OBSTRUCTION TO NAÏVE LIFTABILITY OF DG MODULES

In previous work we introduced the notion of naïve liftability of DG modules. The main purpose of this paper is to explicitly describe the obstruction to naïve liftability along extensions A→B of DG algebras, where B is projective as an underlying graded A-module. In particular, we give an explicit description of a DG B-module homomorphism which defines the obstruction to naïve liftability of a semifree DG B-module N as a certain cohomology class in Ext⁡B1(N,N⊗BJ), where J is the diagonal ideal. Our results on the obstruction class enable us to give concrete examples of DG modules that do and do not satisfy the naïve lifting property.