Regular local ring: Difference between revisions
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Let <math>A</math> be a [[Noetherian Ring|Noetherian]] [[local ring]] with maximal ideal <math>m</math> and residual field <math>k=A/m</math>. The following conditions are equivalent: | Let <math>A</math> be a [[Noetherian Ring|Noetherian]] [[local ring]] with maximal ideal <math>m</math> and residual field <math>k=A/m</math>. The following conditions are equivalent: | ||
# The Krull dimension of <math>A</math> is equal to the dimension of <math> | # The Krull dimension of <math>A</math> is equal to the dimension of the <math>k</math>-vector space <math>m/m^2</math>. | ||
And when these conditions hold, <math>A</math> is called a regular local ring. | And when these conditions hold, <math>A</math> is called a regular local ring. | ||
Revision as of 09:43, 2 December 2007
There are deep connections between algebraic (in fact, scheme-theoretic) notions of smoothness and regularity.
Definition
Let be a Noetherian local ring with maximal ideal and residual field . The following conditions are equivalent:
- The Krull dimension of is equal to the dimension of the -vector space .
And when these conditions hold, is called a regular local ring.
Basic Results on Regular Local Rings
One important criterion for regularity is Serre's Criterion, which states that a Noetherian local ring is regular if and only if its global dimension is finite, in which case it is equal to the krull dimension of .
In a paper of Auslander and Buchsbaum published in 1959, it was shown that every regular local ring is a UFD.
Regular Rings
A regular ring is a Noetherian ring such that the localisation at every prime is a regular local ring.