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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.[1] In symbols, let A be a Noetherian local ring with maximal ideal m, and suppose a1, ..., an 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 of germs at x is regular. (See also: regular scheme.) Regular local rings are not related to von Neumann regular rings.[a]

For Noetherian local rings, there is the following chain of inclusions:

Universally catenary ringsCohen–Macaulay ringsGorenstein ringscomplete intersection ringsregular local rings

Characterizations Edit

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.
  • 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:[2] 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 characteristic zero (it is unclear how one should replace the Frobenius morphism).

Examples Edit

  1. 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.
  2. 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.
  3. 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.
  4. More generally, if k is a field and X1, X2, ..., Xd are indeterminates, then the ring of formal power series k[[X1, X2, ..., Xd]] is a regular local ring having (Krull) dimension d.
  5. If A is a regular local ring, then it follows that the formal power series ring A[[x]] is regular local.
  6. If Z is the ring of integers and X is an indeterminate, the ring Z[X](2, X) (i.e. the ring Z[X] localized in the prime ideal (2, X) ) is an example of a 2-dimensional regular local ring which does not contain a field.
  7. By the structure theorem of Irvin Cohen, 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 Edit

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 Edit

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   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 and Serre's multiplicity conjectures.

Origin of basic notions Edit

Regular local rings were originally defined by Wolfgang Krull in 1937,[3] but they first became prominent in the work of Oscar Zariski a few years later,[4][5] 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.

Regular ring Edit

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 k is a field, the ring of integers, or a principal ideal domain, then the polynomial ring   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[b] but need not be an integral domain. For example, the product of two regular integral domains is regular, but not an integral domain.[6]

See also Edit

Notes Edit

  1. ^ A local von Neumann regular ring is a division ring, so the two conditions are not very compatible.
  2. ^ since a ring is reduced if and only if its localizations at prime ideals are.

Citations Edit

  1. ^ Atiyah & Macdonald 1969, p. 123, Theorem 11.22.
  2. ^ Herrmann, M., S. Ikeda, and U. Orbanz: Equimultiplicity and Blowing Up. An Algebraic Study with an Appendix by B. Moonen. Springer Verlag, Berlin Heidelberg New-York, 1988. Theorem 6.8.
  3. ^ Krull, Wolfgang (1937), "Beiträge zur Arithmetik kommutativer Integritätsbereiche III", Math. Z.: 745–766, doi:10.1007/BF01160110
  4. ^ Zariski, Oscar (1940), "Algebraic varieties over ground fields of characteristic 0", Amer. J. Math., 62: 187–221, doi:10.2307/2371447
  5. ^ Zariski, Oscar (1947), "The concept of a simple point of an abstract algebraic variety", Trans. Amer. Math. Soc., 62: 1–52, doi:10.1090/s0002-9947-1947-0021694-1
  6. ^ Is a regular ring a domain

