RELATIVE DERIVED CATEGORIES, RELATIVE SINGULARITY CATEGORIES AND RELATIVE DEFECT CATEGORIES

We introduce the relative Gorenstein defect category of an abelian category with respect to an admissible subcategory, generalizing the Gorenstein defect categories of P. A. Bergh, D. Jorgensen and S. Oppermann. Under a mild condition of the precovering property for the relative Gorenstein category, we show that the relative Gorenstein defect category is triangle equivalent to the relative singularity category with respect to the relative Gorenstein category. We also introduce relative Ding projective defect categories and, under a similar condition, relate it to the relative singularity category with respect to the relative Ding projective category. Analogous results for relative Ding injective defect categories are also presented.