Strictly unital A-infinity algebras

TL;DR

This paper develops a deformation theory framework for strictly unital A-infinity algebras over commutative rings, generalizing existing results and linking Hochschild cochains to deformation control.

Contribution

It introduces a dg-Lie algebra framework for strictly unital A-infinity structures and extends Positselski's curvature result from fields to arbitrary rings.

Findings

Maurer-Cartan elements characterize strictly unital A-infinity structures

Curvature terms in bar constructions compensate for lack of augmentation over rings

Reduced Hochschild cochains control the deformation functor

Abstract

Given a graded module over a commutative ring, we define a dg-Lie algebra whose Maurer-Cartan elements are the strictly unital A-infinity 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-infinity algebra.