In
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Promi ...
, a regular local ring is a
Noetherian local ring having the property that the minimal number of generators of its
maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals ...
is equal to its
Krull dimension
In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generall ...
. 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
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers ...
''X'' is
nonsingular if and only if the local ring
of
germs at ''x'' is regular. (See also:
regular scheme.) Regular local rings are ''not'' related to
von Neumann regular ring
In mathematics, a von Neumann regular ring is a ring ''R'' (associative, with 1, not necessarily commutative) such that for every element ''a'' in ''R'' there exists an ''x'' in ''R'' with . One may think of ''x'' as a "weak inverse" of the eleme ...
s.
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
is a Noetherian local ring with maximal ideal
, then the following are equivalent definitions
* Let
where
is chosen as small as possible. Then
is regular if
::
,
:where the dimension is the Krull dimension. The minimal set of generators of
are then called a ''regular system of parameters''.
* Let
be the residue field of
. Then
is regular if
::
,
:where the second dimension is the
Krull dimension
In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generall ...
.
* Let
be the
global dimension of
(i.e., the supremum of the
projective dimensions of all
-modules.) Then
is regular if
::
,
:in which case,
.
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'',
) 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
of positive
characteristic ''p'' is regular if and only if the
Frobenius morphism is
flat and
is
reduced. No similar result is known in the characteristic zero (just because how to replace Frobenius is unclear).
Examples
# Every
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
In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.
This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions:
# ''R' ...
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'' is a regular local ring having (Krull) dimension 1.
# If ''p'' is an ordinary prime number, the ring of
p-adic integers 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'' is a regular local ring having (Krull) dimension ''d''.
# If ''A'' is a regular local ring, then it follows that the
formal power series ring ''A'' is regular local.
# If Z is the ring of integers and ''X'' is an indeterminate, the ring Z
'X''sub>(2, ''X'') (i.e. the ring Z
'X'' localised in the prime ideal (2, ''X'') ) is an example of a 2-dimensional regular local ring which does not contain a field.
# By the
structure theorem of
Irvin Cohen
Irvin Sol Cohen (1917 – February 14, 1955) was an American mathematician at the Massachusetts Institute of Technology who worked on local rings. He was a student of Oscar Zariski at Johns Hopkins University.
In his thesis he proved the Co ...
, a
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
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
:
Using another one of the characterizations,
has exactly one prime ideal
, so the ring has Krull dimension
, but
is the zero ideal, so
has
dimension at least
. (In fact it is equal to
since
is a basis.)
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
is a complete regular local ring that contains a field, then
:
,
where
is the
residue field, and
, the Krull dimension.
See also:
Serre's inequality on height In algebra, specifically in the theory of commutative rings, Serre's inequality on height states: given a (Noetherian) regular ring ''A'' and a pair of prime ideals \mathfrak, \mathfrak in it, for each prime ideal \mathfrak r that is a minimal pri ...
and
Serre's multiplicity conjectures In mathematics, Serre's multiplicity conjectures, named after Jean-Pierre Serre, are certain purely algebraic problems, in commutative algebra, motivated by the needs of algebraic geometry. Since André Weil's initial definition of intersection n ...
.
Origin of basic notions
Regular local rings were originally defined by
Wolfgang Krull
Wolfgang Krull (26 August 1899 – 12 April 1971) was a German mathematician who made fundamental contributions to commutative algebra, introducing concepts that are now central to the subject.
Krull was born and went to school in Baden-Baden. H ...
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
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers ...
. Let ''Y'' be an
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers ...
contained in affine ''n''-space over a perfect field, and suppose that ''Y'' is the vanishing locus of the polynomials ''f
1'',...,''f
m''. ''Y'' is nonsingular at ''P'' if ''Y'' satisfies a
Jacobian condition: If ''M'' = (∂''f
i''/∂''x
j'') 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.
Regular ring
In
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Promi ...
, 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
In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generall ...
.
The origin of the term ''regular ring'' lies in the fact that an
affine variety is
nonsingular (that is every point is
regular
The term regular can mean normal or in accordance with rules. It may refer to:
People
* Moses Regular (born 1971), America football player
Arts, entertainment, and media Music
* "Regular" (Badfinger song)
* Regular tunings of stringed instrum ...
) 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
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessari ...
s. 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
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variable ...