Derived noncommutative algebraic geometry
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
 formula_7
if we consider the two terms on the right as a complex, then we get the distinguished triangle
 formula_8
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. 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,
formula_22
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
formula_25
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
formula_31
and is a strong exceptional collection if in addition, for any formula_32 and "any" formula_33, we have
formula_31
We can then decompose our triangulated category into the semiothogonal decomposition
formula_35
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 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
formula_46
in formula_47 where formula_48 are arbitrary coherent sheaves on formula_4.
Another way to reformulate this lemma for formula_50 is by looking at the Koszul complex associated toformula_51where formula_52 are hyperplane divisors of formula_53. This gives the exact complexformula_54which gives a way to construct formula_55 using the sheaves formula_56, since they are the sheaves used in all terms in the above exact sequence, except for
formula_57
which gives a derived equivalence of the rest of the terms of the above complex with formula_55. For formula_59 the Koszul complex above is the exact complexformula_60giving the quasi isomorphism of formula_61 with the complexformula_62 If formula_4 is a smooth projective variety with ample (anti-)canonical sheaf and there is an equivalence of derived categories formula_64, then there is an isomorphism of the underlying varieties. The proof starts out by analyzing two induced Serre functors on formula_65 and finding an ismorphism between them. It particular, it shows there is an object formula_66 which acts like the dualizing sheaf on formula_67. 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_68, for any formula_69, giving an isomorphism of canonical rings
 formula_70
If formula_71 can be shown to be (anti-)ample, then the proj of these rings will give an isomorphism formula_72. All of the details are contained in Dolgachev's notes. This theorem fails in the case formula_4 is Calabi-Yau, since formula_74, 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_76 is it's dual, the Fourier–Mukai transform with kernel formula_77, the Poincare bundle, gives an equivalence
formula_78
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. 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_65 if and only if there is a Hodge isometry formula_81, 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. 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
formula_82
Where an autoequivalence formula_83 is given by an automorphism formula_84, then tensored by a line bundle formula_85 and finally composed with a shift. Note that formula_86 acts on formula_87 via the polarization map, formula_88. The bounded derived category formula_1 was used extensively in SGA6 to construct an intersection theory with formula_90 and formula_91. 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
 formula_93
is there an induced map on the chow motives
 formula_94
such that formula_95 is a summand of formula_96? 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. 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 defines the derived category of singularities as their difference using a quotient of categories. For an ELF scheme formula_4 its derived category of singularities is defined as
 formula_100
for a suitable definition of localization of triangulated categories. Although localization of categories is defined for a class of morphisms formula_101 in the category closed under composition, we can construct such a class from a triangulated subcategory. Given a full triangulated subcategory formula_102 the class of morphisms formula_103, formula_104 in formula_13 where formula_104 fits into a distinguished triangleformula_107with formula_108 and formula_109. It can be checked this forms a multiplicative system using the octahedral axiom for distinguished triangles. Given
 formula_110
with distinguished triangles
 formula_111
 formula_112
where formula_113, then there are distinguished triangles
 formula_114
 formula_115 where formula_116 since formula_117 is closed under extensions. This new category has the following properties 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_136 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. With this definition, there are three categories which can be associated to any point formula_137, a formula_138-graded category formula_139, an exact category formula_140, and a triangulated category formula_141, each of which has objects
formula_142 where formula_143 are multiplication by formula_144.
There is also a shift functor formula_145 send formula_146 toformula_147.The difference between these categories are their definition of morphisms. The most general of which is formula_139 whose morphisms are the formula_138-graded complex
formula_150
where the grading is given by formula_151 and differential acting on degree formula_152 homogeneous elements by
formula_153
In formula_140 the morphisms are the degree formula_155 morphisms in formula_139. Finally, formula_141 has the morphisms in formula_140 modulo the null-homotopies. Furthermore, formula_141 can be endowed with a triangulated structure through a graded cone-construction in formula_140. Given formula_161 there is a mapping code formula_162 with maps
formula_163 where formula_164
and
formula_165 where formula_166
Then, a diagram formula_167 in formula_141 is a distinguished triangle if it is isomorphic to a cone from formula_140. Using the construction of formula_141 we can define the category of D-branes of type B on formula_4 with superpotential formula_172 as the product category
 formula_173
This is related to the singularity category as follows: Given a superpotential formula_172 with isolated singularities only at formula_155, denote formula_176. Then, there is an exact equivalence of categories
 formula_177
given by a functor induced from cokernel functor formula_178 sending a pair formula_179. In particular, since formula_4 is regular, Bertini's theorem shows formula_181 is only a finite product of categories. There is a Fourier-Mukai transform formula_182 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_183 from a separated regular Noetherian scheme of finite Krull dimension, there is an associated scheme formula_184 and morphism formula_185 such that formula_186 where formula_187 are the coordinates of the formula_188-factor. Consider the fibers formula_189, formula_190, and the induced morphism formula_191. And the fiber formula_192. Then, there is an injection formula_193 and a projection formula_194 forming an formula_195-bundle. The Fourier-Mukai transform
formula_196
induces an equivalence of categories
formula_197
called Knörrer periodicity. There is another form of this periodicity where formula_187 is replaced by the polynomial formula_199. These periodicity theorems are the main computational techniques because it allows for a reduction in the analysis of the singularity categories. If we take the Landau–Ginzburg model formula_200 where formula_201, then the only fiber singular fiber of formula_172 is the origin. Then, the D-brane category of the Landau–Ginzburg model is equivalent to the singularity category formula_203. Over the algebra formula_204 there are indecomposible objects
 formula_205
whose morphisms can be completely understood. For any pair formula_206 there are morphisms formula_207 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.
Derived category of singularities
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.