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,
Unfortunately, studying derived categories as geometric objects of themselves does not have a standardized name.
is one of the motivating examples for derived non-commutative schemes due to its easy categorical structure.
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 could repeat this again by tensoring the Euler sequence by the flat sheaf
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.
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.
in a triangulated category is called exceptional if the following property holds where
Beilinson provided the first example of a full strong exceptional collection.
has a resolution whose compositions are tensors of the pullback of the exceptional objects.
which gives a derived equivalence of the rest of the terms of the above complex with
is a smooth projective variety with ample (anti-)canonical sheaf and there is an equivalence of derived categories
[3] The proof starts out by analyzing two induced Serre functors on
can be shown to be (anti-)ample, then the proj of these rings will give an isomorphism
Abelian varieties are a class of examples where a reconstruction theorem could never hold.
, the Poincare bundle,[4] gives an equivalence of derived categories.
This geometry has a full reconstruction theorem using the spectrum of categories.
[6] K3 surfaces are another class of examples where reconstruction fails due to their Calabi-Yau property.
[7] 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.
[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.
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.
in the category closed under composition, we can construct such a class from a triangulated subcategory.
It can be checked this forms a multiplicative system using the octahedral axiom for distinguished triangles.
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.
The Fourier-Mukai transform induces an equivalence of categories called Knörrer periodicity.
[15][16] These periodicity theorems are the main computational techniques because it allows for a reduction in the analysis of the singularity categories.
There are many other cases which can be explicitly computed, using the table of singularities found in Knörrer's original paper.