References Edit

regular, local, ring, commutative, algebra, regular, local, ring, noetherian, local, ring, having, property, that, minimal, number, generators, maximal, ideal, equal, krull, dimension, symbols, noetherian, local, ring, with, maximal, ideal, suppose, minimal, g. 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 1 In symbols let A be a Noetherian local ring with maximal ideal m and suppose a1 an 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 O X x displaystyle mathcal O X x of germs at x is regular See also regular scheme Regular local rings are not related to von Neumann regular rings a For Noetherian local rings there is the following chain of inclusions Universally catenary rings Cohen Macaulay rings Gorenstein rings complete intersection rings regular local ringsContents 1 Characterizations 2 Examples 3 Non examples 4 Basic properties 5 Origin of basic notions 6 Regular ring 7 See also 8 Notes 9 Citations 10 ReferencesCharacterizations EditThere are a number of useful definitions of a regular local ring one of which is mentioned above In particular if A displaystyle A nbsp is a Noetherian local ring with maximal ideal m displaystyle mathfrak m nbsp then the following are equivalent definitions Let m a 1 a n displaystyle mathfrak m a 1 ldots a n nbsp where n displaystyle n nbsp is chosen as small as possible Then A displaystyle A nbsp is regular ifdim A n displaystyle dim A n nbsp dd where the dimension is the Krull dimension The minimal set of generators of a 1 a n displaystyle a 1 ldots a n nbsp are then called a regular system of parameters Let k A m displaystyle k A mathfrak m nbsp be the residue field of A displaystyle A nbsp Then A displaystyle A nbsp is regular ifdim k m m 2 dim A displaystyle dim k mathfrak m mathfrak m 2 dim A nbsp dd where the second dimension is the Krull dimension Let gl dim A sup pd M M is an A module displaystyle mbox gl dim A sup operatorname pd M mid M text is an A text module nbsp be the global dimension of A displaystyle A nbsp i e the supremum of the projective dimensions of all A displaystyle A nbsp modules Then A displaystyle A nbsp is regular ifgl dim A lt displaystyle mbox gl dim A lt infty nbsp dd in which case gl dim A dim A displaystyle mbox gl dim A dim A nbsp Multiplicity one criterion states 2 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 A p dim A displaystyle dim widehat A p dim widehat A nbsp 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 R displaystyle R nbsp of positive characteristic p is regular if and only if the Frobenius morphism R R r r p displaystyle R to R r mapsto r p nbsp is flat and R displaystyle R nbsp is reduced No similar result is known in characteristic zero it is unclear how one should replace the Frobenius morphism Examples EditEvery 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 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 X1 X2 Xd are indeterminates then the ring of formal power series k X1 X2 Xd 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 x is regular local If Z is the ring of integers and X is an indeterminate the ring Z X 2 X i e the ring Z X localized 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 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 EditThe ring A k x x 2 displaystyle A k x x 2 nbsp 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 x k x x 2 x k x x 2 k 0 displaystyle cdots xrightarrow cdot x frac k x x 2 xrightarrow cdot x frac k x x 2 to k to 0 nbsp Using another one of the characterizations A displaystyle A nbsp has exactly one prime ideal m x x 2 displaystyle mathfrak m frac x x 2 nbsp so the ring has Krull dimension 0 displaystyle 0 nbsp but m 2 displaystyle mathfrak m 2 nbsp is the zero ideal so m m 2 displaystyle mathfrak m mathfrak m 2 nbsp has k displaystyle k nbsp dimension at least 1 displaystyle 1 nbsp In fact it is equal to 1 displaystyle 1 nbsp since x m displaystyle x mathfrak m nbsp is a basis Basic properties EditThe 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 m displaystyle A mathfrak m nbsp is a complete regular local ring that contains a field then A k x 1 x d displaystyle A cong k x 1 ldots x d nbsp where k A m displaystyle k A mathfrak m nbsp is the residue field and d dim A displaystyle d dim A nbsp the Krull dimension See also Serre s inequality on height and Serre s multiplicity conjectures Origin of basic notions EditSee also smooth scheme Regular local rings were originally defined by Wolfgang Krull in 1937 3 but they first became prominent in the work of Oscar Zariski a few years later 4 5 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 Regular ring EditFor the unrelated regular rings introduced by John von Neumann see von Neumann 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 k is a field the ring of integers or a principal ideal domain then the polynomial ring k X 1 X n displaystyle k X 1 ldots X n nbsp 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 b but need not be an integral domain For example the product of two regular integral domains is regular but not an integral domain 6 See also EditGeometrically regular ringNotes Edit A local von Neumann regular ring is a division ring so the two conditions are not very compatible since a ring is reduced if and only if its localizations at prime ideals are Citations Edit Atiyah amp Macdonald 1969 p 123 Theorem 11 22 Herrmann M S Ikeda and U Orbanz Equimultiplicity and Blowing Up An Algebraic Study with an Appendix by B Moonen Springer Verlag Berlin Heidelberg New York 1988 Theorem 6 8 Krull Wolfgang 1937 Beitrage zur Arithmetik kommutativer Integritatsbereiche III Math Z 745 766 doi 10 1007 BF01160110 Zariski Oscar 1940 Algebraic varieties over ground fields of characteristic 0 Amer J Math 62 187 221 doi 10 2307 2371447 Zariski Oscar 1947 The concept of a simple point of an abstract algebraic variety Trans Amer Math Soc 62 1 52 doi 10 1090 s0002 9947 1947 0021694 1 Is a regular ring a domainReferences EditAtiyah Michael F Macdonald Ian G 1969 Introduction to Commutative Algebra Addison Wesley MR 0242802 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 1999 ISBN 978 1 4612 0525 8 Chap 5 G Jean Pierre Serre Local algebra Springer Verlag 2000 ISBN 3 540 66641 9 Chap IV D Regular rings at The Stacks Project Retrieved from https en wikipedia org w index php title Regular local ring amp oldid 1177459232, wikipedia, wiki, book, books, library,

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