Derived noncommutative algebraic geometry

Short description: Mathematics study in geometry

In mathematics, derived noncommutative algebraic geometry,[1] the derived version of noncommutative algebraic geometry, is the geometric study of derived categories and related constructions of triangulated categories using categorical tools. Some basic examples include the bounded derived category of coherent sheaves on a smooth variety, Db(X), called its derived category, or the derived category of perfect complexes on an algebraic variety, denoted Dperf(X). For instance, the derived category of coherent sheaves Db(X) on a smooth projective variety can be used as an invariant of the underlying variety for many cases (if X has an ample (anti-)canonical sheaf). Unfortunately, studying derived categories as geometric objects of themselves does not have a standardized name.

## Contents

* 1 Derived category of projective line
* 2 Semiorthogonal decompositions and exceptional collections
* 3 Beilinson's theorem
* 4 Orlov's reconstruction theorem
* 4.1 Sketch of proof
* 4.2 Failure of reconstruction
* 4.3 Equivalences on K3 surfaces
* 4.4 Autoequivalences
* 5 Relation with motives
* 6 Derived category of singularities
* 6.1 Construction of localization
* 6.2 Properties of singularity category
* 6.3 Landau–Ginzburg models
* 6.3.1 Associated categories
* 6.3.2 D-brane category
* 6.4 Computational tools
* 6.4.1 Knörrer periodicity
* 6.5 Computations
* 7 See also
* 8 References
* 8.1 Research articles
* 9 External links

## Derived category of projective line

The derived category of ℙ1 is one of the motivating examples for derived non-commutative schemes due to its easy categorical structure. Recall that the Euler sequence of ℙ1 is the short exact sequence

0->𝒪(−2)->𝒪(−1)⊕2->𝒪->0

if we consider the two terms on the right as a complex, then we get the distinguished triangle

𝒪(−1)⊕2->ϕ𝒪->Cone⁡(ϕ)->+1.

Since Cone⁡(ϕ)≅𝒪(−2)[+1] we have constructed this sheaf 𝒪(−2) using only categorical tools. We could repeat this again by tensoring the Euler sequence by the flat sheaf 𝒪(−1), and apply the cone construction again. If we take the duals of the sheaves, then we can construct all of the line bundles in Coh⁡(ℙ1) using only its triangulated structure. It turns out the correct way of studying derived categories from its objects and triangulated structure is with exceptional collections.

## Semiorthogonal decompositions and exceptional collections

The technical tools for encoding this construction are semiorthogonal decompositions and exceptional collections.[2] A semiorthogonal decomposition of a triangulated category 𝒯 is a collection of full triangulated subcategories 𝒯1,…,𝒯n such that the following two properties hold

(1) For objects Ti∈Ob⁡(𝒯i) we have Hom⁡(Ti,Tj)=0 for i>j

(2) The subcategories 𝒯i generate 𝒯, meaning every object T∈Ob⁡(𝒯) can be decomposed in to a sequence of Ti∈Ob⁡(𝒯),

0=Tn->Tn−1->⋯->T1->T0=T

such that Cone⁡(Ti->Ti−1)∈Ob⁡(𝒯i). Notice this is analogous to a filtration of an object in an abelian category such that the cokernels live in a specific subcategory.

We can specialize this a little further by considering exceptional collections of objects, which generate their own subcategories. An object E in a triangulated category is called exceptional if the following property holds

Hom⁡(E,E[+ℓ])={kif ℓ=00if ℓ≠0

where k is the underlying field of the vector space of morphisms. A collection of exceptional objects E1,…,Er is an exceptional collection of length r if for any i>j and any ℓ, we have

Hom⁡(Ei,Ej[+ℓ])=0

and is a strong exceptional collection if in addition, for any ℓ≠0 and any i,j, we have

Hom⁡(Ei,Ej[+ℓ])=0

We can then decompose our triangulated category into the semiorthogonal decomposition

𝒯=⟨𝒯′,E1,…,Er⟩

where 𝒯′=⟨E1,…,Er⟩⊥, the subcategory of objects in E∈Ob⁡(𝒯) such that Hom⁡(E,Ei[+ℓ])=0. If in addition 𝒯′=0 then the strong exceptional collection is called full.

