Hochschild cohomology, the characteristic morphism and derived deformations

TL;DR

This paper links Hochschild cohomology of abelian categories to obstructions in deforming objects in derived categories, advancing the understanding of derived deformation theory.

Contribution

It demonstrates that the characteristic morphism encodes obstructions to deforming objects in the derived category, bridging previous results on Hochschild cohomology and deformations.

Findings

Hochschild cohomology element maps to deformation obstructions

The characteristic morphism encodes obstructions to object deformations

Provides insights into derived deformation theory

Abstract

A notion of Hochschild cohomology of an abelian category was defined by Lowen and Van den Bergh (2005) and they showed the existence of a characteristic morphism from the Hochschild cohomology into the graded centre of the (bounded) derived category. An element in the second Hochschild cohomology group corresponds to a first order deformation of the abelian category (Lowen and Van den Bergh, 2006). The problem of deforming single objects of the bounded derived category was treated by Lowen (2005). In this paper we show that the image of the Hochschild cohomology element under the characteristic morphism encodes precisely the obstructions to deforming single objects of the bounded derived category. Hence this paper provides a missing link between the above works. Finally we discuss some implications of these facts in the direction of a ``derived deformation theory''.