Homology multipliers and the relation type of parameter ideals

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 dim NCM(R) ≥ 2. The major part of our work is to investigate the remaining situation, i.e., when dim NCM(R) = 1. We introduce the notion of homology multipliers and show that the question has a positive answer when R/Ꮽ(R) is a domain, where Ꮽ( 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.

Uniform bounds in generalized Cohen–Macaulay rings

“…Wang [19] showed that two-dimensional local rings always have this property. Recently, Aberbach, Ghezzi and Ha [1] found out that certain local rings with one-dimensional non-Cohen-Macaulay locus also have this property. Therefore, the class of local rings with this property must be larger than the class of generalized Cohen-Macaulay rings.…”

“…This problem originated from Huneke's question whether equidimensional unmixed local rings have the above property. Recently, Aberbach found a counter-example (see [1]). Results of Wang [19] and of Aberbach, Ghezzi and Ha [1] showed that the class of rings with the above property must be larger than the class of generalized Cohen-Macaulay rings.…”

“…Recently, Aberbach found a counter-example (see [1]). Results of Wang [19] and of Aberbach, Ghezzi and Ha [1] showed that the class of rings with the above property must be larger than the class of generalized Cohen-Macaulay rings. However, we will prove that a local ring is generalized Cohen-Macaulay if and only if there exists an integer r such that for every quotient ring by an ideal generated by a subsystem of parameters, the relation type (or the regularity of the associated graded rings) of parameter ideals is bounded by r. Therefore, if there exists a uniform bound for the relation type of parameter ideals, then the bound must depend on invariants which may increase when we pass to quotient rings of ideals generated by subsystems of parameters.…”

The equations of Rees algebras of equimultiple ideals of deviation one

“…For instance, consider the ideals generated by a system of parameters. C. Huneke asked in [16] whether there is a uniform bound for the relation type of these ideals in a complete local equidimensional Noetherian ring R. The full answer to this question was given in [28], [19] and [1]. Concretely, in [1, Example 2.1], it was shown that if the non-Cohen-Macaulay locus of R has dimension 2 or more, there exist families of parametric ideals of R with unbounded relation type.…”

An approach to the uniform Artin-Rees theorems from the notion of relation type

“…, d − 1. Then there exist an integer s ≥ 1 such that, for every ideal I of A generated by a system of parameters, the relation type of I is bounded above by s.For rings of dimension 3 or more, see the very recent work of Aberbach, Ghezzi and Ha[1].…”