# Deformation theory of objects in homotopy and derived categories III: abelian categories

@article{Efimov2011DeformationTO, title={Deformation theory of objects in homotopy and derived categories III: abelian categories}, author={Alexander I. Efimov and Valery A. Lunts and Dmitri O. Orlov}, journal={Advances in Mathematics}, year={2011}, volume={226}, pages={3857-3911} }

Abstract This is the third paper in a series. In Part I we developed a deformation theory of objects in homotopy and derived categories of DG categories. Here we show how this theory can be used to study deformations of objects in homotopy and derived categories of abelian categories. Then we consider examples from (noncommutative) algebraic geometry. In particular, we study noncommutative Grassmanians that are true noncommutative moduli spaces of structure sheaves of projective subspaces in… Expand

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#### References

SHOWING 1-10 OF 46 REFERENCES

Deformation theory of objects in homotopy and derived categories I: General theory

- Mathematics
- 2009

Abstract This is the first paper in a series. We develop a general deformation theory of objects in homotopy and derived categories of DG categories. Namely, for a DG module E over a DG category we… Expand

Deformation theory of objects in homotopy and derived categories II: pro-representability of the deformation functor

- Mathematics
- 2007

Abstract This is the second paper in a series. In part I we developed deformation theory of objects in homotopy and derived categories of DG categories. Here we extend these (derived) deformation… Expand

Obstruction Theory for Objects in Abelian and Derived Categories

- Mathematics
- 2004

ABSTRACT In this article, we develop the obstruction theory for lifting complexes, up to quasi-isomorphism, to derived categories of flat nilpotent deformations of abelian categories. As a particular… Expand

DG deformation theory of objects in homotopy and derived categories I

- Mathematics
- 2006

We develop a general deformation theory of objects in homotopy and derived categories of DG categories. The main result is a general pro-representability theorem for the corresponding deformation… Expand

Quasi-coherent sheaves in commutative and non-commutative geometry

- Mathematics
- 2003

We give a definition of quasi-coherent modules for any presheaf of sets on the categories of affine commutative and non-commutative schemes. This definition generalizes the usual one. We study the… Expand

Noncommutative Projective Schemes

- Mathematics
- 1994

An analogue of the concept of projective scheme is defined for noncommutative N-graded algebras using the quotient category C of graded right A-modules modulo its full subcategory of torsion modules.… Expand

Deriving DG categories

- Mathematics
- 1994

— We investigate the (unbounded) derived category of a differential Z-graded category (=DG category). As a first application, we deduce a "triangulated analogue" (4.3) of a theorem of Freyd's [5],… Expand

HOMOLOGICAL PROPERTIES OF ASSOCIATIVE ALGEBRAS: THE METHOD OF HELICES

- Mathematics
- 1994

Homological properties of associative algebras arising in the theory of helices are studied. A class of noncommutative algebras is introduced in which it is natural (from the viewpoint of the theory… Expand

DG coalgebras as formal stacks

- Mathematics
- 1998

Abstract The category of unital (unbounded) dg cocommutative coalgebras over a field of characteristic zero is provided with a structure of simplicial closed model category. This generalizes the… Expand

Injective resolutions of BG and derived moduli spaces of local systems

- Mathematics
- 1997

It was suggested on several occasions by Deligne, Drinfeld and Kontsevich that all the moduli spaces arising in the classical problems of deformation theory should be extended to natural \derived"… Expand