A lifting problem for DG modules

TL;DR

This paper investigates conditions under which semi-free DG modules over an extended DG algebra can be lifted to the base algebra, establishing criteria based on vanishing Ext groups and proving uniqueness of such liftings.

Contribution

It provides new criteria for lifting semi-free DG modules over extended DG algebras, specifically linking liftability and uniqueness to Ext group vanishing conditions.

Findings

Liftability of DG modules is guaranteed when Ext^{n+1} vanishes.

Uniqueness of liftings is ensured if Ext^{n} vanishes.

The results apply to DG algebras extended by variables of positive even degree.

Abstract

Let $B = A< X | dX=t >$ be an extended DG algebra by the adjunction of variable of positive even degree $n$, and let $N$ be a semi-free DG $B$-module that is assumed to be bounded below as a graded module. We prove in this paper that $N$ is liftable to $A$ if $Ext_B^{n+1}(N,N)=0$. Furthermore such a lifting is unique up to DG isomorphisms if $Ext_B^{n}(N,N)=0$.