Gorenstein modules, finite index, and finite Cohen-Macaulay type

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.