A characterization of systems of parameters

Abstract

A criterion is given for a set of elements to form a system of parameters on a module over a local ring. Let R be a Noetherian local ring of dimension n. We recall that a sequence of elements xj,... , xn in the maximal ideal m of R is called a system of parameters if the quotient R/(xl,... , xn) has dimension zero, and that the dimension n of R is the smallest integer for which a sequence with this property exists. Similarly, if M is an R-module of dimension k, we say that a sequence of elements x1,... ,Xk of the maximal ideal of R forms a system of parameters for M if the quotient M/(x1, ... ., xk)M has finite length. Assume that M is a finitely generated maximal Cohen-Macaulay module, so that its depth and its Krull dimension are both equal to n. In this note we give a criterion for a sequence of elements in the ideal (x1,... ., xn) to form a system of parameters on M. Let Yl, , Yn be a sequence of n elements in the ideal generated by xl,... , xn. We may theyl write each yi as a sum of multiples of the xj, so that there is a matrix A (aij) such that