Results 1  10
of
20
DG quotients of DG categories
 J. Algebra
"... Abstract. Keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with Verdier’s notion of quotient of a triangulated category modulo a triangulated subcategory. This work is an attempt to further develop his theory. ..."
Abstract

Cited by 79 (0 self)
 Add to MetaCart
Abstract. Keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with Verdier’s notion of quotient of a triangulated category modulo a triangulated subcategory. This work is an attempt to further develop his theory.
A Proof of Tsygan’s formality conjecture for an arbitrary Smooth Manifold
, 2005
"... Proofs of Tsygan’s formality conjectures for chains would unlock important algebraic tools which might lead to new generalizations of the AtiyahPatodiSinger index theorem and the RiemannRochHirzebruch theorem. Despite this pivotal role in the traditional investigations and the efforts of various ..."
Abstract

Cited by 20 (6 self)
 Add to MetaCart
Proofs of Tsygan’s formality conjectures for chains would unlock important algebraic tools which might lead to new generalizations of the AtiyahPatodiSinger index theorem and the RiemannRochHirzebruch theorem. Despite this pivotal role in the traditional investigations and the efforts of various people the most general version of Tsygan’s formality conjecture has not yet been proven. In my thesis I propose Fedosov resolutions for the Hochschild cohomological and homological complexes of the algebra of functions on an arbitrary smooth manifold. Using these resolutions together with Kontsevich’s formality quasiisomorphism for Hochschild cochains of R[[y 1,...,y d]] and Shoikhet’s formality quasiisomorphism for Hochschild chains of R[[y 1,...,y d]] I prove Tsygan’s formality conjecture for Hochschild chains of the algebra of functions on an arbitrary smooth manifold. The construction of the formality quasiisomorphism for Hochschild chains is manifestly functorial for isomorphisms of the pairs (M, ∇), where M is the manifold and ∇ is an affine connection on the
Extended deformation functors
 Int. Math. Res. Not
"... We introduce a precise notion, in terms of some Schlessinger’s type conditions, of extended deformation functors which is compatible with most of recent ideas in the Derived Deformation Theory (DDT) program and with geometric examples. With this notion we develop the (extended) analogue of Schlessin ..."
Abstract

Cited by 17 (9 self)
 Add to MetaCart
We introduce a precise notion, in terms of some Schlessinger’s type conditions, of extended deformation functors which is compatible with most of recent ideas in the Derived Deformation Theory (DDT) program and with geometric examples. With this notion we develop the (extended) analogue of Schlessinger and obstruction theories. The inverse mapping theorem holds for natural transformations of extended deformation functors and all such functors with finite dimensional tangent space are prorepresentable in the homotopy category. Finally we prove that the primary obstruction map induces a structure of graded Lie algebra on the tangent space. Mathematics Subject Classification (1991): 13D10, 14B10, 14D15.
Moduli of objects in dgcategories
, 2006
"... To any dgcategory T (over some base ring k), we define a D −stack MT in the sense of [HAGII], classifying certain T opdgmodules. When T is saturated, MT classifies compact objects in the triangulated category [T] associated to T. The main result of this work states that under certain finiteness ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
To any dgcategory T (over some base ring k), we define a D −stack MT in the sense of [HAGII], classifying certain T opdgmodules. When T is saturated, MT classifies compact objects in the triangulated category [T] associated to T. The main result of this work states that under certain finiteness conditions on T (e.g. if it is saturated) the D −stack MT is locally geometric (i.e. union of open and geometric substacks). As a consequence we prove the algebraicity of the group of autoequivalences of saturated dgcategories. We also obtain the existence of reasonable moduli for perfect complexes on a smooth and proper scheme, as
Two kinds of derived categories, Koszul duality, and comodulecontramodule correspondence
, 2009
"... ..."
Deformations of homotopy algebras
 Comm. Alg
"... 1.1. Let k be a field of characteristic zero, O be a dg operad over k and let A be an Oalgebra. In this note we define formal deformations of A, construct the deformation functor DefA: dgart ≤0 (k) → ∆ 0 Ens ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
1.1. Let k be a field of characteristic zero, O be a dg operad over k and let A be an Oalgebra. In this note we define formal deformations of A, construct the deformation functor DefA: dgart ≤0 (k) → ∆ 0 Ens
Deformations of sheaves of algebras
 Adv. Math
, 2005
"... Abstract. A construction of the tangent dg Lie algebra of a sheaf of operad algebras on a site is presented. The requirements on the site are very mild; the requirements on the algebra are more substantial. A few applications including the description of deformations of a scheme and equivariant defo ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
Abstract. A construction of the tangent dg Lie algebra of a sheaf of operad algebras on a site is presented. The requirements on the site are very mild; the requirements on the algebra are more substantial. A few applications including the description of deformations of a scheme and equivariant deformations are considered. The construction is based upon a model structure on the category of presheaves which should be of an independent interest. 0.1. In this paper we study formal deformations of sheaves of algebras. The most obvious (and very important) example is that of deformations of a scheme X over a field k of characteristic zero. In two different cases, the first when X is smooth, and the second when X is affine, the description is wellknown. In both cases there is a differential graded
Hochschild cohomology and moduli spaces of strongly homotopy associative algebras
 Homology Homotopy Appl
"... Abstract. Motivated by ideas from stable homotopy theory we study the space of strongly homotopy associative multiplications on a twocell chain complex. In the simplest case this moduli space is isomorphic to the set of orbits of a group of invertible power series acting on a certain space. The Hoc ..."
Abstract

Cited by 7 (5 self)
 Add to MetaCart
Abstract. Motivated by ideas from stable homotopy theory we study the space of strongly homotopy associative multiplications on a twocell chain complex. In the simplest case this moduli space is isomorphic to the set of orbits of a group of invertible power series acting on a certain space. The Hochschild cohomology rings of resulting A∞algebras have an interpretation as totally ramified extensions of discrete valuation rings. All A∞algebras are supposed to be unital and we give a detailed analysis of unital structures which is of independent interest. Keywords: A∞algebra, derivation, Hochschild cohomology, formal power series. 1.
Deformation theory of objects in homotopy and derived categories I: General theory
, 2008
"... 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 ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
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 codeformations) of E in the homotopy
KOSZUL DUALITY AND CODERIVED CATEGORIES (AFTER K. LEFÈVRE)
"... Abstract. This is a brief report on a part of Chapter 2 of K. Lefèvre’s thesis [5]. We sketch a framework for Koszul duality [1] where the Koszul dual algebra is replaced by a coalgebra. This allows us to free ourselves from many assumptions (e.g. finiteness assumptions) and leads to clean statement ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
Abstract. This is a brief report on a part of Chapter 2 of K. Lefèvre’s thesis [5]. We sketch a framework for Koszul duality [1] where the Koszul dual algebra is replaced by a coalgebra. This allows us to free ourselves from many assumptions (e.g. finiteness assumptions) and leads to clean statements about