Regular local ring - Knowlpedia

A Noetherian local ring $(R,\mathfrak m)$ is regular if its maximal ideal can be generated by $\dim R$ elements, where $\dim R$ is its Krull dimension.

Dimension criterion and examples

Equivalently,

\dim_{R/\mathfrak m}(\mathfrak m/\mathfrak m^2)=\dim R,

where $\dim R$ is the Krull dimension of $R$ , and $\mathfrak m/\mathfrak m^2$ is viewed as an $R/\mathfrak m$ -vector space. The Noetherian hypothesis is part of this convention. For example, $k[x_1,\ldots,x_n]_{(x_1,\ldots,x_n)}$ is regular of dimension $n$ .

This is the local dimension-theoretic notion used to define a regular scheme and to describe nonsingular fibers.

Reference

Stacks Project, Properties of Schemes, Section 28.9: https://stacks.math.columbia.edu/tag/02IR