Editing Regular local ring (section)

==Basic properties==
The [[Auslander–Buchsbaum theorem]] states that every regular local ring is a [[unique factorization domain]].

Every [[localization of a ring|localization]], as well as the [[completion (ring theory)|completion]], of a regular local ring is regular.

If <math>(A, \mathfrak{m})</math> is a complete regular local ring that contains a field, then
:<math>A \cong k[[x_1, \ldots, x_d]]</math>,
where <math>k = A / \mathfrak{m}</math> is the [[residue field]], and <math>d = \dim A</math>, the Krull dimension.

See also: [[Serre's inequality on height]] and [[Serre's multiplicity conjectures]].