Derived noncommutative algebraic 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, , called its derived category, or the derived category of perfect complexes on an algebraic variety, denoted . For instance, the derived category of coherent sheaves on a smooth projective variety can be used as an invariant of the underlying variety for many cases (if has an ample (anti-)canonical sheaf). Unfortunately, studying derived categories as geometric objects of themselves does not have a standardized name.

Derived category of projective line

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

 

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

 

Since   we have constructed this sheaf   using only categorical tools. We could repeat this again by tensoring the Euler sequence by the flat sheaf  , and apply the cone construction again. If we take the duals of the sheaves, then we can construct all of the line bundles in   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

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

(1) For objects   we have   for  

(2) The subcategories   generate  , meaning every object   can be decomposed in to a sequence of  ,

 

such that  . 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   in a triangulated category is called exceptional if the following property holds

 

where   is the underlying field of the vector space of morphisms. A collection of exceptional objects   is an exceptional collection of length   if for any   and any  , we have

 

and is a strong exceptional collection if in addition, for any   and any  , we have

 

We can then decompose our triangulated category into the semiorthogonal decomposition

 

where  , the subcategory of objects in   such that  . If in addition   then the strong exceptional collection is called full.

Beilinson's theorem

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Beilinson provided the first example of a full strong exceptional collection. In the derived category   the line bundles   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   of   has a resolution whose compositions are tensors of the pullback of the exceptional objects.

Technical Lemma

An exceptional collection of sheaves   on   is full if there exists a resolution

 

in   where   are arbitrary coherent sheaves on  .

Another way to reformulate this lemma for   is by looking at the Koszul complex associated to

 

where   are hyperplane divisors of  . This gives the exact complex

 

which gives a way to construct   using the sheaves  , since they are the sheaves used in all terms in the above exact sequence, except for

 

which gives a derived equivalence of the rest of the terms of the above complex with  . For   the Koszul complex above is the exact complex

 

giving the quasi isomorphism of   with the complex

 

Orlov's reconstruction theorem

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

Sketch of proof

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The proof starts out by analyzing two induced Serre functors on   and finding an isomorphism between them. It particular, it shows there is an object   which acts like the dualizing sheaf on  . 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  , for any  , giving an isomorphism of canonical rings

 

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

Failure of reconstruction

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This theorem fails in the case   is Calabi-Yau, since  , 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   is an abelian variety and   is its dual, the Fourier–Mukai transform with kernel  , the Poincare bundle,[4] gives an equivalence

 

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

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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   is derived equivalent to another K3   if and only if there is a Hodge isometry  , 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

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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

 

Where an autoequivalence   is given by an automorphism  , then tensored by a line bundle   and finally composed with a shift. Note that   acts on   via the polarization map,  .[8]

Relation with motives

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

 

is there an induced map on the chow motives

 

such that   is a summand of  ?[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

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On a smooth variety there is an equivalence between the derived category   and the thick[10][11] full triangulated   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   its derived category of singularities is defined as

 [13]

for a suitable definition of localization of triangulated categories.

Construction of localization

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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   the class of morphisms  ,   in   where   fits into a distinguished triangle

 

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

 

with distinguished triangles

 
 

where  , then there are distinguished triangles

 
  where   since   is closed under extensions. This new category has the following properties
  • It is canonically triangulated where a triangle in   is distinguished if it is isomorphic to the image of a triangle in  
  • The category   has the following universal property: any exact functor   where   where  , then it factors uniquely through the quotient functor  , so there exists a morphism   such that  .

Properties of singularity category

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  • If   is a regular scheme, then every bounded complex of coherent sheaves is perfect. Hence the singularity category is trivial
  • Any coherent sheaf   which has support away from   is perfect. Hence nontrivial coherent sheaves in   have support on  .
  • In particular, objects in   are isomorphic to   for some coherent sheaf  .

Landau–Ginzburg models

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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   together with a morphism   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

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With this definition, there are three categories which can be associated to any point  , a  -graded category  , an exact category  , and a triangulated category  , each of which has objects

  where   are multiplication by  .

There is also a shift functor   send   to

 .

