nLab semiorthogonal decomposition

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Definition

Let AA be a triangulated category. A finite sequence A 1,A 2,,A nA_1,A_2,\ldots, A_n of strictly full subcategories forms a semiorthogonal decomposition of AA if the following conditions hold

(i) Hom(a j,a i)=0Hom(a_j,a_i) = 0 whenever i<ji\lt j, a iA ia_i\in A_i and a jA ja_j\in A_j

(ii) A 1,A 2,,A nA_1,A_2,\ldots, A_n jointly generate AA (the smallest strictly full triangulated category containing all of them is AA)

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

Last revised on August 30, 2024 at 08:11:01. See the history of this page for a list of all contributions to it.