nLab guide to noncommutative algebraic geometry literature

This entry provides commented references on the topic of derived noncommutative algebraic geometry.

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• Dmitri Orlov, Equivalences of derived categories and K3 surfaces, 1996, arXiv.

The original representability theorem: every fully faithful triangulated functor between the derived categories of perfect complexes on two smooth projective varieties is representable by some integral kernel (i.e. is a Fourier-Mukai functor).

• Alexei Bondal, Dmitri Orlov, Reconstruction of a variety from the derived category and groups of auto-equivalences, 1997, arXiv.

If a smooth projective variety has the property that its canonical sheaf is ample? or anti-ample, then one may reconstruct it up to isomorphism from its bounded derived category of coherent sheaves.

• Alexei Bondal, Dmitri Orlov, Derived categories of coherent sheaves on abelian varieties and equivalences between them, arXiv.

A criterion for the equivalence of the derived categories of coherent sheaves on two abelian varieties.

• Alexei Bondal, Michel van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, 2002, arXiv.

The bounded derived category of coherent sheaves on a smooth projective variety admits a strong generator.

• Bertrand Toen, Homotopy theory of dg-categories and derived Morita theory, 2004, arXiv

The homotopy category of the model category of dg-categories, where weak equivalences are quasi-equivalences, is studied. It is shown to admit internal homs and the morphism sets are in bijection with right quasi-representable bimodules. Among other applications is a dg-version of Orlov’s representability theorem: it is shown that morphisms between the dg-categories of perfect complexes on smooth proper schemes are in bijection with perfect complexes on the product. An analogous statement for quasi-coherent complexes? is also shown.

• Dmitri Orlov, Derived categories of coherent sheaves and motives, 2005, arXiv.

The Chow motive of a smooth projective variety may be reconstructed up to Tate twists from the bounded derived category of coherent sheaves.

• Alexander Kuznetsov, Base change for semiorthogonal decompositions, 2007, arXiv.

• David Ben-Zvi, John Francis, David Nadler, Integral transforms and Drinfeld centers in derived algebraic geometry, 2008, (arXiv:0805.0157).

An (infinity,1)-categorical version of Orlov’s representability theorem, which is further generalized to (smooth proper) derived stacks.

• Alexander Kuznetsov, Hochschild homology and semiorthogonal decompositions, 2009, arXiv.

• Valery Lunts, Dmitri Orlov, Uniqueness of enhancement for triangulated categories, 2009, arXiv.

Uniqueness results are established for dg-enhancements of triangulated categories. In particular the results are applied to the derived categories of perfect complexes, coherent sheaves, quasi-coherent sheaves, on schemes. This is then used to deduce from (Toen 2004) representability theorems on the level of triangulated categories, in the non-smooth case.

• Alexander Kuznetsov, Valery Lunts, Categorical resolution of irrational singularities, 2012, arXiv.

Describes how to glue two dg-categories in such a way that the homotopy category of the result has a semiorthogonal decomposition into two components, which are the respective homotopy categories of the original dg-categories.

• Sergei Gorchinskiy?, Dmitri Orlov, Geometric phantom categories, 2012, arXiv.

If an admissible subcategory of the bounded derived category of coherent sheaves on a smooth projective variety is a phantom?, has vanishing Hochschild homology and Grothendieck group, then its noncommutative motive vanishes. In particular its higher K-theory also vanishes.

• Beatriz Rodriguez Gonzalez?, A derivability criterion based on the existence of adjunctions, 2012, arXiv:1202.3359.

It is proved that the internal hom of Ho(DGCat) constructed in (Toën 2004) is in fact the right derived functor of the internal hom of DGCat.

• David Ben-Zvi, David Nadler, Anatoly Preygel, Integral transforms for coherent sheaves (arXiv:1312.7164)

The (infinity,1)-categorical representability theorem of (Ben-Zvi-Francis-Nadler 2008) is extended to the non-smooth case.