Strictly unital A_∞-algebras

Given a graded module over a commutative ring, we define a dg-Lie algebra whose Maurer–Cartan elements are the strictly unital A_∞-algebra structures on that module. We use this to generalize Positselski's result that a curvature term on the bar construction compensates for a lack of augmentation, from a field to arbitrary commutative base ring. We also use this to show that the reduced Hochschild cochains control the strictly unital deformation functor. We motivate these results by giving a full development of the deformation theory of a nonunital A_∞-algebra.