Let be a triangulated category. A finite sequence of strictly full subcategories forms a semiorthogonal decomposition of if the following conditions hold
(i) whenever , and
(ii) jointly generate (the smallest strictly full triangulated category containing all of them is )
Examples
Prominent examples come from the notion of an exceptional collection. The technique is used mainly for the study of derived categories attached to complex algebraic varieties and related “noncommutative” examples like Landau-Ginzburg models.
Valery A. Lunts, Olaf M. Schnürer, Matrix factorizations and semi-orthogonal decompositions for blowing-ups, J. Noncommut. Geom. 10:3 (2016) 907–979 doi