Regular local ring: Difference between revisions
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imported>Giovanni Antonio DiMatteo |
imported>Giovanni Antonio DiMatteo (adding another section) |
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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. | ||
==Definition== | ==Definition== | ||
Let <math>A</math> be a 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>m/m^2</math> as a <math>k</math>-vector space. | # The Krull dimension of <math>A</math> is equal to the dimension of <math>m/m^2</math> as a <math>k</math>-vector space. | ||
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. | ||
==Basic Results on Regular Local Rings== | |||
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 UFD. | |||
==Regular Rings== | |||
A [[regular ring]] is a Noetherian ring such that the localisation at every prime is a regular local ring. | |||
[[Category:CZ Live]] | [[Category:CZ Live]] | ||
[[Category:Mathematics Workgroup]] | [[Category:Mathematics Workgroup]] | ||
[[Category:Stub Articles]] | [[Category:Stub Articles]] | ||
Revision as of 08:40, 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 as a -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.