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

## TL;DR

This paper characterizes Gorenstein local rings with finite Cohen-Macaulay modules not free on the punctured spectrum, linking them to hypersurfaces of specific types and exploring their singular locus properties.

## Contribution

It establishes a complete classification of Gorenstein local rings with finite -representation type in dimension one, connecting it to hypersurfaces of types (A_) and (D_).

## Key findings

- Gorenstein local rings of finite -representation type are hypersurfaces of types (A_) and (D_).
- Finite -representation type relates to the structure of the singular locus.
- The paper discusses the closedness and dimension of the singular locus in these rings.

## Abstract

We say that a Cohen-Macaulay local ring has finite $\operatorname{\mathsf{CM}}_+$-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 $\operatorname{\mathsf{CM}}_+$-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 $\operatorname{\mathsf{CM}}_+$-representation type are exactly the local hypersurfaces of countable $\mathsf{CM}$-representation type, that is, the hypersurfaces of type $(\mathrm{A}_\infty)$ and $(\mathrm{D}_\infty)$. We also discuss the closedness and dimension of the singular locus of a Cohen-Macaulay local ring of finite $\operatorname{\mathsf{CM}}_+$-representation type.