The strong uniform Artin-Rees property in codimension one

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

“…Therefore, in Buchsbaum rings every ideal generated by a system of parameters has relation type 1. Moreover, the relation type is controlled through faithfully flat extensions and the quotient by modules of finite length (see, e.g., [24], [25] and [42]). From this point of view it is natural to ask the following question:…”

“…Our next purpose now is to deepen on this link. In order to do that, we recall some notations and properties beginning with the (module of effective) relations of the Rees module of an ideal I with respect to a module M (see [25]). …”

“…The relation type of F is defined to be rt(F ) = s(E(F )), that is, rt(F ) is the minimum integer r ≥ 1 such that the effective n-relations are zero for all n ≥ r + 1. Using the exact sequence (1), it can be shown that the module of effective n-relations and the relation type do not depend on the chosen symmetric presentation ( [25]). …”

“…For an ideal I of A and an A-module M , the module of effective nrelations and the relation type of I with respect to M are defined to be E(I; M ) n = E(R(I; M )) n and rt(I; M ) = rt(R(I; M )) (see also [8], page 106, and [34] (1), we arrive at the desired exact sequence of A-modules which links Artin-Rees theory with relation type (see [25]):…”

Homology multipliers and the relation type of parameter ideals

“…There are several cases in which the strong uniform Artin-Rees property holds [O'Carroll 1987;1991;Duncan and O'Carroll 1989;Huneke 1992;Planas-Vilanova 2000], but Wang [1997b] has shown that it does not hold in general. See [Planas-Vilanova 2006] for a recent summary and explication of results relating to uniform Artin-Rees theorems.…”