Classifying Thick Subcategories of the Stable Category of Cohen-macaulay Modules

Various classification theorems of thick subcategories of a triangulated category have been obtained in many areas of mathematics. In this paper, as a higher dimensional version of the classification theorem of thick subcategories of the stable category of finitely generated representations of a finite p-group due to Benson, Carlson and Rickard, we consider classifying thick subcategories of the stable category of Cohen-Macaulay modules over a Gorenstein local ring. The main result of this paper yields a complete classification of the thick subcategories of the stable category of CohenMacaulay modules over a local hypersurface in terms of specialization-closed subsets of the prime ideal spectrum of the ring which are contained in its singular locus. We also consider classifying resolving subcategories of the category of finitely generated modules. Our method also gives some information on the structure of Cohen-Macaulay modules that are free on the punctured spectrum.