A Triangulated Category-extension Of Homotopy Categories

Pashall and Scott have studied the structure of the recollement in a triangulated category (cf. [3]), there the recollement means to decompose one triangulated category into two triangulated categories which are related by some good exact functors (cf. [5],[6],[12]). On the other hand , a natural question is how to construct a new triangulated category from two known triangulated categories such that the new triangulated category is a recoUement of the known triangulated categories. In this paper , we study this question in the view of homotopy categories. More precisely , if B and C are two homotopy categories and F is an exact functor from B to C , we construct a new triangulated category D (theorem 1), and prove that D is a recoUement of B and C (theorem 2).