Characterizations Of Finite Resolution Of Objects In Triangulated Categories

Let X be an extension-closed subcategory of a triangulated category such that X admits a weak-cogenerator W.We investigate in the paper finite resolutions by objects in X and ?-resolution dimension of objects.In the first chapter,it is shown that if W is also a weak-generator in ?,then any finit,e resolution by objects in ? of an object can induce another finite resolution such that one of the positions in the resolution belongs to X while all the rest positions are in W.For any object with a finite resolution by objects in ?,we construct a cellular tower with respect to ?,and show that such a cellular tower can be used to detect the value of ?-resolution dimension provided that X is also closed under cocones.In the second chapter,we establish two additive functors between additive quo-tient categories related to X and W.In particular,we show that both two functors possess certain adjointness and that one can characterize the objects with a finite resolution by objects in X having a finite resolution by objects in W(resp.,when an object with a finite resolution by objects in X belongs to X)by vanishing of the counit(resp.,unit)of the adjunction with respect to one of these functors(resp.,another one).