Quillen-Suslin Theorem for connected cochain DG algebras

Abstract

Let $\mathcal{A}$ be a connected cochain DG algebra and $P$ a DG $\mathcal{A}$-module such that its underlying graded module $P^{\#}$ is a finitely generated ${\mathcal{A}}^{\#}$-module. We show that $P$ is semi-free if it is semi-projective and it is categorically free if it is categorically projective. It can be considered as a generalization of the well-known Quillen-Suslin Theorem in DG context. As an application, we show that the ghost length and the cone length of a compact DG module coincide.