The difference between these categories are their definition of morphisms. The most general of which is   whose morphisms are the  -graded complex

 

where the grading is given by   and differential acting on degree   homogeneous elements by

 

In   the morphisms are the degree   morphisms in  . Finally,   has the morphisms in   modulo the null-homotopies. Furthermore,   can be endowed with a triangulated structure through a graded cone-construction in  . Given   there is a mapping code   with maps

  where  

and

  where  

Then, a diagram   in   is a distinguished triangle if it is isomorphic to a cone from  .

D-brane category

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Using the construction of   we can define the category of D-branes of type B on   with superpotential   as the product category

 

This is related to the singularity category as follows: Given a superpotential   with isolated singularities only at  , denote  . Then, there is an exact equivalence of categories

 

given by a functor induced from cokernel functor   sending a pair  . In particular, since   is regular, Bertini's theorem shows   is only a finite product of categories.

Computational tools

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Knörrer periodicity

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There is a Fourier-Mukai transform   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   from a separated regular Noetherian scheme of finite Krull dimension, there is an associated scheme   and morphism   such that   where   are the coordinates of the  -factor. Consider the fibers  ,  , and the induced morphism  . And the fiber  . Then, there is an injection   and a projection   forming an  -bundle. The Fourier-Mukai transform

 

induces an equivalence of categories

 

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

Computations

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If we take the Landau–Ginzburg model   where  , then the only fiber singular fiber of   is the origin. Then, the D-brane category of the Landau–Ginzburg model is equivalent to the singularity category  . Over the algebra   there are indecomposable objects

 

whose morphisms can be completely understood. For any pair   there are morphisms   where

  • for   these are the natural projections
  • for   these are multiplication by  

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

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References

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  1. ^ Shklyarov, D. (2013). "Hirzebruch-Riemann-Roch-type formula for DG algebras". Proceedings of the London Mathematical Society. 106: 1–32. arXiv:0710.1937. doi:10.1112/plms/pds034. S2CID 5541558. 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. arXiv:1808.02287. Bibcode:2018RuMaS..73..865O. doi:10.1070/RM9844. ISSN 0036-0279. S2CID 119173796.) describe this field as the study of derived noncommutative schemes.
  2. ^ a b Liu, Yijia. "Semi-orthogonal Decompositions of Derived Categories". Superschool on Derived Categories. pp. 35, 37, 38, 41.
  3. ^ a b Dolgachev, Igor. Derived categories (PDF). pp. 105–112.
  4. ^ The poincare bundle   on   is a line bundle which is trivial on   and   and has the property   is the line bundle represented by the point  .
  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 – via Project Euclid.
  6. ^ Balmer, Paul (2010). "Tensor triangulated geometry" (PDF). Proceedings of the International Congress of Mathematicians.
  7. ^ Huybrechts, Daniel (2018). "Motives of isogenous K3 surfaces". arXiv:1705.04063 [math.AG].
  8. ^ Brion, Michel. "Notes on Automorphism Groups of Projective Varieties" (PDF). p. 8. Archived (PDF) from the original on 13 February 2020.
  9. ^ Orlov, Dmitri (2011). "Derived categories of coherent sheaves and motives". Russian Mathematical Surveys. 60 (6): 1242–1244. arXiv:math/0512620. doi:10.1070/RM2005v060n06ABEH004292. S2CID 11484447.
  10. ^ Meaning it is closed under extensions. Given any two objects   in the subcategory, any object   fitting into an exact sequence   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  
  11. ^ Thomason, R.W.; Trobaugh, Thomas. "Higher Algebraic K-Theory of Schemes and of Derived Categories" (PDF). Archived (PDF) from the original on 30 January 2019.
  12. ^ Which he uses because of its nice properties: in particular every bounded complex of coherent sheaves   has a resolution from a bounded above complex   such that   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. arXiv:hep-th/0210296. Bibcode:2003JHEP...12..005K. doi:10.1088/1126-6708/2003/12/005. ISSN 1029-8479. S2CID 11337046.
  15. ^ Brown, Michael K.; Dyckerhoff, Tobias (2019-09-15). "Topological K-theory of Equivariant Singularity Categories". p. 11. arXiv:1611.01931 [math.AG].
  16. ^ a b Knörrer, Horst. "Cohen-Macaulay modules on hypersurface singularities I".

Research articles

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