Homology multipliers and the relation type of parameter ideals

Ian M. Aberbach, Laura Ghezzi and Huy Tài Hà

Vol. 226 (2006), No. 1, 1–39

DOI: 10.2140/pjm.2006.226.1

Abstract

The relation type question, raised by C. Huneke, asks whether for a complete equidimensional local ring R there exists a uniform number N such that the relation type of every ideal I ⊂ R generated by a system of parameters is at most N. Wang gave a positive answer to this question when the non-Cohen–Macaulay locus of R (denoted by NCM(R)) has dimension zero. In this paper, we first present an example, due to the first author, which gives a negative answer to the question when dimNCM(R) ≥ 2. The major part of our work is to investigate the remaining situation, i.e., when dimNCM(R) = 1. We introduce the notion of homology multipliers and show that the question has a positive answer when R∕𝒜(R) is a domain, where 𝒜(R) is the ideal generated by all homology multipliers in R. In a more general context, we also discuss many interesting properties of homology multipliers.

Keywords

relation type, uniform bound, Rees algebra, Cohen–Macaulay, finiteness

Mathematical Subject Classification 2000

Primary: 13A30, 13E15, 13H10

Milestones

Received: 23 September 2004

Revised: 25 January 2005

Accepted: 25 January 2005

Published: 1 July 2006