DG-algebras and derived A-infinity algebras

TL;DR

This paper introduces derived A-infinity algebras, extending the concept of A-infinity structures to better model differential graded algebras over arbitrary rings, providing a complete quasi-isomorphism classification.

Contribution

It defines derived A-infinity algebras and proves that any dga over a commutative ring is equivalent to a minimal derived A-infinity algebra, generalizing previous results.

Findings

Minimal derived A-infinity algebra models exist for all dgas over any commutative ring.

Such models are constructed as projective resolutions of homology algebras.

The models fully determine the quasi-isomorphism class of the original dga.

Abstract

A differential graded algebra can be viewed as an A-infinity algebra. By a theorem of Kadeishvili, a dga over a field admits a quasi-isomorphism from a minimal A-infinity algebra. We introduce the notion of a derived A-infinity algebra and show that any dga A over an arbitrary commutative ground ring k is equivalent to a minimal derived A-infinity algebra. Such a minimal derived A-infinity algebra model for A is a k-projective resolution of the homology algebra of A together with a family of maps satisfying appropriate relations. As in the case of A-infinity algebras, it is possible to recover the dga up to quasi-isomorphism from a minimal derived A-infinity algebra model. Hence the structure we are describing provides a complete description of the quasi-isomorphism type of the dga.