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

Semiorthogonal decomposition

Alexei Bondal and Mikhail Kapranov (1989) defined a semiorthogonal decomposition of a triangulated category $𝒯$ to be a sequence ${𝒜}_1,\dots ,{𝒜}_n$ of strictly full triangulated subcategories such that:Template:Sfn

• for all $1\le i<j\le n$ and all objects $A_i\in {𝒜}_i$ and $A_j\in {𝒜}_j$, every morphism from $A_j$ to $A_i$ is zero. That is, there are "no morphisms from right to left".
• $𝒯$ is generated by ${𝒜}_1,\dots ,{𝒜}_n$. That is, the smallest strictly full triangulated subcategory of $𝒯$ containing ${𝒜}_1,\dots ,{𝒜}_n$ is equal to $𝒯$.

The notation $𝒯=⟨{𝒜}_1,\dots ,{𝒜}_n⟩$ is used for a semiorthogonal decomposition.

Having a semiorthogonal decomposition implies that every object of $𝒯$ has a canonical "filtration" whose graded pieces are (successively) in the subcategories ${𝒜}_1,\dots ,{𝒜}_n$. That is, for each object T of $𝒯$, there is a sequence

$0=T_n\to T_{n-1}\to \cdots \to T_0=T$

of morphisms in $𝒯$ such that the cone of $T_i\to T_{i-1}$ is in ${𝒜}_i$, for each i. Moreover, this sequence is unique up to a unique isomorphism.Template:Sfn

One can also consider "orthogonal" decompositions of a triangulated category, by requiring that there are no morphisms from ${𝒜}_i$ to ${𝒜}_j$ for any $i\ne j$. 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 ${\text{D}}^{\text{b}}(X)$ 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 $𝒯=⟨𝒜,ℬ⟩$ as closer to a split exact sequence, because the exact sequence $0\to 𝒜\to 𝒯\to 𝒯/𝒜\to 0$ of triangulated categories is split by the subcategory $ℬ\subset 𝒯$, mapping isomorphically to $𝒯/𝒜$.

Using that observation, a semiorthogonal decomposition $𝒯=⟨{𝒜}_1,\dots ,{𝒜}_n⟩$ implies a direct sum splitting of Grothendieck groups:

$K_0(𝒯)\cong K_0({𝒜}_1)\oplus \cdots \oplus K_0({𝒜}_{𝓃}).$

For example, when $𝒯={\text{D}}^{\text{b}}(X)$ is the bounded derived category of coherent sheaves on a smooth projective variety X, $K_0(𝒯)$ can be identified with the Grothendieck group $K_0(X)$ of algebraic vector bundles on X. In this geometric situation, using that ${\text{D}}^{\text{b}}(X)$ comes from a dg-category, a semiorthogonal decomposition actually gives a splitting of all the algebraic K-groups of X:

$K_i(X)\cong K_i({𝒜}_1)\oplus \cdots \oplus K_i({𝒜}_{𝓃})$

for all i.Template:Sfn

Admissible subcategory

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

A right admissible subcategory $ℬ\subset 𝒯$ determines a semiorthogonal decomposition

$𝒯=⟨{ℬ}^{\perp },ℬ⟩$,

where

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

is the right orthogonal of $ℬ$ in $𝒯$.Template:Sfn Conversely, every semiorthogonal decomposition $𝒯=⟨𝒜,ℬ⟩$ arises in this way, in the sense that $ℬ$ is right admissible and $𝒜={ℬ}^{\perp }$. Likewise, for any semiorthogonal decomposition $𝒯=⟨𝒜,ℬ⟩$, the subcategory $𝒜$ is left admissible, and $ℬ={}^{\perp }𝒜$, where

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

is the left orthogonal of $𝒜$.

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

Moreover, for a regular proper idempotent-complete triangulated category $𝒯$, a full triangulated subcategory is admissible if and only if it is regular and idempotent-complete. These properties are intrinsic to the subcategory.Template:Sfn For example, for X a smooth projective variety and Y a subvariety not equal to X, the subcategory of ${\text{D}}^{\text{b}}(X)$ of objects supported on Y is not admissible.

Exceptional collection

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

The triangulated subcategory generated by an exceptional object E is equivalent to the derived category ${\text{D}}^{\text{b}}(k)$ 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 $E_1,\dots ,E_m$ such that $\mathrm{Hom}(E_j,E_i[t])=0$ for all i < j and all integers t. (That is, there are "no morphisms from right to left".) In a proper triangulated category $𝒯$ 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:

$𝒯=⟨𝒜,E_1,\dots ,E_m⟩,$

where $𝒜=⟨E_1,\dots ,E_m{⟩}^{\perp }$, and $E_i$ denotes the full triangulated subcategory generated by the object $E_i$.Template:Sfn An exceptional collection is called full if the subcategory $𝒜$ is zero. (Thus a full exceptional collection breaks the whole triangulated category up into finitely many copies of ${\text{D}}^{\text{b}}(k)$.)

In particular, if X is a smooth projective variety such that ${\text{D}}^{\text{b}}(X)$ has a full exceptional collection $E_1,\dots ,E_m$, then the Grothendieck group of algebraic vector bundles on X is the free abelian group on the classes of these objects:

$K_0(X)\cong \mathbb{Z}\{E_1,\dots ,E_m\}.$

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

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

${\text{D}}^{\text{b}}({𝐏}^n)=⟨O,O(1),\dots ,O(n)⟩$,

where O(j) for integers j are the line bundles on projective space.Template:Sfn 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.Template:Sfn

More generally, if X is a smooth projective variety of positive dimension such that the coherent sheaf cohomology groups $H^i(X,O_X)$ are zero for i > 0, then the object $O_X$ in ${\text{D}}^{\text{b}}(X)$ is exceptional, and so it induces a nontrivial semiorthogonal decomposition ${\text{D}}^{\text{b}}(X)=⟨(O_X{)}^{\perp },O_X⟩$. 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 $X={\mathrm{Bl}}_Z(Y)$ of a scheme $Y$ at a codimension $k$ locally complete intersection subscheme $Z$ with exceptional locus $\iota :E\simeq {\mathbb{P}}_Z(N_{Z/Y})\to X$. There is a semiorthogonal decomposition $D^b(X)=⟨{\mathrm{\Phi }}_{1-k}(D^b(Z)),\dots ,{\mathrm{\Phi }}_{-1}(D^b(Z)),{\pi }^{*}(D^b(Y))⟩$ where ${\mathrm{\Phi }}_i:D^b(Z)\to D^b(X)$ is the functor ${\mathrm{\Phi }}_i(-)={\iota }_{*}({𝒪}_E(k))\otimes p^{*}(-))$ with $p:X\to Y$ is the natural map.

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 $K_X$ is basepoint-free, every semiorthogonal decomposition ${\text{D}}^{\text{b}}(X)=⟨𝒜,ℬ⟩$ is trivial in the sense that $𝒜$ or $ℬ$ must be zero.Template:Sfn 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

References

• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation