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

Categories of modules and their deformations