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Journals and Conferences
For a ringed space (X, O), we show that the deformations of the abelian category Mod(O) of sheaves of O-modules  are obtained from al-gebroid prestacks, as introduced by Kontsevich. In case X is a quasi-compact separated scheme the same is true for Qch(O), the category of quasi-coherent sheaves on X. It follows in particular that there is a deformation… (More)
In this paper we develop the basic infinitesimal deformation theory of abelian categories. This theory yields a natural generalization of the well-known deformation theory of algebras developed by Gerstenhaber. As part of our deformation theory we define a notion of flatness for abelian categories. We show that various basic properties are preserved under… (More)
In this paper 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 case we also obtain the corresponding obstruction theory for lifting of objects in terms of Yoneda Ext-groups. In appendix we prove the existence of miniversal derived… (More)
A notion of Hochschild cohomology HH * (A) of an abelian category A was defined by Lowen and Van den Bergh (2005) and they showed the existence of a characteristic mor-phism χ from the Hochschild cohomology of A into the graded centre Z * (D b (A)) of the bounded derived category of A. An element c ∈ HH 2 (A) corresponds to a first order deformation A c of… (More)
In this paper we show that, in general, first-order Morita deformations are too limited to capture the second Hochschild cohomology of a differential graded category. For differential graded categories with bounded above cohomology, the Morita deformations do constitute a part of the Hochschild cohomology.
We generalize and clarify Gerstenhaber and Schack's " Special Co-homology Comparison Theorem ". More specifically we obtain a fully faithful functor between the derived categories of bimodules over a prestack over a small category U and the derived category of bimodules over its corresponding fibered category. In contrast to Gerstenhaber and Schack we do… (More)
Since curved dg algebras, and modules over them, have differentials whose square is not zero, these objects have no cohomology, and there is no classical derived category. For different purposes, different notions of " derived " categories have been introduced in the literature. In this note, we show that for some concrete curved dg algebras, these derived… (More)
For a scheme X, we construct a sheaf C of complexes on X such that for every quasi-compact open U ⊂ X, C(U) is quasi-isomorphic to the Hochschild complex of U . Since C is moreover acyclic for taking sections on quasi-compact opens, we obtain a local to global spectral sequence for Hochschild cohomology if X is quasi-compact.