  This paper continues the development of the deformation theory of abelian
categories introduced in a previous paper by the authors. We show first that
the deformation theory of abelian categories is controlled by an obstruction
theory in terms of a suitable notion of Hochschild cohomology for abelian
categories. We then show that this Hochschild cohomology coincides with the one
defined by Gerstenhaber, Schack and Swan in the case of module categories over
diagrams and schemes and also with the Hochschild cohomology for exact
categories introduced recently by Keller. In addition we show in complete
generality that Hochschild cohomology satisfies a Mayer-Vietoris property and
that for constantly ringed spaces it coincides with the cohomology of the
structure sheaf.
