Regular local ring

There are deep connections between algebraic (in fact, scheme-theoretic) notions of smoothness and regularity.

Definition

Let $A$ be a Noetherian local ring with maximal ideal $m$ and residual field $k=A/m$. The following conditions are equivalent:

1. The Krull dimension of $A$ is equal to the dimension of the $k$-vector space $m/m^{2}$.

And when these conditions hold, $A$ is called a regular local ring.

Basic Results on Regular Local Rings

One important criterion for regularity is Serre's Criterion, which states that a Noetherian local ring $A$ is regular if and only if its global dimension is finite, in which case it is equal to the krull dimension of $A$.

In a paper of Auslander and Buchsbaum published in 1959, it was shown that every regular local ring is a UFD.

Regular Rings

A regular ring is a Noetherian ring such that the localisation at every prime is a regular local ring.