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

TL;DR

The paper addresses the problem of classifying extension-closed subcategories of $ ext{CM}_0(R)$ in dimension at most two and of $ ext{D}^{ ext{sg}}_0(R)$ in arbitrary dimension for local hypersurface rings with finite or countable CM-representation type. It uses Knörrer periodicity to reduce to the low-dimensional case and leverages exact-squares, syzygies, and Auslander–Reiten theory to derive explicit Hasse diagrams. The main contributions are complete lists of extension-closed subcategories (five diagrams for $ ext{CM}_0(R)$ in $ ext{dim }R ext{≤}2$ and four diagrams for $ ext{D}^{ ext{sg}}_0(R)$ in general) across ADE and related singularities, including explicit indecomposable CM-modules and their extension relations. This advances the structural understanding of extension-closed subcategories in singularity categories and their stable counterparts, connecting to the broader classification of CM-representation types.

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.

Theorems & Definitions (51)

• Theorem 1.1
• Definition 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Definition 2.5
• Remark 2.6
• Definition 2.7
• Remark 2.8
• Definition 2.9