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, ${\displaystyle {\text{D}}^{\text{b}}(X)}$.

Semiorthogonal decomposition

Alexei Bondal and Mikhail Kapranov (1989) defined a semiorthogonal decomposition of a triangulated category ${\displaystyle {\mathcal {T}}}$ to be a sequence ${\displaystyle {\mathcal {A}}_{1},\ldots ,{\mathcal {A}}_{n}}$ of strictly full triangulated subcategories such that:^[1]

• for all ${\displaystyle 1\leq i<j\leq n}$ and all objects ${\displaystyle A_{i}\in {\mathcal {A}}_{i}}$ and ${\displaystyle A_{j}\in {\mathcal {A}}_{j}}$, every morphism from ${\displaystyle A_{j}}$ to ${\displaystyle A_{i}}$ is zero. That is, there are "no morphisms from right to left".
• ${\displaystyle {\mathcal {T}}}$ is generated by ${\displaystyle {\mathcal {A}}_{1},\ldots ,{\mathcal {A}}_{n}}$. That is, the smallest strictly full triangulated subcategory of ${\displaystyle {\mathcal {T}}}$ containing ${\displaystyle {\mathcal {A}}_{1},\ldots ,{\mathcal {A}}_{n}}$ is equal to ${\displaystyle {\mathcal {T}}}$.

The notation ${\displaystyle {\mathcal {T}}=\langle {\mathcal {A}}_{1},\ldots ,{\mathcal {A}}_{n}\rangle }$ is used for a semiorthogonal decomposition.

Having a semiorthogonal decomposition implies that every object of ${\displaystyle {\mathcal {T}}}$ has a canonical "filtration" whose graded pieces are (successively) in the subcategories ${\displaystyle {\mathcal {A}}_{1},\ldots ,{\mathcal {A}}_{n}}$. That is, for each object T of ${\displaystyle {\mathcal {T}}}$, there is a sequence

${\displaystyle 0=T_{n}\to T_{n-1}\to \cdots \to T_{0}=T}$

of morphisms in ${\displaystyle {\mathcal {T}}}$ such that the cone of ${\displaystyle T_{i}\to T_{i-1}}$ is in ${\displaystyle {\mathcal {A}}_{i}}$, 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 ${\displaystyle {\mathcal {A}}_{i}}$ to ${\displaystyle {\mathcal {A}}_{j}}$ for any ${\displaystyle i\neq j}$. 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 ${\displaystyle {\text{D}}^{\text{b}}(X)}$ 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 ${\displaystyle {\mathcal {T}}=\langle {\mathcal {A}},{\mathcal {B}}\rangle }$ as closer to a split exact sequence, because the exact sequence ${\displaystyle 0\to {\mathcal {A}}\to {\mathcal {T}}\to {\mathcal {T}}/{\mathcal {A}}\to 0}$ of triangulated categories is split by the subcategory ${\displaystyle {\mathcal {B}}\subset {\mathcal {T}}}$, mapping isomorphically to ${\displaystyle {\mathcal {T}}/{\mathcal {A}}}$.

Using that observation, a semiorthogonal decomposition ${\displaystyle {\mathcal {T}}=\langle {\mathcal {A}}_{1},\ldots ,{\mathcal {A}}_{n}\rangle }$ implies a direct sum splitting of Grothendieck groups:

${\displaystyle K_{0}({\mathcal {T}})\cong K_{0}({\mathcal {A}}_{1})\oplus \cdots \oplus K_{0}({\mathcal {A_{n}}}).}$

For example, when ${\displaystyle {\mathcal {T}}={\text{D}}^{\text{b}}(X)}$ is the bounded derived category of coherent sheaves on a smooth projective variety X, ${\displaystyle K_{0}({\mathcal {T}})}$ can be identified with the Grothendieck group ${\displaystyle K_{0}(X)}$ of algebraic vector bundles on X. In this geometric situation, using that ${\displaystyle {\text{D}}^{\text{b}}(X)}$ comes from a dg-category, a semiorthogonal decomposition actually gives a splitting of all the algebraic K-groups of X:

${\displaystyle K_{i}(X)\cong K_{i}({\mathcal {A}}_{1})\oplus \cdots \oplus K_{i}({\mathcal {A_{n}}})}$

for all i.^[3]

