Generating Cohen-Macaulay module categories and Buchweitz-Orlov singularity categories

Cohen-Macaulay representation theory was born in the 1960s as a higher-dimensional version of representation theory of artin algebras. The purpose of this theory is, for a given Cohen-Macaulay ring R, to understand the structure of the category CM(R) of (maximal) Cohen-Macaulay R-modules. In this talk, we will first consider a certain question on how to generate the category CM(R) by using concrete examples. Next we will move on to thinking of a similar question for the singularity category D_{sg}(R) in the sense of Buchweitz and Orlov, and extend theorems of Ballard, Favero and Katzarkov.