Extension-closed subcategories over hypersurfaces of 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 CM₀(𝑅) the full category of CM(𝑅) consisting of modules that are locally free on the punctured spectrum of 𝑅, and by D^sg₀(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 CM₀(𝑅) in dimension at most two, and the extension-closed subcategories of D^sg₀(𝑅) in arbitrary dimension.