Obstruction to naive liftability of DG modules

TL;DR

This paper investigates the obstructions to naive liftability of differential graded modules over algebra extensions, identifying a cohomology class that determines liftability and providing concrete examples of modules with or without this property.

Contribution

It introduces a cohomological obstruction class for naive liftability of DG modules and demonstrates how to determine liftability in specific cases.

Findings

Obstruction to naive liftability is a cohomology class in Ext^1.

Concrete examples of DG modules with and without naive liftability.

Characterization of liftability conditions via the obstruction class.

Abstract

The notion of naive liftability of DG modules is introduced in [9] and [10]. In this paper, we study the obstruction to naive liftability along extensions $A\to B$ of DG algebras, where $B$ is projective as an underlying graded $A$-module. We show that the obstruction to naive liftability of a semifree DG $B$-module $N$ is a certain cohomology class in Ext$^1_B(N,N\otimes_B J)$, where $J$ is the diagonal ideal. Our results on obstruction class enable us to give concrete examples of DG modules that do and do not satisfy the naive lifting property.