Regular scheme

Remarks

A regular local ring is an integrally closed domain, so a regular scheme is normal. Example: $\operatorname{Spec} \Z$ is regular. Its local rings are $\Q$, of dimension $0$ with $\mathfrak{m} = 0$, and $\Z_{(p)}$, of dimension $1$ with $\mathfrak{m}/\mathfrak{m}^2$ spanned by $p$. Non-example: the cusp $\operatorname{Spec} k[x, y]/(y^2 - x^3)$ over a field $k$ is not regular at the origin. There the local ring has dimension $1$, but $\mathfrak{m}/\mathfrak{m}^2$ has basis $x, y$, since $y^2 - x^3 \in \mathfrak{m}^2$; so its dimension is $2 \ne 1$.