Categories of modules and their deformations

TL;DR

This paper develops an obstruction theory using Quillen-Lurie formalism to analyze the lifting of modules over derived stacks, with obstructions in Andre-Quillen cohomology, providing explicit descriptions of module realizations.

Contribution

It introduces a new obstruction theory for modules over geometric -stacks, connecting deformation problems with Andre-Quillen cohomology and explicit Postnikov tower descriptions.

Findings

Obstructions to lifting modules are characterized in Andre-Quillen cohomology.

Explicit descriptions of module realizations involve k-invariants and cotangent complexes.

The framework applies to derived geometric stacks and their classical counterparts.

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.