The Homological Theory of Contravariantly Finite Subcategories:

Let C be an abelian or exact category with enough projectives and let P be the full subcategory of projective objects of C . We consider the stable category C/P modulo projectives, as a left triangulated category [14], [36]. Then there is a triangulated category S(C/P) associated to C/P, which is universal in the following sense. There exists an exact functor S : C/P -t S(C/P) such that any exact functor out of C/P to a triangulated category has a unique exact factorization through S. The triangulated category S(C/P) is called the stabilization of C/P and the functor S is called the stabilization functor. There is also the dual construction of the costabilization R(C/P) of C/P, which is a triangulated category equipped with an exact functor R : R(C/P) -+ C/P, the costabilization functor, such that any exact functor from a triangulated category to C/P has a unique exact factorization through R. If C has enough injectives we can stabilize and costabilize in the above sense the stable category modulo injectives. These constructions have topological origin and make sense for any stable category C / X , where now C is an additive category and X is a contravariantly or covariantly finite subcategory of C in the sense of Auslander-Smal0 [8], assuming that C satisfies some mild condition. The stabilization construction in our setting is due to Heller [33], see also [24], [44], and later was used by Keller-Vossieck in 1361. For the costabilization construction we refer to the work of Grandis [27].