Applications of resolving subcategories to singularity categories and monomorphism categories

In this paper, we study resolving subcategories and singularity categories. First, if the left perpendicular category of a module T over an Artin algebra A is the additive closure of another module M, then the singularity category of A and that of the endomorphism algebra EndA(M) of M 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.