Editing Regular local ring

{{Short description|Type of ring in commutative algebra}}
In [[commutative algebra]], a '''regular local ring''' is a [[Noetherian ring|Noetherian]] [[local ring]] having the property that the minimal number of [[generating set of a module|generators]] of its [[maximal ideal]] is equal to its [[Krull dimension]].{{sfn|Atiyah|Macdonald|1969|p=123|loc=Theorem 11.22}} In symbols, let <math>A</math> be any Noetherian local ring with unique maximal ideal <math>\mathfrak{m}</math>, and suppose <math>a_1,\cdots,a_n</math> is a minimal set of generators of <math>\mathfrak{m}</math>. Then [[Krull's principal ideal theorem]] implies that <math>n\geq\dim A</math>, and <math>A</math> is regular whenever <math>n=\dim A</math>.

The concept is motivated by its geometric meaning. A point <math>x</math> on an [[algebraic variety]] <math>X</math> is [[Singular point of an algebraic variety|nonsingular]] (a [[Smooth scheme|smooth point]]) if and only if the local ring <math>\mathcal{O}_{X, x}</math> of [[germ (mathematics)|germs]] at <math>x</math> is regular. (See also: [[regular scheme]].) Regular local rings are ''not'' related to [[von Neumann regular ring]]s.{{efn|A local von Neumann regular ring is a division ring, so the two conditions are not very compatible.}}

For Noetherian local rings, there is the following chain of inclusions:
{{Commutative local ring classes}}

==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).

==Examples==
# Every [[field (mathematics)|field]] is a regular local ring. These have (Krull) dimension 0. In fact, the fields are exactly the regular local rings of dimension 0.
# Any [[discrete valuation ring]] is a regular local ring of dimension 1 and the regular local rings of dimension 1 are exactly the discrete valuation rings. For example, if ''k'' is a field and ''X'' is an indeterminate, then the ring of [[formal power series]] ''k''{{brackets|''X''}} is a regular local ring having (Krull) dimension 1.
# If ''p'' is an ordinary prime number, the ring of [[p-adic integer]]s is an example of a discrete valuation ring, and consequently a regular local ring. In contrast to the example above, this ring does not contain a field.
# More generally, if ''k'' is a field and ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''d''</sub> are indeterminates, then the ring of formal power series ''k''{{brackets|''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''d''</sub>}} is a regular local ring having (Krull) dimension ''d''.
# Still more generally, if ''A'' is a regular local ring, then the [[formal power series]] ring ''A''{{brackets|''x''}} is regular local.
# If '''Z''' is the ring of integers and ''X'' is an indeterminate, the ring '''Z'''[''X'']<sub>(2, ''X'')</sub> (i.e. the ring '''Z'''[''X''] [[Localization of a ring and a module|localized]] in the prime ideal (2, ''X'') ) is an example of a 2-dimensional regular local ring which does not contain a field.
# By the [[cohen structure theorem|structure theorem]] of [[Irvin Cohen]], a [[completion (ring theory)|complete]] regular local ring of Krull dimension ''d'' that contains a field ''k'' is a power series ring in ''d'' variables over an [[extension field]] of ''k''.

== Non-examples ==
The ring <math>A=k[x]/(x^2)</math> is not a regular local ring since it is finite dimensional but does not have finite global dimension. For example, there is an infinite resolution
:<math>
\cdots \xrightarrow{\cdot x} \frac{k[x]}{(x^2)} \xrightarrow{\cdot x} \frac{k[x]}{(x^2)} \to k \to 0 
</math>

Using another one of the characterizations, <math>A</math> has exactly one prime ideal <math>\mathfrak{m}=\frac{(x)}{(x^2)}</math>, so the ring has Krull dimension <math>0</math>, but <math>\mathfrak{m}^2</math> is the zero ideal, so <math>\mathfrak{m}/\mathfrak{m}^2</math> has <math>k</math> dimension at least <math>1</math>. (In fact it is equal to <math>1</math> since <math>x + \mathfrak{m}</math> is a basis.)

==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]].

