Regular local ring: Difference between revisions
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There are deep connections between algebraic (in fact, scheme-theoretic) notions of smoothness and regularity. | There are deep connections between algebraic (in fact, scheme-theoretic) notions of smoothness and regularity. | ||
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==Definition== | ==Definition== | ||
Let <math>A</math> be a [[Noetherian | 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 the <math>k</math>-vector space <math>m/m^2</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>. | ||
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One important criterion for regularity is [[Serre's Criterion]], which states that a Noetherian local ring <math>A</math> is regular if and only if its [[global dimension]] is finite, in which case it is equal to the krull dimension of <math>A</math>. | One important criterion for regularity is [[Serre's Criterion]], which states that a Noetherian local ring <math>A</math> is regular if and only if its [[global dimension]] is finite, in which case it is equal to the krull dimension of <math>A</math>. | ||
In a paper of Auslander and Buchsbaum published in 1959, it was shown that every regular local ring is a | In a paper of Auslander and Buchsbaum published in 1959, it was shown that every regular local ring is a [[unique factorization domain]]. | ||
==Regular Rings== | ==Regular Rings== | ||
A [[regular ring]] is a Noetherian ring such that the localisation at every prime is a regular local ring. | A [[regular ring]] is a Noetherian ring such that the [[localisation (ring theory)|localisation]] at every prime is a regular local ring. | ||
==References== | |||
[[ | * [[Jean-Pierre Serre]], ''Local algebra'', [[Springer-Verlag]], 2000, ISBN 3-540-66641-9. Chap.IV.D. | ||
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Latest revision as of 00:03, 21 February 2010
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 unique factorization domain.
Regular Rings
A regular ring is a Noetherian ring such that the localisation at every prime is a regular local ring.
References
- Jean-Pierre Serre, Local algebra, Springer-Verlag, 2000, ISBN 3-540-66641-9. Chap.IV.D.