# 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, ..., 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 $\mathcal{O}_{X, x}$ 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 ringsCohen–Macaulay ringsGorenstein ringscomplete intersection ringsRegular local rings

## Characterizations

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

• Let $\mathfrak{m} = (a_1, \ldots, a_n)$ where $n$ is chosen as small as possible. Then $A$ is regular if
$\mbox{dim } A = n\,$,
where the dimension is the Krull dimension. The minimal set of generators of $a_1, \ldots, a_n$ are then called a regular system of parameters.
• Let $k = A / \mathfrak{m}$ be the residue field of $A$. Then $A$ is regular if
$\dim_k \mathfrak{m} / \mathfrak{m}^2 = \dim A\,$,
where the second dimension is the Krull dimension.
• Let $\mbox{gl dim } A := \sup \{ \mbox{pd } M \mbox{ }|\mbox{ } M \mbox{ is an }A\mbox{-module} \}$ be the global dimension of $A$ (i.e., the supremum of the projective dimensions of all $A$-modules.) Then $A$ is regular if
$\mbox{gl dim } A < \infty\,$,
in which case, $\mbox{gl dim } A = \dim A$.

## 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, ..., 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 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) 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.

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

$A \cong k[[x_1, \ldots, x_d]]$,

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