Relative Singularity Categories And Relative Defect Categories

In this paper we study the relative singularity categories and relative defect cate-gories of admissible subcategories of abelian categories. Firstly, we recall the definition of localization, localizing classes and saturated localizing classes, and we show that some well-known results also hold for left (or right) localizing classes. Secondly, given an Artin algebra and a left perpendicular subcategory of its category of finitely generated left modules, we study the relation between the bounded singularity category of the given Artin algebra and the bounded singularity category of (the opposite algebra of) the endomorphism ring of an additive generator of the left perpendicular subcategory, which gives a categorical expla-nation of a recent result of P. Zhang. At last, we define the relative defect category of an admissible subcategory of an abelian category and the main result of this paper shows under rather weak conditions, the relative defect category of an admissible subcategory is triangle equivalent to the relative singularity category of the corresponding Gorenstein category. The work generalizes a result proved independently by F. Kong-P. Zhang and Y.-H. Bao-X.-N. Du-Z.-B. Zhao.