Weakly curved A-infinity algebras over a topological local ring

TL;DR

This paper develops a theory of weakly curved A-infinity algebras over topological local rings, extending Koszul duality and module theory, motivated by applications in Floer-Fukaya theory.

Contribution

It introduces derived categories of the first kind for weakly curved A-infinity algebras over pro-Artinian rings and generalizes Koszul duality to this setting.

Findings

Established derived categories for weakly curved A-infinity algebras.

Extended Koszul duality theory to curved algebra context.

Applied formalism of contramodules and comodules in this framework.

Abstract

We define and study the derived categories of the first kind for curved DG and A-infinity algebras complete over a pro-Artinian local ring with the curvature elements divisible by the maximal ideal of the local ring. We develop the Koszul duality theory in this setting and deduce the generalizations of the conventional results about A-infinity modules to the weakly curved case. The formalism of contramodules and comodules over pro-Artinian topological rings is used throughout the paper. Our motivation comes from the Floer-Fukaya theory.