On the number of generators of ideals in local Cohen-Macaulay rings

Abstract

Of course, for 2-dimensional rings sufficiently high powers of the maximal ideal require arbitrarily large numbers of generators, so there can be no universal bound for the number of generators of an ideal. However our main results (Theorems 1 and 3) give bounds for ideals of fixed colength, and ideals with a fixed number of irreducible components primary to the maximal ideal in a 2-dimensional Cohen-Macaulay ring. We also show how to extend these results to results on the number of generators of ideals primary to the maximal ideal in higher dimensional Cohen-Macaulay rings. For a general survey of this and related problems, we refer the reader to the notes [Sally 21. Related estimates connecting multiplicity and number of generators have been obtained previously by J. Becker; see [12] and, particularly, [13].