The Lie algebra structure of tangent cohomology and deformation theory

Abstract

Tangent cohomology of a commutative algebra is known to have the structure of a graded Lie algebra; we account for this by exhibiting a differential graded Lie algebra (in fact, two of them) equivalent as cochain complex to Harrison's yielding the tangent cohomology. This d.g. Lie algebra, called the tangent Lie algebra, also provides an interpretation of the cohomology in terms of perturbations of multiplicative resolutions and hence clarifies the relation to deformation theory. In particular, the higher order obstructions to deformations appear as Massey-Lie brackets. Moreover, we obtain homological constructions for the base and total spaces of a versal deformation.