## Beilinson's theorem

Beilinson provided the first example of a full strong exceptional collection. In the derived category Db(ℙn) the line bundles 𝒪(−n),𝒪(−n+1),…,𝒪(−1),𝒪 form a full strong exceptional collection.[2] He proves the theorem in two parts. First showing these objects are an exceptional collection and second by showing the diagonal 𝒪Δ of ℙn×ℙn has a resolution whose compositions are tensors of the pullback of the exceptional objects.

Technical Lemma

An exceptional collection of sheaves E1,E2,…,Er on X is full if there exists a resolution

0->p1*E1⊗p2*F1->⋯->p1*En⊗p2*Fn->𝒪Δ->0

in Db(X×X) where Fi are arbitrary coherent sheaves on X.

Another way to reformulate this lemma for

X=ℙn

is by looking at the Koszul complex associated to

> ⨁i=0n𝒪(−Di)→ϕ𝒪

where

are hyperplane divisors of

. This gives the exact complex

> 0->𝒪(−∑i=1nDi)->⋯->⨁i≠j𝒪(−Di−Dj)->⨁i=1n𝒪(−Di)->𝒪->0

which gives a way to construct

𝒪(−n−1)

using the sheaves

𝒪(−n),…,𝒪(−1),𝒪

, since they are the sheaves used in all terms in the above exact sequence, except for

𝒪(−∑i=0nDi)≅𝒪(−n−1)

which gives a derived equivalence of the rest of the terms of the above complex with

𝒪(−n−1)

. For

n=2

the Koszul complex above is the exact complex

> 0->𝒪(−3)->𝒪(−2)⊕𝒪(−2)->𝒪(−1)⊕𝒪(−1)->𝒪->0

giving the quasi isomorphism of

𝒪(−3)

with the complex

> 0->𝒪(−2)⊕𝒪(−2)->𝒪(−1)⊕𝒪(−1)->𝒪->0

## Orlov's reconstruction theorem

If X is a smooth projective variety with ample (anti-)canonical sheaf and there is an equivalence of derived categories F:Db(X)->Db(Y), then there is an isomorphism of the underlying varieties.[3]

### Sketch of proof

The proof starts out by analyzing two induced Serre functors on Db(Y) and finding an isomorphism between them. It particular, it shows there is an object ωY=F(ωX) which acts like the dualizing sheaf on Y. The isomorphism between these two functors gives an isomorphism of the set of underlying points of the derived categories. Then, what needs to be check is an ismorphism F(ωX⊗k)≅ωY⊗k, for any k∈ℕ, giving an isomorphism of canonical rings

A(X)=⨁k=0∞H0(X,ωX⊗k)≅⨁k=0∞H0(Y,ωY⊗k)

If ωY can be shown to be (anti-)ample, then the proj of these rings will give an isomorphism X->Y. All of the details are contained in Dolgachev's notes.

### Failure of reconstruction

This theorem fails in the case X is Calabi–Yau, since ωX≅𝒪X, or is the product of a variety which is Calabi–Yau. Abelian varieties are a class of examples where a reconstruction theorem could never hold. If X is an abelian variety and X^ is its dual, the Fourier–Mukai transform with kernel 𝒫, the Poincare bundle,[4] gives an equivalence

FM𝒫:Db(X)->Db(X^)

of derived categories. Since an abelian variety is generally not isomorphic to its dual, there are derived equivalent derived categories without isomorphic underlying varieties.[5] There is an alternative theory of tensor triangulated geometry where we consider not only a triangulated category, but also a monoidal structure, i.e. a tensor product. This geometry has a full reconstruction theorem using the spectrum of categories.[6]

### Equivalences on K3 surfaces

K3 surfaces are another class of examples where reconstruction fails due to their Calabi–Yau property. There is a criterion for determining whether or not two K3 surfaces are derived equivalent: the derived category of the K3 surface Db(X) is derived equivalent to another K3 Db(Y) if and only if there is a Hodge isometry H2(X,ℤ)->H2(Y,ℤ), that is, an isomorphism of Hodge structure.[3] Moreover, this theorem is reflected in the motivic world as well, where the Chow motives are isomorphic if and only if there is an isometry of Hodge structures.[7]

