On the semifree resolutions of DG algebras over the enveloping DG algebras

The goal of this paper is to construct a semifree resolution for a non-negatively graded strongly commutative DG algebra $B$ over the enveloping DG algebra $B{\otimes }_{A}B$, where $A\subseteq B$ is a DG subalgebra and $B$ is semifree over $A$. Our construction of such a semifree resolution uses the notions of reduced bar resolution and tensor algebra of the shift of the diagonal ideal.