GORENSTEIN PROJECTIVE OBJECTS IN ABELIAN CATEGORIES

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 Y-X, respectively, where Y-X={Y is an element of Ch(A)vertical bar Y is acyclic and Z(n)Y is an element of X}. We point out that under certain hypotheses, these two Gorensein projective objects are related in a nice way. In particular, if P(A) subset of X, we show that X is an element of Ch(A) is Gorenstein projective with respect to Y-X if and only if X-i 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].