Admissible subcategory

One way to produce a semiorthogonal decomposition is from an admissible subcategory. By definition, a full triangulated subcategory ${\displaystyle {\mathcal {A}}\subset {\mathcal {T}}}$ is left admissible if the inclusion functor ${\displaystyle i\colon {\mathcal {A}}\to {\mathcal {T}}}$ has a left adjoint functor, written ${\displaystyle i^{*}}$. Likewise, ${\displaystyle {\mathcal {A}}\subset {\mathcal {T}}}$ is right admissible if the inclusion has a right adjoint, written ${\displaystyle i^{!}}$, and it is admissible if it is both left and right admissible.

A right admissible subcategory ${\displaystyle {\mathcal {B}}\subset {\mathcal {T}}}$ determines a semiorthogonal decomposition

${\displaystyle {\mathcal {T}}=\langle {\mathcal {B}}^{\perp },{\mathcal {B}}\rangle }$,

where

${\displaystyle {\mathcal {B}}^{\perp }:=\{T\in {\mathcal {T}}:\operatorname {Hom} ({\mathcal {B}},T)=0\}}$

is the right orthogonal of ${\displaystyle {\mathcal {B}}}$ in ${\displaystyle {\mathcal {T}}}$.^[2] Conversely, every semiorthogonal decomposition ${\displaystyle {\mathcal {T}}=\langle {\mathcal {A}},{\mathcal {B}}\rangle }$ arises in this way, in the sense that ${\displaystyle {\mathcal {B}}}$ is right admissible and ${\displaystyle {\mathcal {A}}={\mathcal {B}}^{\perp }}$. Likewise, for any semiorthogonal decomposition ${\displaystyle {\mathcal {T}}=\langle {\mathcal {A}},{\mathcal {B}}\rangle }$, the subcategory ${\displaystyle {\mathcal {A}}}$ is left admissible, and ${\displaystyle {\mathcal {B}}={}^{\perp }{\mathcal {A}}}$, where

${\displaystyle {}^{\perp }{\mathcal {A}}:=\{T\in {\mathcal {T}}:\operatorname {Hom} (T,{\mathcal {A}})=0\}}$

is the left orthogonal of ${\displaystyle {\mathcal {A}}}$.

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

Moreover, for a regular proper idempotent-complete triangulated category ${\displaystyle {\mathcal {T}}}$, 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 ${\displaystyle {\text{D}}^{\text{b}}(X)}$ of objects supported on Y is not admissible.

Exceptional collection

Let k be a field, and let ${\displaystyle {\mathcal {T}}}$ be a k-linear triangulated category. An object E of ${\displaystyle {\mathcal {T}}}$ 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 ${\displaystyle {\mathcal {T}}}$. (In the derived category of a smooth complex projective variety X, the first-order deformation space of an object E is ${\displaystyle \operatorname {Ext} _{X}^{1}(E,E)\cong \operatorname {Hom} (E,E[1])}$, and so an exceptional object is in particular rigid. It follows, for example, that there are at most countably many exceptional objects in ${\displaystyle {\text{D}}^{\text{b}}(X)}$, up to isomorphism. That helps to explain the name.)

The triangulated subcategory generated by an exceptional object E is equivalent to the derived category ${\displaystyle {\text{D}}^{\text{b}}(k)}$ 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 ${\displaystyle E_{1},\ldots ,E_{m}}$ such that ${\displaystyle \operatorname {Hom} (E_{j},E_{i}[t])=0}$ for all i < j and all integers t. (That is, there are "no morphisms from right to left".) In a proper triangulated category ${\displaystyle {\mathcal {T}}}$ 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:

${\displaystyle {\mathcal {T}}=\langle {\mathcal {A}},E_{1},\ldots ,E_{m}\rangle ,}$

where ${\displaystyle {\mathcal {A}}=\langle E_{1},\ldots ,E_{m}\rangle ^{\perp }}$, and ${\displaystyle E_{i}}$ denotes the full triangulated subcategory generated by the object ${\displaystyle E_{i}}$.^[7] An exceptional collection is called full if the subcategory ${\displaystyle {\mathcal {A}}}$ is zero. (Thus a full exceptional collection breaks the whole triangulated category up into finitely many copies of ${\displaystyle {\text{D}}^{\text{b}}(k)}$.)

