Hochschild cohomology, the characteristic morphism and derived deformations

Abstract

A notion of Hochschild cohomology HH^*(𝒜) of an abelian category 𝒜 was defined by Lowen and Van den Bergh (Adv. Math. 198 (2005), 172–221). They also showed the existence of a characteristic morphism 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∈HH²(𝒜) corresponds to a first-order deformation 𝒜_c of 𝒜 (Lowen and Van den Bergh, Trans. Amer. Math. Soc. 358 (2006), 5441–5483). The problem of deforming an object M∈D^b(𝒜) to D^b(𝒜_c) was treated by Lowen (Comm. Algebra 33 (2005), 3195–3223). In this paper we show that the element χ(c)_M∈Ext_𝒜²(M,M) is precisely the obstruction to deforming M to D^b(𝒜_c). 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’.