Relative Inverse Limit Perfection of Derived Commutative Rings

Abstract

We study the relative Frobenius map associated with a map of derived commutative rings over a field of positive characteristic. As part of this, we examine a relative analog of perfectness and construct a relative inverse limit perfection which, under suitable conditions on the base, serves as a right adjoint to the inclusion of relatively perfect algebras into the category of all algebras. Specializing to animated rings, we investigate relative versions of semiperfectness and F-finiteness, and use these to show that any map of F-finite animated rings factors into a free map of finite type, followed by a relatively perfect map, followed by a surjective map. We also show that, for a morphism of Noetherian F-finite rings, the vanishing of the cotangent complex implies that the morphism is relatively perfect.