Regular Local Rings

Regular local rings are the algebraic counterpart of smooth points on algebraic varieties. They are the best-behaved class of Noetherian local rings, characterized by having the smallest possible number of generators for their maximal ideal.

Definition and Characterizations

Definition

A Noetherian local ring $(R, \mathfrak{m}, k)$ is regular if the maximal ideal $\mathfrak{m}$ can be generated by $d = \dim R$ elements. Equivalently, $R$ is regular if $\operatorname{edim}(R) = \dim R$ , where $\operatorname{edim}(R) = \dim_k(\mathfrak{m}/\mathfrak{m}^2)$ is the embedding dimension. A regular system of parameters $x_1, \ldots, x_d$ generating $\mathfrak{m}$ is the algebraic analogue of local coordinates.

Since $\operatorname{edim}(R) \geq \dim R$ always holds (by Krull's height theorem), regular local rings are precisely those where equality is achieved.

Definition

A Noetherian ring $R$ is regular if the localization $R_\mathfrak{p}$ is a regular local ring for every prime ideal $\mathfrak{p}$ . A variety $X$ over a field is smooth (or nonsingular) if its local ring $\mathcal{O}_{X,x}$ is regular for every point $x \in X$ .

Examples

ExampleRegular and singular rings

Every field is a regular local ring of dimension $0$ .
A discrete valuation ring (DVR) is a regular local ring of dimension $1$ .
$k[[x_1, \ldots, x_n]]$ and $k[x_1, \ldots, x_n]_{(x_1, \ldots, x_n)}$ are regular of dimension $n$ .
$k[[x, y]]/(y^2 - x^3)$ is not regular: $\dim = 1$ but $\operatorname{edim} = 2$ (the cusp).
$k[[x, y]]/(y^2 - x^2(x+1))$ is not regular: the node has $\operatorname{edim} = 2 > \dim = 1$ .

The Associated Graded Ring

Theorem8.1Regularity via the Associated Graded Ring

A Noetherian local ring $(R, \mathfrak{m})$ is regular if and only if the associated graded ring $\operatorname{gr}_\mathfrak{m}(R) = \bigoplus_{n \geq 0} \mathfrak{m}^n / \mathfrak{m}^{n+1}$ is isomorphic to a polynomial ring $k[X_1, \ldots, X_d]$ where $d = \dim R$ .

RemarkGeometric intuition

Geometrically, $\operatorname{gr}_\mathfrak{m}(R)$ is the coordinate ring of the tangent cone at the point corresponding to $\mathfrak{m}$ . Regularity means the tangent cone is all of affine $d$ -space, i.e., the point is smooth. Singular points have tangent cones that are proper subvarieties of affine space.