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*}

27.9 Regular schemes

Recall, see Algebra, Definition 10.59.9, that a local Noetherian ring $(R, \mathfrak m)$ is said to be regular if $\mathfrak m$ can be generated by $\dim (R)$ elements. Recall that a Noetherian ring $R$ is said to be regular if every local ring $R_{\mathfrak p}$ of $R$ is regular, see Algebra, Definition 10.109.7.

Definition 27.9.1. Let $X$ be a scheme. We say $X$ is regular, or nonsingular if for every $x \in X$ there exists an affine open neighbourhood $U \subset X$ of $x$ such that the ring $\mathcal{O}_ X(U)$ is Noetherian and regular.

Lemma 27.9.2. Let $X$ be a scheme. The following are equivalent:

  1. $X$ is regular,

  2. $X$ is locally Noetherian and all of its local rings are regular, and

  3. $X$ is locally Noetherian and for any closed point $x \in X$ the local ring $\mathcal{O}_{X, x}$ is regular.

Proof. By the discussion in Algebra preceding Algebra, Definition 10.109.7 we know that the localization of a regular local ring is regular. The lemma follows by combining this with Lemma 27.5.2, with the existence of closed points on locally Noetherian schemes (Lemma 27.5.9), and the definitions. $\square$

Lemma 27.9.3. Let $X$ be a scheme. The following are equivalent:

  1. The scheme $X$ is regular.

  2. For every affine open $U \subset X$ the ring $\mathcal{O}_ X(U)$ is Noetherian and regular.

  3. There exists an affine open covering $X = \bigcup U_ i$ such that each $\mathcal{O}_ X(U_ i)$ is Noetherian and regular.

  4. There exists an open covering $X = \bigcup X_ j$ such that each open subscheme $X_ j$ is regular.

Moreover, if $X$ is regular then every open subscheme is regular.

Proof. See Algebra, Lemma 10.151.5. $\square$


Comments (0)


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 02IR. Beware of the difference between the letter 'O' and the digit '0'.