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 exactly the same as its Krull dimension. The minimal number of generators of the maximal ideal is always bounded below by the Krull dimension. In symbols, let A be a local ring with maximal ideal m, and suppose that m is generated by a₁,...,a_n. Then in general n ≥ dim A, and A is defined to be regular if and only if n = dim A.

It is equivalent to say that the dimension of the vector space m/m², considered as a vector space of the residue field k=A/m of A, is equal to the dimension of A. See system of parameters.

Regular local rings were originally defined by Wolfgang Krull, but they first became prominent in the work of Oscar Zariski, 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, 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. 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. Geometrically, this corresponds to the intuition that if a surface contains a curve, and that curve is smooth, then the surface is smooth near the curve. Again, this lay unsolved until the introduction of homological techniques. However, Jean-Pierre Serre found a homological characterization of regular local rings: A local ring A is regular if and only if A has finite global dimension. 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 rings, not just local ones: A ring A is said to be regular if its localizations at all of its prime ideals are regular local rings. It is equivalent to say that A has finite global dimension.

If A is a regular ring, then it follows that the polynomial ring A[x] and the formal power series ring A[[x]] are both regular.

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[[X₁, X₂, ..., X_d]] is a regular local ring having (Krull) dimension d.
5. If Z is the ring of integers and X is an indeterminate, the ring Z[X]_{(2, X)} is an example of a 2-dimensional regular local ring which does not contain a field.