nLab semiorthogonal decomposition

Definition

Let $A$ be a triangulated category. A finite sequence $A_1,A_2,\ldots, A_n$ of strictly full subcategories forms a semiorthogonal decomposition of $A$ if the following conditions hold

(i) $Hom(a_j,a_i) = 0$ whenever $i\lt j$, $a_i\in A_i$ and $a_j\in A_j$

(ii) $A_1,A_2,\ldots, A_n$ jointly generate $A$ (the smallest strictly full triangulated category containing all of them is $A$)

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.

Literature

• wikipedia: Semiorthogonal decomposition
• Dmitri Orlov, Smooth and proper noncommutative schemes and gluing of DG categories, Adv. Math. 302 (2016) 59-105 doi arXiv:1402.7364
• Alexei Bondal, Dmitri Orlov, Semiorthogonal decomposition for algebraic varieties, arxiv:alg-geom/9506012
• Alexander Kuznetsov, Base change for semiorthogonal decompositions, Compositio Mathematica 147:03 (2011) 852–876 arXiv:0711.1734, doi
• Alexander Kuznetsov, Semiorthogonal decompositions in families, arXiv:2111.00527
• Alexander Kuznetsov, Semiorthogonal decompositions in algebraic geometry, (ICM 2014) arXiv:1404.3143
• Valery A. Lunts, Olaf M. Schnürer, Matrix factorizations and semi-orthogonal decompositions for blowing-ups, J. Noncommut. Geom. 10:3 (2016) 907–979 doi

category: algebraic geometry, triangulated categories