System of parameters

Summary

A system of parameters of a Noetherian local ring of Krull dimension $d$ is a list of $d$ elements of the maximal ideal $\mathfrak m$. The ideal that they generate must be primary to $\mathfrak m$. The dimension theorem for local rings says that such a list always exists and that no shorter list can do the same job. Systems of parameters play the role of local coordinates, and they are the starting point of the theory of Cohen–Macaulay rings and multiplicities.