Hochschild cohomology and moduli spaces of strongly homotopy associative algebras

Abstract

Motivated by ideas from stable homotopy theory we study the space of strongly homotopy associative multiplications on a two-cell chain complex. In the simplest case this moduli space is isomorphic to the set of orbits of a group of invertible power series acting on a certain space. The Hochschild cohomology rings of resulting $A_\infty$-algebras have an interpretation as totally ramified extensions of discrete valuation rings. All $A_\infty$-algebras are supposed to be unital and we give a detailed analysis of unital structures which is of independent interest.