System of parameters explained

In mathematics, a system of parameters for a local Noetherian ring of Krull dimension d with maximal ideal m is a set of elements x₁, ..., x_d that satisfies any of the following equivalent conditions:

1. m is a minimal prime over (x₁, ..., x_d).
2. The radical of (x₁, ..., x_d) is m.
3. Some power of m is contained in (x₁, ..., x_d).
4. (x₁, ..., x_d) is m-primary.

Every local Noetherian ring admits a system of parameters.^[1]

It is not possible for fewer than d elements to generate an ideal whose radical is m because then the dimension of R would be less than d.

If M is a k-dimensional module over a local ring, then x₁, ..., x_k is a system of parameters for M if the length of is finite.

Notes and References

1. Web site: Math 711: Lecture of September 5, 2007 . University of Michigan. September 5, 2007 . May 31, 2022.