Categories of modules and their deformations

Abstract

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.