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

@article{Efimov2009DeformationTO,
  title={Deformation theory of objects in homotopy and derived categories I: General theory},
  author={Alexander I. Efimov and Valery A. Lunts and Dmitri O. Orlov},
  journal={Advances in Mathematics},
  year={2009},
  volume={222},
  pages={359-401}
}
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 define four deformation functors Def h ( E ) , coDef h ( E ) , Def ( E ) , coDef ( E ) . The first two functors describe the deformations (and co-deformations) of E in the homotopy category, and the last two – in the derived category. We study their properties and relations. These functors are… Expand
Deformation theory of objects in homotopy and derived categories II: pro-representability of the deformation functor
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) deformationExpand
Deformation theory of objects in homotopy and derived categories III: abelian categories
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 toExpand
DEFORMATIONS OF OBJECTS IN DERIVED CATEGORIES AND NONCOMMUTATIVE GRASSMANIANS
This paper is the part of our joint papers with D.O. Orlov and V.A. Lunts (ELOI), (ELOII), (ELOIII). All results in this paper are obtained by the author. In (ELOI) we developed deformation theory ofExpand
Moduli Spaces, Non-Commutative Geometry and Deformed Differential Categories
The following research plays a central role in deformation theory. If x, is a moduli space over a field k, of characteristic zero, then a formal neighborhood of any point xϵx, is controlled by aExpand
The derived deformation theory of a point
We provide a prorepresenting object for the noncommutative derived deformation problem of deforming a module $X$ over a differential graded algebra. Roughly, we show that the correspondingExpand
Cohomology rings, differential graded algebras and derived equivalences
We first construct derived equivalences of di fferential graded algebras which are endomorphism algebras o f the objects from a triangle in the homotopy category of di fferential graded algebras.Expand
Smoothness of equivariant derived categories
We introduce the notion of (homological) G-smoothness for a complex G-variety X, where G is a connected affine algebraic group. This is based on the notion of smoothness for dg algebras and uses aExpand
DIFFERENTIAL GRADED ENDOMORPHISM ALGEBRAS, COHOMOLOGY RINGS AND DERIVED EQUIVALENCES
Abstract In this paper, we will consider derived equivalences for differential graded endomorphism algebras by Keller's approaches. First, we construct derived equivalences of differential gradedExpand
Moduli Problems for Ring Spectra
In algebraic geometry, it is common to study a geometric object X (such as a scheme) by means of the functor R 7! Hom(SpecR,X) represented by X. In this paper, we consider functors which are definedExpand
Uniqueness of enhancement for triangulated categories
The paper contains general results on the uniqueness of a DG enhancement for trian- gulated categories. As a consequence we obtain such uniqueness for the unbounded categories of quasi-coherentExpand
...
1
2
3
4
...

References

SHOWING 1-10 OF 43 REFERENCES
Deformation theory of objects in homotopy and derived categories II: pro-representability of the deformation functor
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) deformationExpand
Deformation theory of objects in homotopy and derived categories III: abelian categories
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 toExpand
Obstruction Theory for Objects in Abelian and Derived Categories
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 particularExpand
DG coalgebras as formal stacks
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 theExpand
Deriving DG categories
— 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
The Lie algebra structure of tangent cohomology and deformation theory
Abstract Tangent cohomology of a commutative algebra is known to have the structure of a graded Lie algebra; we account for this by exhibiting a differential graded Lie algebra (in fact, two of them)Expand
Noncommutative deformations of modules
The classical deformation theory for modules on a k-algebra, where k is a Þeld, is generalized. For any k-algebra, and for any Þnite family of r modules, we consider a deformation functor deÞned onExpand
Lie theory for nilpotent L∞-algebras
The Deligne groupoid is a functor from nilpotent differential graded Lie algebras concentrated in positive degrees to groupoids; in the special case of Lie algebras over a field of characteristicExpand
Derived Hilbert schemes
We construct the derived version of the Hilbert scheme parametrizing subschemes in a given projective scheme X with given Hilbert polynomial h. This is a dg-manifold (smooth dg-scheme) RHilb_h(X)Expand
Rational homotopy theory
1 The Sullivan model 1.1 Rational homotopy theory of spaces We will restrict our attention to simply-connected spaces. Much of this goes through with nilpotent spaces, but this will keep thingsExpand
...
1
2
3
4
5
...