Connections and na\"{i}ve lifting of DG modules

TL;DR

This paper extends the concept of connections from noncommutative geometry to DG homological algebra and provides criteria for lifting DG modules along algebra homomorphisms.

Contribution

It introduces a generalized notion of connections in DG algebra and characterizes naive liftability of DG modules via these connections.

Findings

Connections are generalized to DG homological algebra.

Necessary and sufficient conditions for naive liftability are established.

Results apply to DG algebra homomorphisms with projective modules.

Abstract

In this paper, we generalize the notion of connections, which was introduced by Alain Connes in noncommutative differential geometry, to the differential graded (DG) homological algebra setting. Then, along a DG algebra homomorphism $A \to B$, where $B$ is assumed to be projective as an underlying graded $A$-module, we give necessary and sufficient conditions for a semifree DG $B$-module to be na\"{i}vely liftable to $A$ in terms of connections.