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. In symbols, let A be a Noetherian local ring with maximal ideal m, and suppose a₁, ..., 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 X is nonsingular if and only if the local ring ${\displaystyle {\mathcal {O}}_{X,x}}$ of germs at x is regular. (See also: regular scheme.) Regular local rings are not related to von Neumann regular rings.[1]

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

Characterizations

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 ${\displaystyle {\mathfrak {m}}=(a_{1},\ldots ,a_{n})}$ where ${\displaystyle n}$ is chosen as small as possible. Then ${\displaystyle A}$ is regular if

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

where the dimension is the Krull dimension. The minimal set of generators of ${\displaystyle a_{1},\ldots ,a_{n}}$ are then called a regular 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 the Krull dimension.

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

${\displaystyle {\mbox{gl dim }}A<\infty \,}$,

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

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, ${\displaystyle \dim {\widehat {A}}/p=\dim {\widehat {A}}}$) 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 ${\displaystyle R}$ of positive characteristic p is regular if and only if the Frobenius morphism ${\displaystyle R\to R,r\mapsto r^{p}}$ is flat and ${\displaystyle R}$ is reduced. No similar result is known in the characteristic zero (just because how to replace Frobenius is unclear).

Examples

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, ..., X_d are indeterminates, then the ring of formal power series k[[''X''₁, ''X''₂, ..., ''X''_''d'']] 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] localised 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 equicharacteristic regular local ring of Krull dimension d and containing a field is a power series ring over a field.

Non-Examples

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

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

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.)

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

${\displaystyle A\cong k[[x_{1},\ldots ,x_{d}]]}$,

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

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

Origin of basic notions

See 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,...,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 allows us to define regularity for all commutative rings, not just local ones: A commutative ring A is said to be a regular ring if its localizations at all of its prime ideals are regular local rings. If A is finite dimensional, it is equivalent to say that A has finite global dimension.

Notes

1. A local von Neumann regular ring is a division ring, so the two conditions are not very compatible.
2. Herrmann–Ikeda–Orbanz, Theorem 6.8. harvnb error: no target: CITEREFHerrmann–Ikeda–Orbanz (help)
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

References

• Kunz, Characterizations of regular local rings of characteristic p. Amer. J. Math. 91 (1969), 772–784.
• 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.
• Jean-Pierre Serre, Local algebra, Springer-Verlag, 2000, ISBN 3-540-66641-9. Chap.IV.D.