Flatness and Shipley’s algebraicization theorem

We provide an enhancement of Shipley's algebraicization theorem which behaves better in the context of commutative algebras. This involves defining flat model structures as in Shipley and Pavlov-Scholbach, and showing that the functors still provide Quillen equivalences in this refined context. The use of flat model structures allows one to identify the algebraic counterparts of change of groups functors, as demonstrated in forthcoming work of the author.