Dominant Local Rings and Subcategory Classification

Abstract

AbstractWe introduce a new notion of commutative noetherian local rings, which we call dominant. We explore fundamental properties of dominant local rings and compare them with other local rings. We also provide several methods to get a new dominant local ring from a given one. Finally, we classify resolving subcategories of the module category $\operatorname {\textsf {mod}} R$ and thick subcategories of the derived category $\textsf {D}^{b}(R)$ and the singularity category $\textsf {D}^{sg}(R)$ for a local ring $R$ whose certain localizations are dominant local rings. Our results recover and refine all the known classification theorems described in this context.