Exceptional collection

Summary

An exceptional collection in a linear triangulated category is a finite ordered list of exceptional objects. Each of them has only scalar endomorphisms and no self-extensions. There are no morphisms, in any degree, from later objects to earlier ones. A full exceptional collection splits the category into a semiorthogonal decomposition whose pieces are copies of the derived category of vector spaces. The basic example is Beilinson's collection $\mathcal{O}, \mathcal{O}(1), \dots, \mathcal{O}(n)$ on projective space.