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*}

This is taken from a forthcoming paper by János Kollár entitled “Variants of normality for Noetherian schemes”.

Lemma 10.118.2 (Kollár). Let $(R, \mathfrak m)$ be a local Noetherian ring. Then exactly one of the following holds:

  1. $(R, \mathfrak m)$ is Artinian,

  2. $(R, \mathfrak m)$ is regular of dimension $1$,

  3. $\text{depth}(R) \geq 2$, or

  4. there exists a finite ring map $R \to R'$ which is not an isomorphism whose kernel and cokernel are annihilated by a power of $\mathfrak m$ such that $\mathfrak m$ is not an associated prime of $R'$.

Proof. Observe that $(R, \mathfrak m)$ is not Artinian if and only if $V(\mathfrak m) \subset \mathop{\mathrm{Spec}}(R)$ is nowhere dense. See Proposition 10.59.6. We assume this from now on.

Let $J \subset R$ be the largest ideal killed by a power of $\mathfrak m$. If $J \not= 0$ then $R \to R/J$ shows that $(R, \mathfrak m)$ is as in (4).

Otherwise $J = 0$. In particular $\mathfrak m$ is not an associated prime of $R$ and we see that there is a nonzerodivisor $x \in \mathfrak m$ by Lemma 10.62.18. If $\mathfrak m$ is not an associated prime of $R/xR$ then $\text{depth}(R) \geq 2$ by the same lemma. Thus we are left with the case when there is an $y \in R$, $y \not\in xR$ such that $y \mathfrak m \subset xR$.

If $y \mathfrak m \subset x \mathfrak m$ then we can consider the map $\varphi : \mathfrak m \to \mathfrak m$, $f \mapsto yf/x$ (well defined as $x$ is a nonzerodivisor). By the determinantal trick of Lemma 10.15.2 there exists a monic polynomial $P$ with coefficients in $R$ such that $P(\varphi ) = 0$. We conclude that $P(y/x) = 0$ in $R_ x$. Let $R' \subset R_ x$ be the ring generated by $R$ and $y/x$. Then $R \subset R'$ and $R'/R$ is a finite $R$-module annihilated by a power of $x$. Thus $R$ is as in (4).

Otherwise there is a $t \in \mathfrak m$ such that $y t = u x$ for some unit $u$ of $R$. After replacing $t$ by $u^{-1}t$ we get $yt = x$. In particular $y$ is a nonzerodivisor. For any $t' \in \mathfrak m$ we have $y t' = x s$ for some $s \in R$. Thus $y (t' - s t ) = x s - x s = 0$. Since $y$ is not a zero-divisor this implies that $t' = ts$ and so $\mathfrak m = (t)$. Thus $(R, \mathfrak m)$ is regular of dimension 1. $\square$


Comments (3)

Comment #2227 by David Savitt on

I guess you need a little argument to see that (2) and (3) each exclude (4). [For (2), is maximal CM hence free of rank one. For (3), use the long exact sequence of associated to .]

Comment #2230 by on

Yes, this proof is really a bit too terse. I shall work on it when I go through the comments in a few weeks.

Comment #3818 by Sandor Kovacs on

At the end of the fourth paragraph (starting with "If then...") it should be saying that "R′/R is a finite R-module annihilated by a power of " not that "R′/R is a finite R-module annihilated by a power of x".


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