Regular system of parameters

Definition

Definition with symbols

Let $R$ be a regular local ring, viz a local ring in which the unique maximal ideal ${\mathcal {m}}$. Let $d$ be the Krull dimension of $R$. A regular system of parameters for $R$ is a collection of elements $a_{1},a_{2},\ldots ,a_{d}\in {\mathcal {m}}$ satisfying the following equivalent conditions:

• $a_{1},a_{2},\ldots ,a_{d}$ generate ${\mathcal {m}}$ as an ideal
• The images of $a_{1},a_{2},\ldots ,a_{d}$ form a basis for ${\mathcal {m}}$ as a vector space

Equivalence of definitions

The equivalence of definitions follows from Nakayama's lemma. Further information: Size of minimal generating set in Noetherian local ring is unique