==Origin of basic notions==
{{see also|smooth scheme}}
Regular local rings were originally defined by [[Wolfgang Krull]] in 1937,<ref>{{Citation | last1=Krull | first1=Wolfgang | author1-link= Wolfgang Krull | title=Beiträge zur Arithmetik kommutativer Integritätsbereiche III | journal=Math. Z. | year=1937 | volume=42 | pages=745–766 | doi = 10.1007/BF01160110}}</ref> but they first became prominent in the work of [[Oscar Zariski]] a few years later,<ref>{{Citation | last1=Zariski | first1=Oscar | author1-link=Oscar Zariski | title=Algebraic varieties over ground fields of characteristic 0 | journal=Amer. J. Math. | year=1940 | volume=62 | pages=187–221 | doi=10.2307/2371447| jstor=2371447 }}</ref><ref>{{Citation | last1=Zariski | first1=Oscar | author1-link=Oscar Zariski | title=The concept of a simple point of an abstract algebraic variety | journal=Trans. Amer. Math. Soc. | year=1947 | volume=62 | pages=1–52 | doi=10.1090/s0002-9947-1947-0021694-1| doi-access=free }}</ref> who showed that geometrically, a regular local ring corresponds to a smooth point on an [[algebraic variety]].  Let ''Y'' be an [[algebraic variety]] contained in affine ''n''-space over a [[perfect field]], and suppose that ''Y'' is the vanishing locus of the polynomials ''f<sub>1</sub>'',...,''f<sub>m</sub>''.  ''Y'' is nonsingular at ''P'' if ''Y'' satisfies a [[Jacobian variety|Jacobian condition]]: If ''M'' = (∂''f<sub>i</sub>''/∂''x<sub>j</sub>'') is the matrix of partial derivatives of the defining equations of the variety, then the rank of the matrix found by evaluating ''M'' at ''P'' is ''n'' &minus; dim ''Y''.  Zariski proved that ''Y'' is nonsingular at ''P'' if and only if the local ring of ''Y'' at ''P'' is regular. (Zariski observed that this can fail over non-perfect fields.)  This implies that smoothness is an intrinsic property of the variety, in other words it does not depend on where or how the variety is embedded in affine space.  It also suggests that regular local rings should have good properties, but before the introduction of techniques from [[homological algebra]] very little was known in this direction.  Once such techniques were introduced in the 1950s, Auslander and Buchsbaum proved that every regular local ring is a [[unique factorization domain]].

Another property suggested by geometric intuition is that the localization of a regular local ring should again be regular.   Again, this lay unsolved until the introduction of homological techniques.  It was [[Jean-Pierre Serre]] who found a homological characterization of regular local rings: A local ring ''A'' is regular if and only if ''A'' has finite [[global dimension]], i.e. if every ''A''-module has a projective resolution of finite length.  It is easy to show that the property of having finite global dimension is preserved under localization, and consequently that localizations of regular local rings at prime ideals are again regular.  

This justifies the definition of ''regularity'' for non-local commutative rings given in the next section.

==Regular ring==
{{for|the unrelated regular rings introduced by John von Neumann|von Neumann regular ring}}
In [[commutative algebra]], a '''regular ring''' is a commutative [[Noetherian ring]], such that the [[localization of a ring|localization]] at every [[prime ideal]] is a regular local ring: that is, every such localization has the property that the minimal number of generators of its maximal ideal is equal to its [[Krull dimension]].

The origin of the term ''regular ring'' lies in the fact that an [[affine variety]] is [[nonsingular variety|nonsingular]] (that is every point is [[regular point of an algebraic variety|regular]]) if and only if its [[ring of regular functions]] is regular.

For regular rings, Krull dimension agrees with [[global homological dimension]].

[[Jean-Pierre Serre]] defined a regular ring as a commutative noetherian ring of ''finite'' global homological dimension. His definition is stronger than the definition above, which allows regular rings of infinite Krull dimension.

Examples of regular rings include fields (of dimension zero) and [[Dedekind domain]]s.  If ''A'' is regular then so is ''A''[''X''], with dimension one greater than that of ''A''. 

In particular if {{mvar|k}} is a field, the ring of integers, or a [[principal ideal domain]], then the [[polynomial ring]] <math>k[X_1, \ldots,X_n]</math> is regular. In the case of a field, this is [[Hilbert's syzygy theorem]].

Any localization of a regular ring is regular as well.

A regular ring is [[reduced ring|reduced]]{{efn|since a ring is reduced if and only if its localizations at prime ideals are.}} but need not be an integral domain. For example, the product of two regular integral domains is regular, but not an integral domain.<ref>[https://math.stackexchange.com/q/18657 Is a regular ring a domain]</ref>

== See also ==
*[[Geometrically regular ring]]
*[[quasi-free ring]]

==Notes==
{{Notelist}}

==Citations==
{{Reflist}}

==References==
*{{Citation | last1=Atiyah | first1=Michael F. | author1-link=Michael Atiyah | last2=Macdonald | first2=Ian G.|author2-link=Ian G. Macdonald | title=Introduction to Commutative Algebra | publisher=[[Addison-Wesley]] | mr=0242802 | year=1969}}
* Kunz, Characterizations of regular local rings of characteristic p. Amer. J. Math. 91 (1969), 772–784.
* [[Tsit-Yuen Lam]], ''Lectures on Modules and Rings'', [[Springer-Verlag]], 1999, {{isbn|978-1-4612-0525-8}}.  Chap.5.G.
* [[Jean-Pierre Serre]], ''Local algebra'', [[Springer-Verlag]], 2000, {{ISBN|3-540-66641-9}}.  Chap.IV.D.
* [http://stacks.math.columbia.edu/tag/065U Regular rings at The Stacks Project]

{{DEFAULTSORT:Regular Local Ring}}
[[Category:Algebraic geometry]]
[[Category:Ring theory]]