion of $S$ with respect to the maximal ideal $\mathfrak m_ i$. Hence it suffices to show that $S_{\mathfrak m_ i}$ is analytically unramified. In other words, we have reduced to the case where $R$ is a Noetherian normal Nagata domain.

Assume $R$ is a Noetherian, normal, local Nagata domain. Pick a nonzero $x \in \mathfrak m$ in the maximal ideal. We are going to apply Lemma 10.156.12. We have to check properties (1), (2), (3)(a) and (3)(b). Property (1) is clear. We have that $R/xR$ has no embedded primes by Lemma 10.151.6. Thus property (2) holds. The same lemma also tells us each associated prime $\mathfrak p$ of $R/xR$ has height $1$. Hence $R_{\mathfrak p}$ is a $1$-dimensional normal domain hence regular (Lemma 10.118.7). Thus (3)(a) holds. Finally (3)(b) holds by induction hypothesis, since $R/\mathfrak p$ is Nagata (by Lemma 10.156.5 or directly from the definition). Thus we conclude $R$ is analytically unramified. $\square$

Comments (7)

Comment #1513 by kollar on

It may make sense to add the partial converse: if R is analytically unramified then S (the normalization) is finite over R. See 32.2 of Nagata's book.

Comment #1515 by on

Yes, you are right. In fact this is Lemma 51.11.5 some algebraic version of which should probably be put much earlier...

Comment #1519 by kollar on

I think one can make it into an equivalence: R is Nagata iff every domain that is finite over R is analytically unramified.

Comment #1524 by kollar on

I should add that this ties it very nicely with G-rings: A local ring is nagata iff the formal fibers are reduced.

Comment #1526 by on

@#1524: This is not true because there exists a discrete valuation ring which is not Nagata, see Example 10.156.17. I think you meant to say the formal fibres are geometrically reduced. We have a tiny bit of discussion of formal fibres and G-rings in Section 15.49. To adequately cover your suggestion we would need to rewrite most of that section so it applies to a property of formal fibres, such as geometrically reduced, geometrically normal, Gorenstein, CM, CI, geometrically regular, etc, ec. We will do this if we ever need it.

Comment #1553 by on

@#1524: OK, I decided to add some of this material. So now your remark is Lemma 15.51.4. I agree that the proof is not optimal... The change in the latex code is here.

There are also:

  • 2 comment(s) on Section 10.156: Nagata 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 0331. Beware of the difference between the letter 'O' and the digit '0'.