##  [Regular Local Ring](/regular-local-ring-0) 

 Definition

A Noetherian local ring (R, m) whose Krull dimension equals the minimal number of generators of its maximal ideal m; equivalently R has finite global (homological) dimension equal to that Krull dimension.

 

 

 

 

 

 





## Principle

Principle

Regularity is organized by the equality between intrinsic geometric dimension (Krull dimension) and embedding dimension (the minimal number of generators of m); this equality encodes homological finiteness and the absence of certain relations among generators.

 

 

 

 

 





## Demonstration

Demonstration

The formal power series ring k[[x1,...,xn]] or the localization k[x1,...,xn]_{(x1,...,xn)} are regular local rings of dimension n; their maximal ideal is generated by the coordinate parameters and global homological dimension equals n.

 

 

 

 

## Misapplication

Misapplication

Declaring a ring regular because its maximal ideal can be generated by n elements without verifying Noetherian hypotheses or equality with Krull dimension; confusing a regular element or a regular sequence with the ring being a regular local ring.

 

 

 

 

 





## Consequence

Consequence

Modules over a regular local ring have finite projective dimensions bounded by the ring dimension; regular local rings are Cohen–Macaulay and their spectra are smooth at the closed point, enabling predictable deformation and intersection behavior.

 

 

 

 

## Reversal

Reversal

An irregular (singular) local ring has embedding dimension strictly larger than its Krull dimension; such rings exhibit homological pathologies (infinite projective dimensions for some modules) and are local models of singularities.

 

 

 

 

 





## Boundary

Boundary

Applies only to Noetherian local rings; it excludes non-Noetherian rings, non-local rings (unless applied to each localization at maximal ideals), and statements about global smoothness of non-affine schemes without local verification.

 

 

 

 

 





## Semantic Tension

Semantic Tension

The term 'regular' also denotes smoothness in geometry and nonvanishing conditions in linear algebra; in this algebraic sense 'regular' is a precise homological/geometric equality rather than a generic notion of 'well-behaved'.

 

 

 

 

 





## Synthesis

Synthesis

A regular local ring is a Noetherian local algebraic model whose dimension equals the minimal number of local parameters, equivalently characterized by finite homological dimension; it is the algebraic avatar of a nonsingular point.