Regular local ring

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

Definition

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} be a Noetherian local ring with maximal ideal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} and residual field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k=A/m} . The following conditions are equivalent:

1. The Krull dimension of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is equal to the dimension of the $k$-vector space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m/m^2} .

And when these conditions hold, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is regular if and only if its global dimension is finite, in which case it is equal to the krull dimension of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

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.