Regular local ring: Difference between revisions

Revision as of 09:31, 2 December 2007

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

Let $A$ be a Noetherian local ring with maximal ideal $m$ and residual field $k=A/m$ . The following conditions are equivalent:

The Krull dimension of $A$ is equal to the dimension of $m/m^{2}$ as a $k$ -vector space.

And when these conditions hold, $A$ is called a regular local ring.

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 ==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:
+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:
-# 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.
+# 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.
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