GORENSTEIN PROJECTIVE OBJECTS IN ABELIAN CATEGORIES

ABSTRACT. Let A be an abelian category with enough projective objects and let X be a full subcategory of A. We define Gorenstein projective objects with respect to X and Yx, respectively, where Yx={Y e Ch(A) Y is acyclic and ZnY e X}. We point out that under certain hypotheses, these two Gorensein projective objects are related in a nice way. In particular, if P(A) C X, we show that Xe Ch(A) is Gorenstein projective with respect to yx if and only if X ® is Gorenstein projective with respect to X for each i, when X is a self-orthogonal class or X is Hom(-, X)-exact. Subsequently, we consider the relationships of Gorenstein projective dimensions between them. As an application, if A is of finite left Gorenstein projective global dimension with respect to X and contains an injective cogenerator, then we find a new model structure on Ch(A) by Hovey's results in [14].