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 any Noetherian local ring with unique maximal ideal m, and suppose a1, ..., an is a minimal set of generators of m. Then Krull's principal ideal theorem implies that n ≥ dim A, and A is regular whenever n = dim A.
The concept is motivated by its geometric meaning. A point x on an algebraic variety X is nonsingular (a smooth point) if and only if the local ring
l{O}X,
For Noetherian local rings, there is the following chain of inclusions:
There are a number of useful definitions of a regular local ring, one of which is mentioned above. In particular, if
A
ak{m}
ak{m}=(a1,\ldots,an)
n
A
\dimA=n
where the dimension is the Krull dimension. The minimal set of generators of
a1,\ldots,an
k=A/ak{m}
A
A
\dimkak{m}/ak{m}2=\dimA
where the second dimension is the Krull dimension.
gldimA:=\sup\{\operatorname{pd}M\midMisanA-module\}
A
A
A
gldimA<infty
in which case,
gldimA=\dimA
Multiplicity one criterion states:[1] 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,
\dim\widehat{A}/p=\dim\widehat{A}
In the positive characteristic case, there is the following important result due to Kunz: A Noetherian local ring
R
R\toR,r\mapstorp
R
The ring
A=k[x]/(x2)
… \xrightarrow{ ⋅ x}
| k[x] | |
| (x2) |
\xrightarrow{ ⋅ x}
| k[x] | |
| (x2) |
\tok\to0
Using another one of the characterizations,
A
|
0
ak{m}2
ak{m}/ak{m}2
k
1
1
x+ak{m}
The Auslander–Buchsbaum theorem states that every regular local ring is a unique factorization domain.
Every localization, as well as the completion, of a regular local ring is regular.
If
(A,ak{m})
A\congk[[x1,\ldots,xd]]
k=A/ak{m}
d=\dimA
See also: Serre's inequality on height and Serre's multiplicity conjectures.
See also: smooth scheme. 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 justifies the definition of regularity for non-local commutative rings given in the next section.
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.
k[X1,\ldots,Xn]
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.[2]