Vol. 226, No. 1, 2006

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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
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
Authors
Ian M. Aberbach
Mathematics Department
University of Missouri
Columbia, MO 65211
United States
http://www.math.missouri.edu/people/iaberbach.html
Laura Ghezzi
Mathematics Department
Florida International University
University Park, Miami FL 33199
United States
http://www.fiu.edu/~ghezzil/
Huy Tài Hà
Tulane University
Department of Mathematics
6823 St. Charles Ave.
New Orleans, LA 70118
United States
http://www.math.tulane.edu/~tai/