Normalization of Modules

We introduce techniques to derive estimates for the degrees of the generators of the integral closure of several classes of Rees algebras of modules, and to bound the length of normalization processes. In the case of regular base rings, the bounds are expressed in terms of Buchsbaum–Rim multiplicities and a module version of Briançon–Skoda numbers. Normalization of modules is the study of the integral closure of the Rees algebra of a module. The questions that arise are natural extensions of those that occur in the normalization of ordinary blowups and also of the so–called multi Rees algebras of ideals. In this paper, we focus on numerical aspects of the process of normalization of a module. Let R be a Noetherian ring and E a finitely generated torsionfree R–module having a rank. To enable the extension, we consider two technical devices to attach an ideal I of a ring S to the R–module E so that the comparison can be made between the Rees algebra of E and the Rees algebra of I. One of the ideals that can be used is the ideal generated by the module in the polynomial ring which contains the Rees algebra of the module. The other is known as the generic Bourbaki ideal of E. More details on these ideals will be given later. 2000 AMS Mathematics Subject Classification. Primary 13B22; Secondary 13H10, 13H15.