In mathematics, a semiorthogonal decomposition is a way to divide a triangulated category into simpler pieces. One way to produce a semiorthogonal decomposition is from an exceptional collection, a special sequence of objects in a triangulated category. For an algebraic variety X, it has been fruitful to study semiorthogonal decompositions of the bounded derived category of coherent sheaves, D b ( X ) {\displaystyle {\text{D}}^{\text{b}}(X)} .
Semiorthogonal decomposition Alexei Bondal and Mikhail Kapranov (1989) defined a semiorthogonal decomposition of a triangulated category T {\displaystyle {\mathcal {T}}} to be a sequence A 1 , … , A n {\displaystyle {\mathcal {A}}_{1},\ldots ,{\mathcal {A}}_{n}} of strictly full triangulated subcategories such that:
for all 1 ≤ i < j ≤ n {\displaystyle 1\leq i<j\leq n} and all objects A i ∈ A i {\displaystyle A_{i}\in {\mathcal {A}}_{i}} and A j ∈ A j {\displaystyle A_{j}\in {\mathcal {A}}_{j}} , every morphism from A j {\displaystyle A_{j}} to A i {\displaystyle A_{i}} is zero. That is, there are "no morphisms from right to left".
T {\displaystyle {\mathcal {T}}} is generated by A 1 , … , A n {\displaystyle {\mathcal {A}}_{1},\ldots ,{\mathcal {A}}_{n}} . That is, the smallest strictly full triangulated subcategory of T {\displaystyle {\mathcal {T}}} containing A 1 , … , A n {\displaystyle {\mathcal {A}}_{1},\ldots ,{\mathcal {A}}_{n}} is equal to T {\displaystyle {\mathcal {T}}} . The notation T = ⟨ A 1 , … , A n ⟩ {\displaystyle {\mathcal {T}}=\langle {\mathcal {A}}_{1},\ldots ,{\mathcal {A}}_{n}\rangle } is used for a semiorthogonal decomposition. Having a semiorthogonal decomposition implies that every object of T {\displaystyle {\mathcal {T}}} has a canonical "filtration" whose graded pieces are (successively) in the subcategories A 1 , … , A n {\displaystyle {\mathcal {A}}_{1},\ldots ,{\mathcal {A}}_{n}} . That is, for each object T of T {\displaystyle {\mathcal {T}}} , there is a sequence
0 = T n → T n − 1 → ⋯ → T 0 = T {\displaystyle 0=T_{n}\to T_{n-1}\to \cdots \to T_{0}=T}
… excerpt ends here. Continue reading the full article.