### Autoequivalences

One nice application of the proof of this theorem is the identification of autoequivalences of the derived category of a smooth projective variety with ample (anti-)canonical sheaf. This is given by

Auteq⁡(Db(X))≅(Pic⁡(X)⋊Aut⁡(X))×ℤ

Where an autoequivalence F is given by an automorphism f:X->X, then tensored by a line bundle ℒ∈Pic⁡(X) and finally composed with a shift. Note that Aut⁡(X) acts on Pic⁡(X) via the polarization map, g↦g*(L)⊗L−1.[8]

## Relation with motives

The bounded derived category Db(X) was used extensively in SGA6 to construct an intersection theory with K(X) and GrγK(X)⊗ℚ. Since these objects are intimately relative with the Chow ring of X, its chow motive, Orlov asked the following question: given a fully faithful functor

F:Db(X)->Db(Y)

is there an induced map on the chow motives

f:M(X)->M(Y)

such that M(X) is a summand of M(Y)?[9] In the case of K3 surfaces, a similar result has been confirmed since derived equivalent K3 surfaces have an isometry of Hodge structures, which gives an isomorphism of motives.

## Derived category of singularities

On a smooth variety there is an equivalence between the derived category Db(X) and the thick[10][11] full triangulated Dperf(X) of perfect complexes. For separated, Noetherian schemes of finite Krull dimension (called the ELF condition)[12] this is not the case, and Orlov defines the derived category of singularities as their difference using a quotient of categories. For an ELF scheme X its derived category of singularities is defined as

Dsg(X):=Db(X)/Dperf(X)[13]

for a suitable definition of localization of triangulated categories.

### Construction of localization

Although localization of categories is defined for a class of morphisms

in the category closed under composition, we can construct such a class from a triangulated subcategory. Given a full triangulated subcategory

𝒩⊂𝒯

the class of morphisms

Σ(𝒩)

where

fits into a distinguished triangle

> X→sY->N->X[+1]

with

X,Y∈𝒯

and

N∈𝒩

. It can be checked this forms a multiplicative system using the octahedral axiom for distinguished triangles. Given

X→sY→s′Z

with distinguished triangles

X→sY->N->X[+1]

Y→s′Z->N′->Y[+1]

where N,N′∈𝒩, then there are distinguished triangles

X->Z->M->X[+1]

N->M->N′->N[+1] where M∈𝒩 since 𝒩 is closed under extensions. This new category has the following properties

* It is canonically triangulated where a triangle in 𝒯/𝒩 is distinguished if it is isomorphic to the image of a triangle in 𝒯
* The category 𝒯/𝒩 has the following universal property: any exact functor F:𝒯->𝒯′ where F(N)≅0 where N∈𝒩, then it factors uniquely through the quotient functor Q:𝒯->𝒯/𝒩, so there exists a morphism F~:𝒯/𝒩->𝒯′ such that F~∘Q≃F.

### Properties of singularity category

* If X is a regular scheme, then every bounded complex of coherent sheaves is perfect. Hence the singularity category is trivial
* Any coherent sheaf ℱ which has support away from Sing⁡(X) is perfect. Hence nontrivial coherent sheaves in Dsg(X) have support on Sing⁡(X).
* In particular, objects in Dsg(X) are isomorphic to ℱ[+k] for some coherent sheaf ℱ.

### Landau–Ginzburg models

Kontsevich proposed a model for Landau–Ginzburg models which was worked out to the following definition:[14] a Landau–Ginzburg model is a smooth variety X together with a morphism W:X->𝔸1 which is flat. There are three associated categories which can be used to analyze the D-branes in a Landau–Ginzburg model using matrix factorizations from commutative algebra.

#### Associated categories

With this definition, there are three categories which can be associated to any point w0∈𝔸1, a ℤ/2-graded category DGw0(W), an exact category Pairw0(W), and a triangulated category DBw0(W), each of which has objects

P‾=(p1:P1->P0,p0:P0->P1) where p0∘p1,p1∘p0 are multiplication by W−w0.

