GORENSTEIN MODULES, FINITE INDEX, AND FINITE COHEN–MACAULAY TYPE

ABSTRACT

A Gorenstein module over a local ring R is a maximal Cohen–Macaulay module of finite injective dimension. We use existence of Gorenstein modules to extend a result due to S. Ding: A Cohen–Macaulay ring of finite index, with a Gorenstein module, is Gorenstein on the punctured spectrum. We use this to show that a Cohen–Macaulay local ring of finite Cohen–Macaulay type is Gorenstein on the punctured spectrum. Finally, we show that for a large class of rings (including all excellent rings), the Gorenstein locus of a finitely generated module is an open set in the Zariski topology.

ACKNOWLEDGMENTS

This work formed part of my doctoral dissertation at the University of Nebraska. It was made possible in part by the support of the Maude Hammond Fling Graduate Fellowship. I am grateful to my advisor, Roger Wiegand, for belaying. I also thank Craig Huneke, Luchezar Avramov, and Mark Walker for their patience and encouragement in conversations.