Editing Regular local ring (section)

==Characterizations==
There are a number of useful definitions of a regular local ring, one of which is mentioned above. In particular, if <math>A</math> is a Noetherian local ring with maximal ideal <math>\mathfrak{m}</math>, then the following are equivalent definitions:
* Let <math>\mathfrak{m} = (a_1, \ldots, a_n)</math> where <math>n</math> is chosen as small as possible. Then <math>A</math> is regular if
::<math>\dim A = n\,</math>,
:where the dimension is the Krull dimension. The minimal set of generators of <math>a_1, \ldots, a_n</math> are then called a ''regular system of parameters''.
* Let <math>k = A / \mathfrak{m}</math> be the residue field of <math>A</math>. Then <math>A</math> is regular if
::<math>\dim_k \mathfrak{m} / \mathfrak{m}^2 = \dim A\,</math>,
:where the second dimension is the [[Krull dimension]].
* Let <math>\mbox{gl dim } A := \sup \{ \operatorname{pd} M \mid M \text{ is an }A\text{-module} \}</math> be the [[global dimension]] of <math>A</math> (i.e., the supremum of the [[projective dimension]]s of all <math>A</math>-modules.) Then <math>A</math> is regular if
::<math>\mbox{gl dim } A < \infty\,</math>,
:in which case, <math>\mbox{gl dim } A = \dim A</math>.

'''Multiplicity one criterion''' states:<ref>Herrmann, M., S. Ikeda, and U. Orbanz: Equimultiplicity and Blowing Up. An Algebraic Study with an Appendix by B. Moonen. Springer Verlag, Berlin Heidelberg New-York, 1988. Theorem 6.8.</ref> if the [[completion (algebra)|completion]] of a Noetherian local ring ''A'' is unimixed (in the sense that there is no embedded prime divisor of the zero ideal and for each minimal prime ''p'', <math>\dim \widehat{A}/p = \dim \widehat{A}</math>) and if the [[Hilbert–Samuel multiplicity|multiplicity]] of ''A'' is one, then ''A'' is regular. (The converse is always true: the multiplicity of a regular local ring is one.) This criterion corresponds to a geometric intuition in algebraic geometry that a local ring of an [[scheme-theoretic intersection|intersection]] is regular if and only if the intersection is a [[Transversality (mathematics)|transversal intersection]].

In the positive [[characteristic (algebra)|characteristic]] case, there is the following important result due to Kunz: A Noetherian local ring <math>R</math> of positive characteristic ''p'' is regular if and only if the [[Frobenius morphism]] <math>R \to R, r \mapsto r^p</math> is [[flat ring homomorphism|flat]] and <math>R</math> is [[reduced ring|reduced]]. No similar result is known in characteristic zero (it is unclear how one should replace the Frobenius morphism).