There is also a shift functor

[+1]

send

> P‾[+1]=(−p0:P0->P1,−p1:P1->P0)

> .

The difference between these categories are their definition of morphisms. The most general of which is

DGw0(W)

whose morphisms are the

ℤ/2

-graded complex

Hom⁡(P‾,Q‾)=⨁i,jHom⁡(Pi,Qj)

where the grading is given by (i−j)mod2 and differential acting on degree d homogeneous elements by

Df=q∘f−(−1)df∘p

In Pairw0(W) the morphisms are the degree 0 morphisms in DGw0(W). Finally, DBw0(W) has the morphisms in Pairw0(W) modulo the null-homotopies. Furthermore, DBw0(W) can be endowed with a triangulated structure through a graded cone-construction in Pairw0(W). Given f‾:P‾->Q‾ there is a mapping code C(f) with maps

c1:Q1⊕P0->Q0⊕P1 where c1=[q0f10−p1]

and

c0:Q0⊕P1->Q1⊕P0 where c0=[q1f00−p0]

Then, a diagram P‾->Q‾->R‾->P‾[+1] in DBw0(W) is a distinguished triangle if it is isomorphic to a cone from Pairw0(W).

#### D-brane category

Using the construction of DBw0(W) we can define the category of D-branes of type B on X with superpotential W as the product category

DB(W)=∏w∈𝔸1DBw0(W).

This is related to the singularity category as follows: Given a superpotential W with isolated singularities only at 0, denote X0=W−1(0). Then, there is an exact equivalence of categories

DBw0(W)≅Dsg(X0)

given by a functor induced from cokernel functor Cok sending a pair P‾↦Coker⁡(p1). In particular, since X is regular, Bertini's theorem shows DB(W) is only a finite product of categories.

### Computational tools

#### Knörrer periodicity

There is a Fourier-Mukai transform ΦZ on the derived categories of two related varieties giving an equivalence of their singularity categories. This equivalence is called Knörrer periodicity. This can be constructed as follows: given a flat morphism f:X->𝔸1 from a separated regular Noetherian scheme of finite Krull dimension, there is an associated scheme Y=X×𝔸2 and morphism g:Y->𝔸1 such that g=f+xy where xy are the coordinates of the 𝔸2-factor. Consider the fibers X0=f−1(0), Y0=g−1(0), and the induced morphism x:Y0->𝔸1. And the fiber Z=x−1(0). Then, there is an injection i:Z->Y0 and a projection q:Z->X0 forming an 𝔸1-bundle. The Fourier-Mukai transform

ΦZ(⋅)=𝐑i*q*(⋅)

induces an equivalence of categories

Dsg(X0)->Dsg(Y0)

called Knörrer periodicity. There is another form of this periodicity where xy is replaced by the polynomial x2+y2.[15][16] These periodicity theorems are the main computational techniques because it allows for a reduction in the analysis of the singularity categories.

### Computations

If we take the Landau–Ginzburg model (ℂ2k+1,W) where W=z0n+z12+⋯+z2k2, then the only fiber singular fiber of W is the origin. Then, the D-brane category of the Landau–Ginzburg model is equivalent to the singularity category Dsing(Spec⁡(ℂ[z]/(zn))). Over the algebra A=ℂ[z]/(zn) there are indecomposable objects

Vi=Coker⁡(A→ziA)=A/zi

whose morphisms can be completely understood. For any pair i,j there are morphisms αji:Vi->Vj where

* for i≥j these are the natural projections
* for i<j these are multiplication by zj−i

where every other morphism is a composition and linear combination of these morphisms. There are many other cases which can be explicitly computed, using the table of singularities found in Knörrer's original paper.[16]

## See also

* Semiorthogonal decomposition
* Fourier–Mukai transform
* Bridgeland stability condition
* Homological mirror symmetry

