Let \( R \) be a principal ideal domain, and let \(( {\mathbb {T}}(V_{n+1}\oplus V_{\le n}),\partial )\) and \(({\mathbb {T}}(W_{n+1}\oplus V_{\le n}),\delta )\) be two free differential graded \( R \) -algebras such that \( \partial = \delta \) on \( v \in V_{\le n} \) . This paper is dedicated to exploring the problem of constructing a DGA-map \(\alpha :( {\mathbb {T}}(V_{n+1}\oplus V_{\le n}),\partial )\rightarrow ({\mathbb {T}}(W_{n+1}\oplus V_{\le n}),\delta )\) such that the chain map \( \tilde{\alpha }_{*}\) , induced by \(\alpha \) on the indecomposables, satisfies \( \tilde{\alpha } = id \) on \(V_{\le n}\) . Our focus is to provide an algebraic condition under which \(( {\mathbb {T}}(V_{n+1}\oplus V_{\le n}),\partial )\) and \(({\mathbb {T}}(W_{n+1}\oplus V_{\le n}),\delta )\) become quasi-isomorphic.
