Semiorthogonal decomposition

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, formula_1.
Semiorthogonal decomposition.
Alexei Bondal and Mikhail Kapranov (1989) defined a semiorthogonal decomposition of a triangulated category formula_2 to be a sequence formula_3 of strictly full triangulated subcategories such that:
The notation formula_14 is used for a semiorthogonal decomposition.
Having a semiorthogonal decomposition implies that every object of formula_2 has a canonical "filtration" whose graded pieces are (successively) in the subcategories formula_3. That is, for each object "T" of formula_2, there is a sequence
of morphisms in formula_2 such that the cone of formula_20 is in formula_21, for each "i". Moreover, this sequence is unique up to a unique isomorphism.
One can also consider "orthogonal" decompositions of a triangulated category, by requiring that there are no morphisms from formula_21 to formula_23 for any formula_24. However, that property is too strong for most purposes. For example, for an (irreducible) smooth projective variety "X" over a field, the bounded derived category formula_1 of coherent sheaves never has a nontrivial orthogonal decomposition, whereas it may have a semiorthogonal decomposition, by the examples below. 
A semiorthogonal decomposition of a triangulated category may be considered as analogous to a finite filtration of an abelian group. Alternatively, one may consider a semiorthogonal decomposition formula_26 as closer to a split exact sequence, because the exact sequence formula_27 of triangulated categories is split by the subcategory formula_28, mapping isomorphically to formula_29.
Using that observation, a semiorthogonal decomposition formula_14 implies a direct sum splitting of Grothendieck groups:
For example, when formula_32 is the bounded derived category of coherent sheaves on a smooth projective variety "X", formula_33 can be identified with the Grothendieck group formula_34 of algebraic vector bundles on "X". In this geometric situation, using that formula_1 comes from a dg-category, a semiorthogonal decomposition actually gives a splitting of all the algebraic K-groups of "X":
for all "i".
Admissible subcategory.
One way to produce a semiorthogonal decomposition is from an admissible subcategory. By definition, a full triangulated subcategory formula_37 is left admissible if the inclusion functor formula_38 has a left adjoint functor, written formula_39. Likewise, formula_37 is right admissible if the inclusion has a right adjoint, written formula_41, and it is admissible if it is both left and right admissible.
A right admissible subcategory formula_42 determines a semiorthogonal decomposition
where
is the right orthogonal of formula_45 in formula_2. Conversely, every semiorthogonal decomposition formula_47 arises in this way, in the sense that formula_45 is right admissible and formula_49. Likewise, for any semiorthogonal decomposition formula_47, the subcategory formula_51 is left admissible, and formula_52, where
is the left orthogonal of formula_51.
If formula_2 is the bounded derived category of a smooth projective variety over a field "k", then every left or right admissible subcategory of formula_2 is in fact admissible. By results of Bondal and Michel Van den Bergh, this holds more generally for formula_2 any regular proper triangulated category that is idempotent-complete.
Moreover, for a regular proper idempotent-complete triangulated category formula_2, a full triangulated subcategory is admissible if and only if it is regular and idempotent-complete. These properties are intrinsic to the subcategory. For example, for "X" a smooth projective variety and "Y" a subvariety not equal to "X", the subcategory of formula_1 of objects supported on "Y" is not admissible.
Exceptional collection.
Let "k" be a field, and let formula_2 be a "k"-linear triangulated category. An object "E" of formula_2 is called exceptional if Hom("E","E") = "k" and Hom("E","E"["t"]) = 0 for all nonzero integers "t", where ["t"] is the shift functor in formula_2. (In the derived category of a smooth complex projective variety "X", the first-order deformation space of an object "E" is formula_63, and so an exceptional object is in particular rigid. It follows, for example, that there are at most countably many exceptional objects in formula_1, up to isomorphism. That helps to explain the name.)
The triangulated subcategory generated by an exceptional object "E" is equivalent to the derived category formula_65 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 formula_66 such that formula_67 for all "i" < "j" and all integers "t". (That is, there are "no morphisms from right to left".) In a proper triangulated category formula_2 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 formula_70, and formula_71 denotes the full triangulated subcategory generated by the object formula_71. An exceptional collection is called full if the subcategory formula_51 is zero. (Thus a full exceptional collection breaks the whole triangulated category up into finitely many copies of formula_65.)
In particular, if "X" is a smooth projective variety such that formula_1 has a full exceptional collection formula_66, 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 formula_78 for all formula_79; moreover, the cycle class map formula_80 must be an isomorphism.
Examples.
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. Full exceptional collections have also been constructed on all smooth projective toric varieties, del Pezzo surfaces, many projective homogeneous varieties, and some other Fano varieties.
More generally, if "X" is a smooth projective variety of positive dimension such that the coherent sheaf cohomology groups formula_82 are zero for "i" > 0, then the object formula_83 in formula_1 is exceptional, and so it induces a nontrivial semiorthogonal decomposition formula_85. This applies to every Fano variety over a field of characteristic zero, for example. It also applies to some other varieties, such as Enriques surfaces and some surfaces of general type.
On the other hand, many naturally occurring triangulated categories are "indecomposable". In particular, for a smooth projective variety "X" whose canonical bundle formula_86 is basepoint-free, every semiorthogonal decomposition formula_87 is trivial in the sense that formula_51 or formula_45 must be zero. For example, this applies to every variety which is Calabi–Yau in the sense that its canonical bundle is trivial. 