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Abstract

This paper establishes a precise correspondence between semi-orthogonal decompositions (SOD) and finite t-stabilities on a triangulated category $\mathcal{D}$. By means of a reduction technique to certain quotient categories, we characterize the connectedness of the mutation graph of the finest $\infty$-admissible SODs of $\mathcal{D}$. Moreover, when $\mathcal{D}$ admits a Serre functor and satisfies a mild condition, we establish one-to-one correspondences among (1) finest SODs, (2) finite finest t-stabilities, (3) finite finest admissible filtrations, and (4) full exceptional sequences. These correspondences are proved to be compatible with mutations. As a by-product, classifications of SODs for the projective plane, weighted projective lines, and finite acyclic quivers are obtained.