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.

Boris Shoikhet noticed that the proof of lemma 1 in section 2.3 of [1] contains an error. In this note I give a correct proof of this lemma which was suggested to me by Dmitry Tamarkin. The correction does not change the results of [1]. 1

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.

Abstract. Motivated by ideas from stable homotopy theory we study the space of strongly homotopy associative multiplications on a two-cell 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.

To any dg-category T (over some base ring k), we define a D −-stack MT in the sense of [HAGII], classifying certain T op-dg-modules. 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 sub-stacks). As a consequence we prove the algebraicity of the group of auto-equivalences of saturated dg-categories. We also obtain the existence of reasonable moduli for perfect complexes on a smooth and proper scheme, as

1.1. Let k be a field of characteristic zero, O be a dg operad over k and let A be an O-algebra. In this note we define formal deformations of A, construct the deformation functor DefA: dgart ≤0 (k) → ∆ 0 Ens

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 well-known. In both cases there is a differential graded

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

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

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