In particular, if X is a smooth projective variety such that ${\displaystyle {\text{D}}^{\text{b}}(X)}$ has a full exceptional collection ${\displaystyle E_{1},\ldots ,E_{m}}$, then the Grothendieck group of algebraic vector bundles on X is the free abelian group on the classes of these objects:

${\displaystyle K_{0}(X)\cong \mathbb {Z} \{E_{1},\ldots ,E_{m}\}.}$

A smooth complex projective variety X with a full exceptional collection must have trivial Hodge theory, in the sense that ${\displaystyle h^{p,q}(X)=0}$ for all ${\displaystyle p\neq q}$; moreover, the cycle class map ${\displaystyle CH^{*}(X)\otimes \mathbb {Q} \to H^{*}(X,\mathbb {Q} )}$ must be an isomorphism.^[8]

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

${\displaystyle {\text{D}}^{\text{b}}(\mathbf {P} ^{n})=\langle O,O(1),\ldots ,O(n)\rangle }$,

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 ${\displaystyle H^{i}(X,O_{X})}$ are zero for i > 0, then the object ${\displaystyle O_{X}}$ in ${\displaystyle {\text{D}}^{\text{b}}(X)}$ is exceptional, and so it induces a nontrivial semiorthogonal decomposition ${\displaystyle {\text{D}}^{\text{b}}(X)=\langle (O_{X})^{\perp },O_{X}\rangle }$. 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 ${\displaystyle X=\operatorname {Bl} _{Z}(Y)}$ of a scheme ${\displaystyle Y}$ at a codimension ${\displaystyle k}$ locally complete intersection subscheme ${\displaystyle Z}$ with exceptional locus ${\displaystyle \iota :E\simeq \mathbb {P} _{Z}(N_{Z/Y})\to X}$. There is a semiorthogonal decomposition ${\displaystyle D^{b}(X)=\langle \Phi _{1-k}(D^{b}(Z)),\ldots ,\Phi _{-1}(D^{b}(Z)),\pi ^{*}(D^{b}(Y))\rangle }$ where ${\displaystyle \Phi _{i}:D^{b}(Z)\to D^{b}(X)}$ is the functor ${\displaystyle \Phi _{i}(-)=\iota _{*}({\mathcal {O}}_{E}(k))\otimes p^{*}(-))}$ with ${\displaystyle p:X\to Y}$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 ${\displaystyle K_{X}}$ is basepoint-free, every semiorthogonal decomposition ${\displaystyle {\text{D}}^{\text{b}}(X)=\langle {\mathcal {A}},{\mathcal {B}}\rangle }$ is trivial in the sense that ${\displaystyle {\mathcal {A}}}$ or ${\displaystyle {\mathcal {B}}}$ 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

• Derived noncommutative algebraic geometry

Notes

References

• Bondal, Alexei; Kapranov, Mikhail (1990), "Representable functors, Serre functors, and reconstructions", Mathematics of the USSR-Izvestiya, 35: 519–541, doi:10.1070/IM1990v035n03ABEH000716, MR 1039961
• Huybrechts, Daniel (2006), Fourier–Mukai transforms in algebraic geometry, Oxford University Press, ISBN 978-0199296866, MR 2244106
• Kuznetsov, Alexander (2007), "Homological projective duality", Publications Mathématiques de l'IHÉS, 105: 157–220, arXiv:math/0507292, doi:10.1007/s10240-007-0006-8, MR 2354207
• Kuznetsov, Alexander (2014), "Semiorthogonal decompositions in algebraic geometry", Proceedings of the International Congress of Mathematicians (Seoul, 2014), vol. 2, Seoul: Kyung Moon Sa, pp. 635–660, arXiv:1404.3143, MR 3728631
• Marcolli, Matilde; Tabuada, Gonçalo (2015), "From exceptional collections to motivic decompositions via noncommutative motives", Journal für die reine und angewandte Mathematik, 701: 153–167, arXiv:1202.6297, doi:10.1515/crelle-2013-0027, MR 3331729
• Orlov, Dmitri (2016), "Smooth and proper noncommutative schemes and gluing of DG categories", Advances in Mathematics, 302: 59–105, arXiv:1402.7364, doi:10.1016/j.aim.2016.07.014, MR 3545926