  Using Quillen-Lurie deformation theory formalism we develop an obstruction
theory for studying the stable $\infty$-category of modules over a given
geometric $\infty$-stack.
  The obstruction theory studies the problem of lifting compact objects to the
stable $\infty$-category of quasi-coherent modules over a derived geometric
stack from the category of modules over its underlying classical stack. The
obstructions live in Andre-Quillen cohomology. An explicit description of the
space of realizations of a given module over X as a colimit of perfect modules
can be given in terms of the k-invariants of a postnikov tower of X and the
cotangent complex of the moduli functor of perfect modules.
