Incommutative algebra, aregular local ring is aNoetherianlocal ring having the property that the minimal number ofgenerators of itsmaximal ideal is equal to itsKrull dimension.^[1] In symbols, letA be any Noetherian local ring with unique maximal ideal m, and supposea₁, ...,a_n is a minimal set of generators of m. ThenKrull's principal ideal theorem implies thatn ≥ dimA, andA is regular whenevern = dimA.

The concept is motivated by its geometric meaning. A pointx on analgebraic varietyX isnonsingular (asmooth point) if and only if the local ring${\displaystyle {\mathcal{O}}_{X,x}}$ ofgerms atx is regular. (See also:regular scheme.) Regular local rings arenot related tovon 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 rings

There are a number of useful definitions of a regular local ring, one of which is mentioned above. In particular, if${\displaystyle A}$ is a Noetherian local ring with maximal ideal${\displaystyle \mathfrak{m}}$, then the following are equivalent definitions:

• Let${\displaystyle \mathfrak{m}=(a_1,\dots ,a_n)}$ where${\displaystyle n}$ is chosen as small as possible. Then${\displaystyle A}$ is regular if

${\displaystyle \dim A=n}$,

where the dimension is the Krull dimension. The minimal set of generators of${\displaystyle a_1,\dots ,a_n}$ are then called aregular system of parameters.

• Let${\displaystyle k=A/\mathfrak{m}}$ be the residue field of${\displaystyle A}$. Then${\displaystyle A}$ is regular if

${\displaystyle {\dim }_k\mathfrak{m}/{\mathfrak{m}}^2=\dim A}$,

where the second dimension is theKrull dimension.

• Let${\displaystyle \text{gl dim }A:=sup\{\mathrm{pd}M\mid M\text{ is an }A\text{-module}\}}$ be theglobal dimension of${\displaystyle A}$ (i.e., the supremum of theprojective dimensions of all${\displaystyle A}$-modules.) Then${\displaystyle A}$ is regular if

,

in which case,${\displaystyle \text{gl dim }A=\dim A}$.

Multiplicity one criterion states:^[2] if thecompletion of a Noetherian local ringA is unimixed (in the sense that there is no embedded prime divisor of the zero ideal and for each minimal primep,${\displaystyle \dim \hat{A}/p=\dim \hat{A}}$) and if themultiplicity ofA is one, thenA 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 anintersection is regular if and only if the intersection is atransversal intersection.

In the positivecharacteristic case, there is the following important result due to Kunz: A Noetherian local ring${\displaystyle R}$ of positive characteristicp is regular if and only if theFrobenius morphism${\displaystyle R\to R,r\mapsto r^p}$ isflat and${\displaystyle R}$ isreduced. No similar result is known in characteristic zero (it is unclear how one should replace the Frobenius morphism).

1. Everyfield is a regular local ring. These have (Krull) dimension 0. In fact, the fields are exactly the regular local rings of dimension 0.
2. Anydiscrete valuation ring is a regular local ring of dimension 1 and the regular local rings of dimension 1 are exactly the discrete valuation rings. For example, ifk is a field andX is an indeterminate, then the ring offormal power seriesk[[X]] is a regular local ring having (Krull) dimension 1.
3. Ifp is an ordinary prime number, the ring ofp-adic integers is an example of a discrete valuation ring, and consequently a regular local ring. In contrast to the example above, this ring does not contain a field.
4. More generally, ifk is a field andX₁,X₂, ...,X_d are indeterminates, then the ring of formal power seriesk[[X₁,X₂, ...,X_d]] is a regular local ring having (Krull) dimensiond.
5. Still more generally, ifA is a regular local ring, then theformal power series ringA[[x]] is regular local.
6. IfZ is the ring of integers andX is an indeterminate, the ringZ[X]_(2,X) (i.e. the ringZ[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 thestructure theorem ofIrvin Cohen, acomplete regular local ring of Krull dimensiond that contains a fieldk is a power series ring ind variables over anextension field ofk.

The ring${\displaystyle A=k[x]/(x^2)}$ 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

${\displaystyle \cdots \stackrel{\cdot x}{\to }\frac{k[x]}{(x^2)}\stackrel{\cdot x}{\to }\frac{k[x]}{(x^2)}\to k\to 0}$

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

TheAuslander–Buchsbaum theorem states that every regular local ring is aunique factorization domain.

Everylocalization, as well as thecompletion, of a regular local ring is regular.

If${\displaystyle (A,\mathfrak{m})}$ is a complete regular local ring that contains a field, then

${\displaystyle A\cong k[[x_1,\dots ,x_d]]}$,

where${\displaystyle k=A/\mathfrak{m}}$ is theresidue field, and${\displaystyle d=\dim A}$, the Krull dimension.

See also:Serre's inequality on height andSerre's multiplicity conjectures.

Regular local rings were originally defined byWolfgang Krull in 1937,^[3] but they first became prominent in the work ofOscar Zariski a few years later,^[4][5] who showed that geometrically, a regular local ring corresponds to a smooth point on analgebraic variety. LetY be analgebraic variety contained in affinen-space over aperfect field, and suppose thatY is the vanishing locus of the polynomialsf₁,...,f_m.Y is nonsingular atP ifY satisfies aJacobian condition: IfM = (∂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 evaluatingM atP isn − dimY. Zariski proved thatY is nonsingular atP if and only if the local ring ofY atP 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 fromhomological 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 aunique 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 wasJean-Pierre Serre who found a homological characterization of regular local rings: A local ringA is regular if and only ifA has finiteglobal dimension, i.e. if everyA-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 ofregularity for non-local commutative rings given in the next section.

Incommutative algebra, aregular ring is a commutativeNoetherian ring, such that thelocalization at everyprime 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 itsKrull dimension.

The origin of the termregular ring lies in the fact that anaffine variety isnonsingular (that is every point isregular) if and only if itsring of regular functions is regular.

For regular rings, Krull dimension agrees withglobal homological dimension.

Jean-Pierre Serre defined a regular ring as a commutative noetherian ring offinite 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) andDedekind domains. IfA is regular then so isA[X], with dimension one greater than that ofA.

In particular ifk is a field, the ring of integers, or aprincipal ideal domain, then thepolynomial ring${\displaystyle k[X_1,\dots ,X_n]}$ is regular. In the case of a field, this isHilbert's syzygy theorem.

Any localization of a regular ring is regular as well.

A regular ring isreduced^[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]

• Geometrically regular ring
• quasi-free ring

• Atiyah, 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

• Algebraic geometry
• Ring theory

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