Regular Ring
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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. Prominent ...
, a regular local ring is a
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic n ...
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 cont ...
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 generally t ...
. 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 In commutative algebra, Krull's principal ideal theorem, named after Wolfgang Krull (1899–1971), gives a bound on the height of a principal ideal in a commutative Noetherian ring. The theorem is sometimes referred to by its German name, ''Krull ...
''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. Mo ...
''X'' is
nonsingular In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplica ...
if and only if the local ring \mathcal_ of germs at ''x'' is regular. (See also:
regular scheme In algebraic geometry, a regular scheme is a locally Noetherian scheme whose local rings are regular everywhere. Every smooth scheme is regular, and every regular scheme of finite type over a perfect field is smooth.. For an example of a regul ...
.) 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 element ...
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 A is a Noetherian local ring with maximal ideal \mathfrak, then the following are equivalent definitions * Let \mathfrak = (a_1, \ldots, a_n) where n is chosen as small as possible. Then A is regular if ::\mbox A = n\,, :where the dimension is the Krull dimension. The minimal set of generators of a_1, \ldots, a_n are then called a ''regular system of parameters''. * Let k = A / \mathfrak be the residue field of A. Then A is regular if ::\dim_k \mathfrak / \mathfrak^2 = \dim A\,, :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 generally t ...
. * Let \mbox A := \sup \ be the
global dimension In ring theory and homological algebra, the global dimension (or global homological dimension; sometimes just called homological dimension) of a ring ''A'' denoted gl dim ''A'', is a non-negative integer or infinity which is a homological invariant ...
of A (i.e., the supremum of the
projective dimension In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characterizat ...
s of all A-modules.) Then A is regular if ::\mbox A < \infty\,, :in which case, \mbox A = \dim A. 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'', \dim \widehat/p = \dim \widehat) and if the
multiplicity Multiplicity may refer to: In science and the humanities * Multiplicity (mathematics), the number of times an element is repeated in a multiset * Multiplicity (philosophy), a philosophical concept * Multiplicity (psychology), having or using mult ...
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 R of positive characteristic ''p'' is regular if and only if the
Frobenius morphism In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism m ...
R \to R, r \mapsto r^p is
flat Flat or flats may refer to: Architecture * Flat (housing), an apartment in the United Kingdom, Ireland, Australia and other Commonwealth countries Arts and entertainment * Flat (music), a symbol () which denotes a lower pitch * Flat (soldier), ...
and R is reduced. No similar result is known in the characteristic zero (just because how to replace Frobenius is unclear).


Examples

# Every
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
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 In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...
''k'' is a regular local ring having (Krull) dimension 1. # If ''p'' is an ordinary prime number, the ring of
p-adic integer In mathematics, the -adic number system for any prime number  extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extensio ...
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'' is a regular local ring having (Krull) dimension ''d''. # If ''A'' is a regular local ring, then it follows that the
formal power series In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...
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 Cohen ...
, a
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
regular local ring of Krull dimension ''d'' that contains a field ''k'' is a power series ring in ''d'' variables over an
extension field In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
of ''k''.


Non-examples

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


Basic properties

The
Auslander–Buchsbaum theorem In commutative algebra, the Auslander–Buchsbaum theorem states that regular local rings are unique factorization domains. The theorem was first proved by . They showed that regular local ring In abstract algebra, more specifically ring theory, ...
states that every regular local ring is a
unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an ...
. Every
localization Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is a ...
of a regular local ring is regular. The completion of a regular local ring is regular. If (A, \mathfrak) is a complete regular local ring that contains a field, then :A \cong k x_1, \ldots, x_d, where k = A / \mathfrak is the
residue field In mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is a ...
, and d = \dim A, 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 , birth_date = , birth_place = Kobrin, Russian Empire , death_date = , death_place = Brookline, Massachusetts, United States , nationality = American , field = Mathematics , work_institutions = ...
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. Mo ...
. 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. Mo ...
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 Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
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 In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an ...
. 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 Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
who found a homological characterization of regular local rings: A local ring ''A'' is regular if and only if ''A'' has finite
global dimension In ring theory and homological algebra, the global dimension (or global homological dimension; sometimes just called homological dimension) of a ring ''A'' denoted gl dim ''A'', is a non-negative integer or infinity which is a homological invariant ...
, 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. Prominent ...
, a regular ring is a commutative
Noetherian ring In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noether ...
, such that the
localization Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is a ...
at every
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
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 generally t ...
. The origin of the term ''regular ring'' lies in the fact that an
affine variety In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ideal. ...
is
nonsingular In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplica ...
(that is every point is regular) if and only if its
ring of regular functions In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime idea ...
is regular. For regular rings, Krull dimension agrees with
global homological dimension In ring theory and homological algebra, the global dimension (or global homological dimension; sometimes just called homological dimension) of a ring (mathematics), ring ''A'' denoted gl dim ''A'', is a non-negative integer or infinity which is a ho ...
.
Jean-Pierre Serre Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
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 necessarily ...
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 In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, ...
, 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 variables) ...
k _1, \ldots,X_n/math> is regular. In the case of a field, this is
Hilbert's syzygy theorem In mathematics, Hilbert's syzygy theorem is one of the three fundamental theorems about polynomial rings over fields, first proved by David Hilbert in 1890, which were introduced for solving important open questions in invariant theory, and are at ...
. 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.Is a regular ring a domain
/ref>


See also

*
Geometrically regular ring In algebraic geometry, a geometrically regular ring is a Noetherian ring over a field that remains a regular ring after any finite extension of the base field. Geometrically regular schemes are defined in a similar way. In older terminology, points ...


Notes


Citations


References

* * 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 Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...
, 1999, . Chap.5.G. *
Jean-Pierre Serre Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
, ''Local algebra'',
Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...
, 2000, . Chap.IV.D.
Regular rings at The Stacks Project
{{DEFAULTSORT:Regular Local Ring Algebraic geometry Ring theory