Regular local ring: Difference between revisions

Revision as of 00:13, 22 November 2007

There are deep connections between algebraic (in fact, scheme-theoretic) notions of smoothness and regularity.

Serre's Regularity Criterion states that a Noetherian local ring $A$ is regular if and only if its global dimension is finite, in which case it is equal to the Krull dimension of $A$ .

Revision as of 21:39, 21 November 2007 (view source) imported>Giovanni Antonio DiMatteo (→‎Definition: minor edit) ← Older edit		Revision as of 00:13, 22 November 2007 (view source) imported>Eric Winesett mNo edit summary Newer edit →
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	==Definition==		==Definition==

	Serre's Regularity Criterion states that a [[Noetherian Ring\|Noetherian]] [[~~Local Ring\|~~local ring]] <math>A</math> is regular if and only if its [[~~Global Dimension\|~~global dimension]] is finite, in which case it is equal to the [[Krull dimension]] of <math>A</math>.		Serre's Regularity Criterion states that a [[Noetherian Ring\|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>.