A comparison of dg algebra resolutions with prime residual characteristic

In this article we fix a prime integer $p$ and compare certain dg algebra resolutions over a local ring whose residue field has characteristic $p$. Namely, we show that given a closed surjective map between such algebras there is a precise description for the minimal model in terms of the acyclic closure, and that the latter is a quotient of the former. A first application is that the homotopy Lie algebra of a closed surjective map with residual characteristic $p$ is abelian. We also use these calculations to show deviations enjoy rigidity properties which detect the (quasi-)complete intersection property.