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

Source abstract

Let $k$ be an algebraically closed uncountable field of characteristic zero. Let $R$ be a complete local hypersurface over $k$. Denote by $\operatorname{CM}(R)$ the category of maximal Cohen–Macaulay $R$-modules and by $\operatorname{D^{sg}}(R)$ the singularity category of $R$. Denote by $\operatorname{CM}_{0}(R)$ the full category of $\operatorname{CM}(R)$ consisting of modules that are locally free on the punctured spectrum of $R$, and by $\operatorname{D^{sg}_{0}}(R)$ the full subcategory of $\operatorname{D^{sg}}(R)$ 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 $\operatorname{CM}_{0}(R)$ in dimension at most two, and the extension-closed subcategories of $\operatorname{D^{sg}_{0}}(R)$ in arbitrary dimension.