Homology multipliers and the relation type of parameter ideals

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.