Semi-orthogonal decompositions via t-stabilities
Mingfa Chen
TL;DR
The paper develops a comprehensive framework linking semi-orthogonal decompositions with finite t-stabilities on triangulated categories, establishing precise one-to-one correspondences between finest SODs, finite finest t-stabilities, finite finest admissible filtrations, and full exceptional sequences (under Serre-functor assumptions) and showing these correspondences are mutation-compatible. It introduces a reduction approach via Verdier quotients to study mutation graphs, providing a concrete connectedness criterion that reduces global questions to quotient categories. The results yield classifications of SODs for key categories (Proj$^2$, weighted projective lines, and finite acyclic quivers) and describe the structure of their mutation graphs, including braid-relations for mutations. The methods combine SOD theory, t-stability, and exceptional sequences to offer a unifying, mutation-friendly perspective with broad applications in representation theory and algebraic geometry.
Abstract
This paper presents a precise relationship between semi-orthogonal decompositions (SOD) and finite t-stabilities on a triangulated category $\mathcal{D}$. By means of a reduction method to certain quotient categories, we provide a characterization of 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 show 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 applications, we obtain a classification of SODs for the projective plane, weighted projective lines, and finite acyclic quivers via the t-stability approach.