Realizing modules over the homology of a DGA

TL;DR

This paper establishes a categorical equivalence between A_{n+1}-module structures on a homology module over a DGA and Postnikov systems, linking obstruction theories for module realization.

Contribution

It constructs an equivalence of categories connecting A_{n+1}-module structures with Postnikov systems, clarifying the obstruction theories for module realization over a DGA.

Findings

Equivalence between A_{n+1}-module structures and Postnikov systems.

Obstruction theories for realizing modules coincide.

Compatibility of these equivalences for different n values.

Abstract

Let A be a DGA over a field and X a module over H_*(A). Fix an $A_\infty$-structure on H_*(A) making it quasi-isomorphic to A. We construct an equivalence of categories between A_{n+1}-module structures on X and length n Postnikov systems in the derived category of A-modules based on the bar resolution of X. This implies that quasi-isomorphism classes of A_n-structures on X are in bijective correspondence with weak equivalence classes of rigidifications of the first n terms of the bar resolution of X to a complex of A-modules. The above equivalences of categories are compatible for different values of n. This implies that two obstruction theories for realizing X as the homology of an A-module coincide.