Derived noncommutative algebraic geometry

In mathematics, derived noncommutative algebraic geometry, 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, formula_1, called its derived category, or the derived category of perfect complexes on an algebraic variety, denoted formula_2. For instance, the derived category of coherent sheaves formula_1 on a smooth projective variety can be used as an invariant of the underlying variety for many cases (if formula_4 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.
The derived category of formula_5 is one of the motivating examples for derived non-commutative schemes due to its easy categorical structure. Recall that the Euler sequence of formula_5 is the short exact sequence
if we consider the two terms on the right as a complex, then we get the distinguished triangle
Since formula_9 we have constructed this sheaf formula_10 using only categorical tools. We could repeat this again by tensoring the Euler sequence by the flat sheaf formula_11, and apply the cone construction again.If we take the duals of the sheaves, then we can construct all of the line bundles in formula_12 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. A semiorthogonal decomposition of a triangulated category formula_13 is a collection of full triangulated subcategories formula_14 such that the following two properties hold
(1) For objects formula_15 we have formula_16 for formula_17
(2) The subcategories formula_18 generate formula_13, meaning every object formula_20 can be decomposed in to a sequence of formula_21,
such that formula_23. 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 formula_24 in a triangulated category is called exceptional if the following property holds
where formula_26 is the underlying field of the vector space of morphisms. A collection of exceptional objects formula_27 is an exceptional collection of length formula_28 if for any formula_17 and any formula_30, we have
and is a strong exceptional collection if in addition, for any formula_32 and "any" formula_33, we have
We can then decompose our triangulated category into the semiothogonal decomposition
where formula_36, the subcategory of objects in formula_37 such that formula_38. If in addition formula_39 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 formula_40 the line bundles formula_41 form a full strong exceptional collection. He proves the theorem in two parts. First showing these objects are an exceptional collection and second by showing the diagonal formula_42 of formula_43 has a resolution whose compositions are tensors of the pullback of the exceptional objects.
Technical Lemma
An exceptional collection of sheaves formula_44 on formula_4 is full if there exists a resolution
in formula_47 where formula_48 are arbitrary coherent sheaves on formula_4.
Orlov's reconstruction theorem.
If formula_4 is a smooth projective variety with ample (anti-)canonical sheaf and there is an equivalence of derived categories formula_51, then there is an isomorphism of the underlying varieties.
Sketch of proof.
The proof starts out by analyzing two induced Serre functors on formula_52 and finding an ismorphism between them. It particular, it shows there is an object formula_53 which acts like the dualizing sheaf on formula_54. 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 formula_55, for any formula_56, giving an isomorphism of canonical rings
If formula_58 can be shown to be (anti-)ample, then the proj of these rings will give an isomorphism formula_59. All of the details are contained in Dolgachev's notes.
Failure of reconstruction.
This theorem fails in the case formula_4 is Calabi-Yau, since formula_61, 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 formula_4 is an abelian variety and formula_63 is it's dual, the Fourier–Mukai transform with kernel formula_64, the Poincare bundle, 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. 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.
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 formula_1 is derived equivalent to another K3 formula_52 if and only if there is a Hodge isometry formula_68, that is, an isomorphism of Hodge structure. 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.
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
Where an autoequivalence formula_70 is given by an automorphism formula_71, then tensored by a line bundle formula_72 and finally composed with a shift. Note that formula_73 acts on formula_74 via the polarization map, formula_75.
Relation with motives.
The bounded derived category formula_1 was used extensively in SGA6 to construct an intersection theory with formula_77 and formula_78. Since these objects are intimately relative with the Chow ring of formula_4, its chow motive, Orlov asked the following question: given a fully-faithful functor
is there an induced map on the chow motives
such that formula_82 is a summand of formula_83? 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 formula_1 and the thick full triangulated formula_2 of perfect complexes. For separated, Noetherian schemes of finite Krull dimension (called the ELF condition) this is not the case, and Orlov takes advantage of this fact through the definition of the derived category of singularities. For a ELF scheme formula_4 its derived category of singularities is defined as
for a suitable definition of localization of triangulated categories.
Construction of localization.
Although localization of categories is defined for a class of morphisms formula_88 in the category closed under composition, we can construct such a class from a triangulated subcategory. Given a full triangulated subcategory formula_89 the class of morphisms formula_90, formula_91 in formula_13 where formula_91 fits into a distinguished triangleformula_94with formula_95 and formula_96. It can be checked this forms a multiplicative system using the octahedral axiom for distinguished triangles. Given
with distinguished triangles
where formula_100, then there are distinguished triangles
Landau–Ginzburg models.
Kontsevich proposed a model for Landau–Ginzburg models which was worked out to the following definition: a Landau–Ginzburg model is a smooth variety formula_4 together with a morphism formula_123 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 formula_124, a formula_125-graded category formula_126, an exact category formula_127, and a triangulated category formula_128, each of which has objects
There is also a shift functor formula_132 send formula_133 toformula_134.The difference between these categories are their definition of morphisms. The most general of which is formula_126 whose morphisms are the formula_125-graded complex
where the grading is given by formula_138 and differential acting on degree formula_139 homogeneous elements by
In formula_127 the morphisms are the degree formula_142 morphisms in formula_126. Finally, formula_128 has the morphisms in formula_127 modulo the null-homotopies. Furthermore, formula_128 can be endowed with a triangulated structure through a graded cone-construction in formula_127. Given formula_148 there is a mapping code formula_149 with maps
and
Then, a diagram formula_154 in formula_128 is a distinguished triangle if it is isomorphic to a cone from formula_127.
D-brane category.
Using the construction of formula_128 we can define the category of D-branes of type B on formula_4 with superpotential formula_159 as the product category
This is related to the singularity category as follows: Given a superpotential formula_159 with isolated singularities only at formula_142, denote formula_163. Then, there is an exact equivalence of categories
given by a functor induced from cokernel functor formula_165 sending a pair formula_166. In particular, since formula_4 is regular, Bertini's theorem shows formula_168 is only a finite product of categories.
Computational tools.
Knörrer periodicity.
There is a Fourier-Mukai transform formula_169 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 formula_170 from a separated regular Noetherian scheme of finite Krull dimension, there is an associated scheme formula_171 and morphism formula_172 such that formula_173 where formula_174 are the coordinates of the formula_175-factor. Consider the fibers formula_176, formula_177, and the induced morphism formula_178. And the fiber formula_179. Then, there is an injection formula_180 and a projection formula_181 forming an formula_182-bundle. The Fourier-Mukai transform
induces an equivalence of categories
called Knörrer periodicity. There is another form of this periodicity where formula_174 is replaced by the polynomial formula_186. 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 formula_187 where formula_188, then the only fiber singular fiber of formula_159 is the origin. Then, the D-brane category of the Landau–Ginzburg model is equivalent to the singularity category formula_190. Over the algebra formula_191 there are indecomposible objects
whose morphisms can be completely understood. For any pair formula_193 there are morphisms formula_194 where
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.