## References

1. ↑ Shklyarov, D. (2013). "Hirzebruch-Riemann-Roch-type formula for DG algebras". Proceedings of the London Mathematical Society 106: 1–32. doi:10.1112/plms/pds034. The reference notes that the name "derived noncommutative algebraic geometry" may not be standard. Some authors (e.g., Orlov, Dmitri (October 2018). "Derived noncommutative schemes, geometric realizations, and finite dimensional algebras". Russian Mathematical Surveys 73 (5): 865–918. doi:10.1070/RM9844. ISSN 0036-0279. Bibcode: 2018RuMaS..73..865O. ) describe this field as the study of derived noncommutative schemes.
2. ↑ 2.0 2.1 Liu, Yijia. "Semi-orthogonal Decompositions of Derived Categories". Superschool on Derived Categories. pp. 35, 37, 38, 41.
3. ↑ 3.0 3.1 Dolgachev, Igor. Derived categories. pp. 105–112. http://www.math.lsa.umich.edu/~idolga/derived9.pdf.
4. ↑ The poincare bundle 𝒫 on X×X^ is a line bundle which is trivial on {0}×X^ and X×{0} and has the property 𝒫|X×{x} is the line bundle represented by the point x∈X^.
5. ↑ Mukai, Shigeru (1981). "Duality between D(X) and D(X^) with its application to Picard sheaves". Nagoya Math. J. 81: 153–175. doi:10.1017/S002776300001922X. https://projecteuclid.org/euclid.nmj/1118786312.
6. ↑ Balmer, Paul (2010). "Tensor triangulated geometry". Proceedings of the International Congress of Mathematicians. https://www.math.ucla.edu/~balmer/Pubfile/TTG.pdf.
7. ↑ Huybrechts, Daniel (2018). "Motives of isogenous K3 surfaces". arXiv:1705.04063 [math.AG].
8. ↑ Brion, Michel. "Notes on Automorphism Groups of Projective Varieties". p. 8. https://www-fourier.ujf-grenoble.fr/~mbrion/autos.pdf.
9. ↑ Orlov, Dmitri (2011). "Derived categories of coherent sheaves and motives". Russian Mathematical Surveys 60 (6): 1242–1244. doi:10.1070/RM2005v060n06ABEH004292.
10. ↑ Meaning it is closed under extensions. Given any two objects ℰ′,ℰ″ in the subcategory, any object ℰ fitting into an exact sequence 0->ℰ′->ℰ->ℰ″->0 is also in the subcategory. In the triangulated case, this translates to the same conditions, but instead of an exact sequence, it is a distinguished triangle ℰ′->ℰ->ℰ″->ℰ[+1]
11. ↑ Thomason, R.W.; Trobaugh, Thomas. "Higher Algebraic K-Theory of Schemes and of Derived Categories". https://www.gwern.net/docs/math/1990-thomason.pdf.
12. ↑ Which he uses because of its nice properties: in particular every bounded complex of coherent sheaves C∙ has a resolution from a bounded above complex P∙→≃C∙ such that P∙ is a complex of locally free sheaves of finite type.
13. ↑ Orlov, Dmitri (2003). "Triangulated Categories of Singularities and D-Branes in Landau–Ginzburg Models". arXiv:math/0302304.
14. ↑ Kapustin, Anton; Li, Yi (2003-12-03). "D-Branes in Landau–Ginzburg Models and Algebraic Geometry". Journal of High Energy Physics 2003 (12): 005. doi:10.1088/1126-6708/2003/12/005. ISSN 1029-8479. Bibcode: 2003JHEP...12..005K.
15. ↑ Brown, Michael K.; Dyckerhoff, Tobias (2019-09-15). "Topological K-theory of Equivariant Singularity Categories". p. 11\. arXiv:1611.01931 [math.AG].
16. ↑ 16.0 16.1 Knörrer, Horst. [159,"panX":0.538,"panY":0.719,"view":"info","zoom":0.583} "Cohen-Macaulay modules on hypersurface singularities I"]. https://gdz.sub.uni-goettingen.de/id/PPN356556735_0088?tify={"pages":[159],"panX":0.538,"panY":0.719,"view":"info","zoom":0.583}.

### Research articles

* A noncommutative version of Beilinson's theorem
* Derived Categories of Toric Varieties
* Derived Categories of Toric Varieties II

## External links

* Derived Categories notes - http://www.math.lsa.umich.edu/~idolga/derived9.pdf

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