Semiregularity maps and deformations of modules over Lie algebroids

Let A\subset L be a flat inclusion of Lie algebroids, i.e., a Lie pair, on a smooth separated scheme over a field of characteristic 0. For every locally free A-module M we define its semiregularity maps and prove that, under some additional assumptions, they annihilate obstructions to deformations of M. In case A=0 and L=tangent bundle, this construction gives to the usual Buchweitz-Flenner's semiregularity maps for coherent sheaves.