Regular local ring

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 nonsingular if and only if the local ring formula_1 of germs at "x" is regular. (See also: regular scheme.) Regular local rings are "not" related to von Neumann regular rings.
For Noetherian local rings, there is the following chain of inclusions:
Characterizations.
There are a number of useful definitions of a regular local ring, one of which is mentioned above. In particular, if formula_2 is a Noetherian local ring with maximal ideal formula_3, then the following are equivalent definitions
Multiplicity one criterion states: if the 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", formula_19) and if the 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 intersection is regular if and only if the intersection is a transversal intersection.
In the positive characteristic case, there is the following important result due to Kunz: A Noetherian local ring formula_20 of positive characteristic "p" is regular if and only if the Frobenius morphism formula_21 is flat and formula_20 is reduced. No similar result is known in the characteristic zero (just because how to replace Frobenius is unclear).
Non-Examples.
The ring formula_23 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
Basic properties.
The Auslander–Buchsbaum theorem states that every regular local ring is a unique factorization domain.
Every localization of a regular local ring is regular.
The completion of a regular local ring is regular.
If formula_34 is a complete regular local ring that contains a field, then
where formula_9 is the residue field, and formula_37, the Krull dimension.
See also: Serre's inequality on height and Serre's multiplicity conjectures.
Origin of basic notions.
Regular local rings were originally defined by Wolfgang Krull in 1937, but they first became prominent in the work of Oscar Zariski a few years later, 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. 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 allows us to define regularity for all commutative rings, not just local ones: A commutative ring "A" is said to be a regular ring if its localizations at all of its prime ideals are regular local rings. 
If "A" is finite dimensional, it is equivalent to say that "A" has finite global dimension.
Regular ring.
In commutative algebra, a regular ring is a commutative Noetherian ring, such that the 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 (that is every point is 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 domains. If "A" is regular then so is "A"["X"], with dimension one greater than that of "A". 
In particular if is a field, the ring of integers, or a principal ideal domain, then the polynomial ring formula_38 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 but need not be an integral domain. For example, the product of two regular integral domains is regular, but not an integral domain.