Length of triangulated categories

Abstract

We introduce the notion of composition series of triangulated categories, which generalizes full exceptional sequences. The lengths of composition series yield invariants for triangulated categories. We study composition series of derived categories for some classes of projective varieties and finite-dimensional algebras. We prove that certain negative rational curves on rational surfaces cause composition series of different lengths in the derived categories of the surfaces. On the other hand, we show that for derived categories of finite-dimensional hereditary algebras, for nontrivial admissible subcategories of ${\rm D}^{\rm b}(\mathbb{P}^2)$ and for derived categories of some singular varieties, all composition series have the same length.