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.
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 characteristicp 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).
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 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 seriesk[[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] localised in the prime ideal (2, X) ) is an example of a 2-dimensional regular local ring which does not contain a field.
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.)
Regular local rings were originally defined by Wolfgang Krull 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. 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.
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.