Systems of Parameters

A system of parameters provides a finite set of generators for an ideal of definition, connecting the algebraic notion of Krull dimension with the geometric notion of the number of equations needed to cut out a point.

Definition and Existence

Definition

Let $(R, \mathfrak{m})$ be a Noetherian local ring of Krull dimension $d$ . A system of parameters for $R$ is a set of $d$ elements $x_1, \ldots, x_d \in \mathfrak{m}$ such that $\mathfrak{m}$ is a minimal prime over $(x_1, \ldots, x_d)$ . Equivalently, $\dim R/(x_1, \ldots, x_d) = 0$ , meaning $(x_1, \ldots, x_d)$ is an $\mathfrak{m}$ -primary ideal.

The existence of systems of parameters follows from the Krull dimension theorem: $d$ elements suffice to reduce the dimension to zero.

Definition

A system of parameters $x_1, \ldots, x_d$ is a regular system of parameters if it generates the maximal ideal $\mathfrak{m}$ , i.e., $\mathfrak{m} = (x_1, \ldots, x_d)$ . A local ring admitting a regular system of parameters is called a regular local ring.

Dimension and Systems of Parameters

Theorem6.5Characterization of Dimension

For a Noetherian local ring $(R, \mathfrak{m})$ , the following quantities are equal:

$\dim R =$ Krull dimension
$\delta(R) =$ minimum number of generators of an $\mathfrak{m}$ -primary ideal
$d(R) =$ degree of the Hilbert-Samuel polynomial $P_\mathfrak{m}(n) = \ell(R/\mathfrak{m}^{n+1})$ for large $n$

This is the dimension theorem of commutative algebra.

ExampleRegular vs. non-regular parameters

In $R = k[[x, y]]/(y^2 - x^3)$ (cuspidal curve), $\dim R = 1$ . The image of $x$ alone forms a system of parameters since $(x)$ is $\mathfrak{m}$ -primary. However, $\mathfrak{m} = (\bar{x}, \bar{y})$ requires two generators, so $R$ is not regular. In contrast, $k[[t]]$ with $\mathfrak{m} = (t)$ is regular of dimension $1$ .

Hilbert-Samuel Function

RemarkHilbert-Samuel polynomial

The Hilbert-Samuel function $H_\mathfrak{m}(n) = \ell(R/\mathfrak{m}^{n+1})$ agrees with a polynomial $P_\mathfrak{m}(n)$ for $n \gg 0$ . This polynomial has degree $d = \dim R$ and leading coefficient $e(\mathfrak{m}) / d!$ , where $e(\mathfrak{m})$ is the multiplicity of $R$ . For a regular local ring, $e(\mathfrak{m}) = 1$ , and $P_\mathfrak{m}(n) = \binom{n+d}{d}$ .

Systems of parameters provide the algebraic framework for understanding the "number of independent equations" needed to define a geometric object, bridging commutative algebra with algebraic geometry and singularity theory.