Semi-orthogonal decompositions via t-stabilities

In this paper we introduce a local-refinement procedure to investigate finite t-stabilities on a triangulated category, and show a direct sufficient condition for a finite t-stability to be finite finest. We classify all finite finest t-stabilities for certain triangulated categories, including those from the projective plane, weighted projective lines, and finite acyclic quivers. As applications, we obtain a simple classification of semi-orthogonal decompositions (SOD) for these categories. Furthermore, we study the connectedness of SODs by a reduction method and give an easily verified criterion for it.