Regular local ring: Difference between revisions
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imported>Aleksander Stos m (categories) |
imported>Giovanni Antonio DiMatteo (adding some stuff) |
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==Definition== | ==Definition== | ||
Let <math>A</math> be a Noetherian local ring with maximal ideal <math>\mathfrac{m}</math> and residual field <math>k=A/\mathfrac{m}</math>. The following conditions are equivalent: | |||
# The Krull dimension of <math>A</math> is equal to the dimension of <math>\mathfrac{m}/\mathfrac{m}^2</math> as a <math>k</math>-vector space. | |||
[[Category:CZ Live]] | [[Category:CZ Live]] | ||
[[Category:Mathematics Workgroup]] | [[Category:Mathematics Workgroup]] | ||
[[Category:Stub Articles]] | [[Category:Stub Articles]] | ||
Revision as of 08:30, 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 Failed to parse (unknown function "\mathfrac"): {\displaystyle \mathfrac{m}} and residual field Failed to parse (unknown function "\mathfrac"): {\displaystyle k=A/\mathfrac{m}} . The following conditions are equivalent:
- The Krull dimension of is equal to the dimension of Failed to parse (unknown function "\mathfrac"): {\displaystyle \mathfrac{m}/\mathfrac{m}^2} as a -vector space.
