Regular local ring

From Citizendium
Jump to navigation Jump to search
This article is a stub and thus not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.

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


Definition

Let be a Noetherian local ring with maximal ideal and residual field . The following conditions are equivalent:

  1. The Krull dimension of is equal to the dimension of the -vector space .

And when these conditions hold, 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 is regular if and only if its global dimension is finite, in which case it is equal to the krull dimension of .

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

Regular Rings

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

References