Vol. 63, No. 2, 1976

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Bounds for numbers of generators of Cohen-Macaulay ideals

Judith D. Sally

Vol. 63 (1976), No. 2, 517–520

Let (R,m) be a local Cohen-Macaulay ring of dimension d and multiplicity e(R) = e. A natural question to ask about an m-primary ideal I is whether there is any relation between the number of generators of I and the least power t of m contained in I. (t will be called the nilpotency degree of R∕I) It is quite straight forward to obtain a bound for v(I), the number of generators in a minimal basis of I, in terms of t and e. However, there are several interesting applications. The first is the existence of a bound for the number of generators of any Cohen-Macaulay ideal I, i.e. any ideal I such that R∕I is Cohen-Macaulay, in terms of e(R∕I), e(R) and height I. The second application is a bound in terms of d and e for the reduction exponent of m.

Mathematical Subject Classification 2000
Primary: 13H15
Received: 1 December 1975
Published: 1 April 1976
Judith D. Sally