Applications of resolving subcategories to singularity categories and monomorphism categories

Abstract

In this paper, we study resolving subcategories and singularity categories. First, if the left perpendicular category of a module [Formula: see text] over an Artin algebra [Formula: see text] is the additive closure of another module [Formula: see text], then the singularity category of [Formula: see text] and that of the endomorphism algebra [Formula: see text] of [Formula: see text] are closed related. This gives a categorical version of a recent result of Zhang ([31], Theorem 2]). Second, we apply the resolution theorem for derived categories to elliptic curves, the monomorphism subcategory of a Gorenstein algebra and of a kind of Eilenberg–Moore category. As consequences, their singularity categories are equivalent, which explains why monomorphism categories are closely related to singularity categories.