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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 , where is a DG subalgebra and B is semifree over A. Our construction of such a semifree resolution, that we denote by , uses the notions of reduced bar resolution and tensor algebra of the shift of the diagonal ideal. As an application of , we provide a new characterization of naïve liftability of semifree DG B-modules.