Naïve liftings of DG modules

Let n be a positive integer, and let A be a strongly commutative differential graded (DG) algebra over a commutative ring R. Assume that (a)B= A[X₁, … , X_n] is a polynomial extension of A, where X₁, … , X_n are variables of positive degrees; or(b)A is a divided power DG R-algebra and B= A⟨ X₁, … , X_n⟩ is a free extension of A obtained by adjunction of variables X₁, … , X_n of positive degrees. In this paper, we study naïve liftability of DG modules along the natural injection A→ B using the notions of diagonal ideals and homotopy limits. We prove that if N is a bounded below semifree DG B-module such that ExtBi(N,N)=0 for all i⩾ 1 , then N is naïvely liftable to A. This implies that N is a direct summand of a DG B-module that is liftable to A. Also, the relation between naïve liftability of DG modules and the Auslander-Reiten Conjecture has been described.