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, .

Semiorthogonal decomposition

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Alexei Bondal and Mikhail Kapranov (1989) defined a semiorthogonal decomposition of a triangulated category   to be a sequence   of strictly full triangulated subcategories such that:[1]

  • for all   and all objects   and  , every morphism from   to   is zero. That is, there are "no morphisms from right to left".
  •   is generated by  . That is, the smallest strictly full triangulated subcategory of   containing   is equal to  .

The notation   is used for a semiorthogonal decomposition.

Having a semiorthogonal decomposition implies that every object of   has a canonical "filtration" whose graded pieces are (successively) in the subcategories  . That is, for each object T of  , there is a sequence

 

of morphisms in   such that the cone of   is in  , for each i. Moreover, this sequence is unique up to a unique isomorphism.[2]

One can also consider "orthogonal" decompositions of a triangulated category, by requiring that there are no morphisms from   to   for any  . 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   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   as closer to a split exact sequence, because the exact sequence   of triangulated categories is split by the subcategory  , mapping isomorphically to  .

Using that observation, a semiorthogonal decomposition   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,   can be identified with the Grothendieck group   of algebraic vector bundles on X. In this geometric situation, using that   comes from a dg-category, a semiorthogonal decomposition actually gives a splitting of all the algebraic K-groups of X:

 

for all i.[3]

Admissible subcategory

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One way to produce a semiorthogonal decomposition is from an admissible subcategory. By definition, a full triangulated subcategory   is left admissible if the inclusion functor   has a left adjoint functor, written  . Likewise,   is right admissible if the inclusion has a right adjoint, written  , and it is admissible if it is both left and right admissible.

A right admissible subcategory   determines a semiorthogonal decomposition

 ,

where

 

is the right orthogonal of   in  .[2] Conversely, every semiorthogonal decomposition   arises in this way, in the sense that   is right admissible and  . Likewise, for any semiorthogonal decomposition  , the subcategory   is left admissible, and  , where

 

is the left orthogonal of  .

If   is the bounded derived category of a smooth projective variety over a field k, then every left or right admissible subcategory of   is in fact admissible.[4] By results of Bondal and Michel Van den Bergh, this holds more generally for   any regular proper triangulated category that is idempotent-complete.[5]

Moreover, for a regular proper idempotent-complete triangulated category  , a full triangulated subcategory is admissible if and only if it is regular and idempotent-complete. These properties are intrinsic to the subcategory.[6] For example, for X a smooth projective variety and Y a subvariety not equal to X, the subcategory of   of objects supported on Y is not admissible.

Exceptional collection

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Let k be a field, and let   be a k-linear triangulated category. An object E of   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  . (In the derived category of a smooth complex projective variety X, the first-order deformation space of an object E is  , and so an exceptional object is in particular rigid. It follows, for example, that there are at most countably many exceptional objects in  , up to isomorphism. That helps to explain the name.)

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   such that   for all i < j and all integers t. (That is, there are "no morphisms from right to left".) In a proper triangulated category   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  , and   denotes the full triangulated subcategory generated by the object  .[7] An exceptional collection is called full if the subcategory   is zero. (Thus a full exceptional collection breaks the whole triangulated category up into finitely many copies of  .)

In particular, if X is a smooth projective variety such that   has a full exceptional collection  , 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   for all  ; moreover, the cycle class map   must be an isomorphism.[8]

Examples

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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.[9] 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.[10]

More generally, if X is a smooth projective variety of positive dimension such that the coherent sheaf cohomology groups   are zero for i > 0, then the object   in   is exceptional, and so it induces a nontrivial semiorthogonal decomposition  . 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.

A source of examples is Orlov's blowup formula concerning the blowup   of a scheme   at a codimension   locally complete intersection subscheme   with exceptional locus  . There is a semiorthogonal decomposition   where   is the functor   with   is the natural map.[11]

While these examples encompass a large number of well-studied derived categories, many naturally occurring triangulated categories are "indecomposable". In particular, for a smooth projective variety X whose canonical bundle   is basepoint-free, every semiorthogonal decomposition   is trivial in the sense that   or   must be zero.[12] For example, this applies to every variety which is Calabi–Yau in the sense that its canonical bundle is trivial.

See also

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Notes

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  1. ^ Huybrechts 2006, Definition 1.59.
  2. ^ a b Bondal & Kapranov 1990, Proposition 1.5.
  3. ^ Orlov 2016, Section 1.2.
  4. ^ Kuznetsov 2007, Lemmas 2.10, 2.11, and 2.12.
  5. ^ Orlov 2016, Theorem 3.16.
  6. ^ Orlov 2016, Propositions 3.17 and 3.20.
  7. ^ Huybrechts 2006, Lemma 1.58.
  8. ^ Marcolli & Tabuada 2015, Proposition 1.9.
  9. ^ Huybrechts 2006, Corollary 8.29.
  10. ^ Kuznetsov 2014, Section 2.2.
  11. ^ Orlov, D O (1993-02-28). "PROJECTIVE BUNDLES, MONOIDAL TRANSFORMATIONS, AND DERIVED CATEGORIES OF COHERENT SHEAVES". Russian Academy of Sciences. Izvestiya Mathematics. 41 (1): 133–141. doi:10.1070/im1993v041n01abeh002182. ISSN 1064-5632.
  12. ^ Kuznetsov 2014, Section 2.5.

References

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