Extension-closed subcategories over hypersurfaces of finite or countable CM-representation type

TL;DR

This paper classifies extension-closed subcategories of certain Cohen-Macaulay modules and singularity categories over hypersurfaces with finite or countable CM-representation type, especially in low dimensions.

Contribution

It provides a complete classification of extension-closed subcategories in CM_0(R) and D^{sg}_0(R) under specific representation type assumptions.

Findings

Classifies extension-closed subcategories in CM_0(R) for dimension ≤ 2.

Classifies extension-closed subcategories in D^{sg}_0(R) in all dimensions.

Focuses on hypersurfaces with finite or countable CM-representation type.

Abstract

Let k be an algebraically closed uncountable field of characteristic zero. Let R be a complete local hypersurface over k. Denote by CM(R) the category of maximal Cohen-Macaulay R-modules and by D^{sg}(R) the singularity category of R. Denote by CM_0(R) the full category of CM(R) consisting of modules that are locally free on the punctured spectrum of R, and by D^{sg}_0(R) the full subcategory of D^{sg}(R) consisting of objects that are locally zero on the punctured spectrum of R. In this paper, under the assumption that R has finite or countable CM-representation type, we completely classify the extension-closed subcategories of CM_0(R) in dimension at most two, and the extension-closed subcategories of D^{sg}_0(R) in arbitrary dimension.