semiorthogonal decomposition#Admissible subcategory

Alexei Bondal and Mikhail Kapranov (1989) defined a semiorthogonal decomposition of a triangulated category $\backslash mathcal\{T\}$ to be a sequence $\backslash mathcal\{A\}\_1,\backslash ldots,\backslash mathcal\{A\}\_n$ of strictly full triangulated subcategories such that:{{sfn|Huybrechts|2006|loc=Definition 1.59}}

• for all
• $\backslash mathcal\{T\}$ is generated by $\backslash mathcal\{A\}\_1,\backslash ldots,\backslash mathcal\{A\}\_n$. That is, the smallest strictly full triangulated subcategory of $\backslash mathcal\{T\}$ containing $\backslash mathcal\{A\}\_1,\backslash ldots,\backslash mathcal\{A\}\_n$ is equal to $\backslash mathcal\{T\}$.

The notation $\backslash mathcal\{T\}=\backslash langle\backslash mathcal\{A\}\_1,\backslash ldots,\backslash mathcal\{A\}\_n\backslash rangle$ is used for a semiorthogonal decomposition.

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

:$0=T\_n\backslash to\; T\_\{n-1\}\backslash to\backslash cdots\backslash to\; T\_0=T$

of morphisms in $\backslash mathcal\{T\}$ such that the cone of $T\_i\backslash to\; T\_\{i-1\}$ is in $\backslash mathcal\{A\}\_i$, for each i. Moreover, this sequence is unique up to a unique isomorphism.{{sfn|Bondal|Kapranov|1990|loc=Proposition 1.5}}

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

Using that observation, a semiorthogonal decomposition $\backslash mathcal\{T\}=\backslash langle\backslash mathcal\{A\}\_1,\backslash ldots,\backslash mathcal\{A\}\_n\backslash rangle$ implies a direct sum splitting of Grothendieck groups:

:$K\_0(\backslash mathcal\{T\})\backslash cong\; K\_0(\backslash mathcal\{A\}\_1)\backslash oplus\backslash cdots\backslash oplus\; K\_0(\backslash mathcal\{A\_n\}).$

For example, when $\backslash mathcal\{T\}=\backslash text\{D\}^\{\backslash text\{b\}\}(X)$ is the bounded derived category of coherent sheaves on a smooth projective variety X, $K\_0(\backslash mathcal\{T\})$ can be identified with the Grothendieck group $K\_0(X)$ of algebraic vector bundles on X. In this geometric situation, using that $\backslash text\{D\}^\{\backslash text\{b\}\}(X)$ comes from a dg-category, a semiorthogonal decomposition actually gives a splitting of all the algebraic K-groups of X:

:$K\_i(X)\backslash cong\; K\_i(\backslash mathcal\{A\}\_1)\backslash oplus\backslash cdots\backslash oplus\; K\_i(\backslash mathcal\{A\_n\})$

for all i.{{sfn|Orlov|2016|loc=Section 1.2}}

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

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

where

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

is the right orthogonal of $\backslash mathcal\{B\}$ in $\backslash mathcal\{T\}$.{{sfn|Bondal|Kapranov|1990|loc=Proposition 1.5}} Conversely, every semiorthogonal decomposition $\backslash mathcal\{T\}=\backslash langle\; \backslash mathcal\{A\},\backslash mathcal\{B\}\backslash rangle$ arises in this way, in the sense that $\backslash mathcal\{B\}$ is right admissible and $\backslash mathcal\{A\}=\backslash mathcal\{B\}^\{\backslash perp\}$. Likewise, for any semiorthogonal decomposition $\backslash mathcal\{T\}=\backslash langle\; \backslash mathcal\{A\},\backslash mathcal\{B\}\backslash rangle$, the subcategory $\backslash mathcal\{A\}$ is left admissible, and $\backslash mathcal\{B\}=\{\}^\{\backslash perp\}\backslash mathcal\{A\}$, where

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

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

If $\backslash mathcal\{T\}$ is the bounded derived category of a smooth projective variety over a field k, then every left or right admissible subcategory of $\backslash mathcal\{T\}$ is in fact admissible.{{sfn|Kuznetsov|2007|loc=Lemmas 2.10, 2.11, and 2.12}} By results of Bondal and Michel Van den Bergh, this holds more generally for $\backslash mathcal\{T\}$ any regular proper triangulated category that is idempotent-complete.{{sfn|Orlov|2016|loc=Theorem 3.16}}

Moreover, for a regular proper idempotent-complete triangulated category $\backslash mathcal\{T\}$, a full triangulated subcategory is admissible if and only if it is regular and idempotent-complete. These properties are intrinsic to the subcategory.{{sfn|Orlov|2016|loc=Propositions 3.17 and 3.20}} For example, for X a smooth projective variety and Y a subvariety not equal to X, the subcategory of $\backslash text\{D\}^\{\backslash text\{b\}\}(X)$ of objects supported on Y is not admissible.

