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,
Alexei Bondal and Mikhail Kapranov (1989) defined a semiorthogonal decomposition of a triangulated category
of strictly full triangulated subcategories such that:[1] The notation
has a canonical "filtration" whose graded pieces are (successively) in the subcategories
[2] One can also consider "orthogonal" decompositions of a triangulated category, by requiring that there are no morphisms from
For example, for an (irreducible) smooth projective variety X over a field, the bounded derived category
A semiorthogonal decomposition of a triangulated category may be considered as analogous to a finite filtration of an abelian group.
implies a direct sum splitting of Grothendieck groups: For example, when
is the bounded derived category of coherent sheaves on a smooth projective variety X,
comes from a dg-category, a semiorthogonal decomposition actually gives a splitting of all the algebraic K-groups of X: for all i.
[3] One way to produce a semiorthogonal decomposition is from an admissible subcategory.
is the bounded derived category of a smooth projective variety over a field k, then every left or right admissible subcategory of
[4] By results of Bondal and Michel Van den Bergh, this holds more generally for
any regular proper triangulated category that is idempotent-complete.
, a full triangulated subcategory is admissible if and only if it is regular and idempotent-complete.
[6] For example, for X a smooth projective variety and Y a subvariety not equal to X, the subcategory of
(In the derived category of a smooth complex projective variety X, the first-order deformation space of an object E is
The triangulated subcategory generated by an exceptional object E is equivalent to the derived category
of finite-dimensional k-vector spaces, the simplest triangulated category in this context.
(For example, every object of that subcategory is isomorphic to a finite direct sum of shifts of E.) Alexei Gorodentsev and Alexei Rudakov (1987) defined an exceptional collection to be a sequence of exceptional objects
over k, such as the bounded derived category of coherent sheaves on a smooth projective variety, every exceptional collection generates an admissible subcategory, and so it determines a semiorthogonal decomposition: where
denotes the full triangulated subcategory generated by the object
(Thus a full exceptional collection breaks the whole triangulated category up into finitely many copies of
, then the Grothendieck group of algebraic vector bundles on X is the free abelian group on the classes of these objects: A smooth complex projective variety X with a full exceptional collection must have trivial Hodge theory, in the sense that
[8] The original example of a full exceptional collection was discovered by Alexander Beilinson (1978): the derived category of projective space over a field has the full exceptional collection where O(j) for integers j are the line bundles on projective space.
[10] More generally, if X is a smooth projective variety of positive dimension such that the coherent sheaf cohomology groups
This applies to every Fano variety over a field of characteristic zero, for example.
[12] For example, this applies to every variety which is Calabi–Yau in the sense that its canonical bundle is trivial.