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Quadratic algorithm

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In [[commutative algebra]], a '''regular local ring''' is a [[Noetherian]] [[local ring]] having the property that the minimal number of generators of its [[maximal ideal]] is equal to its [[Krull dimension]]. In symbols, let ''A'' be a Noetherian local ring with maximal ideal m, and suppose ''a''1, ..., ''a''''n'' is a minimal set of generators of m. Then by [[Krull's principal ideal theorem]] ''n'' ≥ dim ''A'', and ''A'' is defined to be regular if ''n'' = dim ''A''.

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

==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>\mbox{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 \{ \mbox{pd } M \mbox{ }|\mbox{ } M \mbox{ is an A-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>.

==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. Specifically, if ''k'' is a field and ''X'' is an indeterminate, then the ring of [[formal power series]] ''k''[[''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, which does not contain a field.
# More generally, if ''k'' is a field and ''X''1, ''X''2, ..., ''X''''d'' are indeterminates, then the ring of formal power series ''k''[[''X''1, ''X''2, ..., ''X''''d'']] is a regular local ring having (Krull) dimension ''d''.
# If ''A'' is a local ring, then it follows that the [[formal power series]] ring ''A''[[''x'']] is regular local.
# If '''Z''' is the ring of integers and ''X'' is an indeterminate, the ring '''Z'''[''X''](2, ''X'') is an example of a 2-dimensional regular local ring which does not contain a field.

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

Every [[localization of a ring|localization]] of a regular local ring is regular.

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.

==Origin of basic notions==
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 | pages=745–766}}</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}}</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}}</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 ''f1'',...,''fm''. ''Y'' is nonsingular at ''P'' if ''Y'' satisfies a [[Jacobian]] condition: If ''M'' = (∂''fi''/∂''xj'') 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'' − 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. However, [[Jean-Pierre Serre]] found a homological characterization of regular local rings: A local ring ''A'' is regular if and only if ''A'' has finite [[global dimension]]. 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 allows us to define regularity for all rings, not just local ones: A ring ''A'' is said to be a '''[[regular ring]]''' if its localizations at all of its prime ideals are regular local rings. It is equivalent to say that ''A'' has finite global dimension.

==Notes==
<references />

==References==
* [[Jean-Pierre Serre]], ''Local algebra'', [[Springer-Verlag]], 2000, ISBN 3-540-66641-9. Chap.IV.D.

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

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en>FrescoBot: Bot: link syntax and minor changes

207.38.151.173: /* Quadratic algorithm */