Maximal Cohen–Macaulay modules that are not locally free on the punctured spectrum

Abstract

We say that a Cohen–Macaulay local ring has finite CM+-representation type if there exist only finitely many isomorphism classes of indecomposable maximal Cohen–Macaulay modules that are not locally free on the punctured spectrum. In this paper, we consider finite CM+-representation type from various points of view, relating it with several conjectures on finite/countable Cohen–Macaulay representation type. We prove in dimension one that the Gorenstein local rings of finite CM+-representation type are exactly the local hypersurfaces of countable CM-representation type, that is, the hypersurfaces of type (A∞) and (D∞). We also discuss the closedness and dimension of the singular locus of a Cohen–Macaulay local ring of finite CM+-representation type.