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Concept
Regular local ring
Summary
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 of germs at x is regular. (See also: regular scheme.) Regular local rings are not related to von Neumann regular rings.
For Noetherian local rings, there is the following chain of inclusions:
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.
Let be the residue field of . Then is regular if
where the second dimension is the Krull dimension.
Let be the global dimension of (i.e., the supremum of the projective dimensions of all -modules.) Then is regular if
in which case, .
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 characteristic p is regular if and only if the Frobenius morphism is flat and is reduced.
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