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

The triangulated subcategory generated by an exceptional object E is equivalent to the derived category $\backslash text\{D\}^\{\backslash text\{b\}\}(k)$ 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 $E\_1,\backslash ldots,E\_m$ such that $\backslash operatorname\{Hom\}(E\_j,E\_i[t])=0$ for all i < j and all integers t. (That is, there are "no morphisms from right to left".) In a proper triangulated category $\backslash mathcal\{T\}$ 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:

:$\backslash mathcal\{T\}=\backslash langle\backslash mathcal\{A\},E\_1,\backslash ldots,E\_m\backslash rangle,$

where $\backslash mathcal\{A\}=\backslash langle\; E\_1,\backslash ldots,E\_m\backslash rangle^\{\backslash perp\}$, and $E\_i$ denotes the full triangulated subcategory generated by the object $E\_i$.{{sfn|Huybrechts|2006|loc=Lemma 1.58}} An exceptional collection is called full if the subcategory $\backslash mathcal\{A\}$ is zero. (Thus a full exceptional collection breaks the whole triangulated category up into finitely many copies of $\backslash text\{D\}^\{\backslash text\{b\}\}(k)$.)

In particular, if X is a smooth projective variety such that $\backslash text\{D\}^\{\backslash text\{b\}\}(X)$ has a full exceptional collection $E\_1,\backslash ldots,E\_m$, then the Grothendieck group of algebraic vector bundles on X is the free abelian group on the classes of these objects:

:$K\_0(X)\backslash cong\; \backslash Z\backslash \{E\_1,\backslash ldots,E\_m\backslash \}.$

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 $p\backslash neq\; q$; moreover, the cycle class map $CH^*(X)\backslash otimes\backslash Q\backslash to\; H^*(X,\backslash Q)$ must be an isomorphism.{{sfn|Marcolli|Tabuada|2015|loc=Proposition 1.9}}

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

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

where O(j) for integers j are the line bundles on projective space.{{sfn|Huybrechts|2006|loc=Corollary 8.29}} 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.{{sfn|Kuznetsov|2014|loc=Section 2.2}}

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 $\backslash text\{D\}^\{\backslash text\{b\}\}(X)$ is exceptional, and so it induces a nontrivial semiorthogonal decomposition $\backslash text\{D\}^\{\backslash text\{b\}\}(X)=\backslash langle\; (O\_X)^\{\backslash perp\},O\_X\backslash rangle$. 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 $X\; =\; \backslash operatorname\{Bl\}\_Z(Y)$ of a scheme $Y$ at a codimension $k$ locally complete intersection subscheme $Z$ with exceptional locus $\backslash iota:\; E\; \backslash simeq\; \backslash mathbb\{P\}\_Z(N\_\{Z/Y\})\backslash to\; X$. There is a semiorthogonal decomposition $D^b(X)\; =\; \backslash langle\; \backslash Phi\_\{1-k\}(D^b(Z)),\; \backslash ldots,\; \backslash Phi\_\{-1\}(D^b(Z)),\; \backslash pi^*(D^b(Y))\backslash rangle$ where $\backslash Phi\_i:D^b(Z)\; \backslash to\; D^b(X)$ is the functor $\backslash Phi\_i(-)\; =\; \backslash iota\_*(\backslash mathcal\{O\}\_E(k))\backslash otimes\; p^*(-))$ with $p\; :\; X\; \backslash to\; Y$ is the natural map.{{Citation |last=Orlov |first=D O |date=1993-02-28 |title=Projective Bundles, Monoidal Transformations, and Derived Categories of Coherent Sheaves |url=http://dx.doi.org/10.1070/im1993v041n01abeh002182 |journal=Russian Academy of Sciences. Izvestiya Mathematics |volume=41 |issue=1 |pages=133–141 |doi=10.1070/im1993v041n01abeh002182 |bibcode=1993IzMat..41..133O |issn=1064-5632|url-access=subscription }}

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 $\backslash text\{D\}^\{\backslash text\{b\}\}(X)=\backslash langle\backslash mathcal\{A\},\backslash mathcal\{B\}\backslash rangle$ is trivial in the sense that $\backslash mathcal\{A\}$ or $\backslash mathcal\{B\}$ must be zero.{{sfn|Kuznetsov|2014|loc=Section 2.5}} For example, this applies to every variety which is Calabi–Yau in the sense that its canonical bundle is trivial.

• Derived noncommutative algebraic geometry

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Category:Algebraic geometry