Connections and naïve lifting of DG modules
Saeed Nasseh, Maiko Ono, Yuji Yoshino
TL;DR
This work extends Connes' notion of connections to the differential graded (DG) setting and analyzes when semifree DG $B$-modules lift naively to $A$ along a DG algebra map $A \to B$, with $B$ projective as a graded $A$-module. By developing connections along DG $B^e$-modules and, separately, along DG $B$-modules, the authors establish a fundamental exact sequence of connections and identify obstruction classes—through Atiyah-type maps and Kodaira–Spencer theory—that govern liftability. The central result (Theorem \textit{naive}) gives equivalent conditions for naive liftability in terms of the existence of $oxed{\delta}$-connections with zero differential, the splitting of fundamental sequences $\mathcal{E}_N(X)$ for all DG $B^e$-modules $X$, and the presence of ${}^*\mathrm{Der}_A(B,-)$-connections, tying lifting to derivations and curvature. They further extend the analysis to DG $B$-modules with a parallel set of equivalences (Theorem \textit{fesOX}) involving the Atiyah map, bar{\delta}-connections, and the Kodaira–Spencer morphism $\kappa$, illustrating a DG-analogue of obstructions to lifting and revealing connections between Ext-vanishing phenomena and semisimple extensions. The results provide concrete, calculable criteria linking lifting problems, differential module structures, and Kodaira–Spencer-type obstructions, with potential applications to quasi-smooth DG extensions and noncommutative differential geometry in homological algebra.
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 naïvely liftable to $A$ in terms of connections.