The deformation theory of sheaves of commutative rings

We define a sheaf of abelian groups whose cohomology is represented by the cotangent complex, permitting a rapid introduction to the theory of the cotangent complex in the same generality as it was defined by Illusie, but avoiding simplicial methods. We show how obstructions to some standard deformation problems arise as the classes of torsors under and gerbes banded by this sheaf. This generalizes results of Quillen, Rim, and Gaitsgory.