Let
G be a finite group and let
ca6a529b65644a7086d88c821" title="Click to view the MathML source">F be a family of subgroups of
G. We introduce a class of
G -equivariant spectra that we call
ca6a529b65644a7086d88c821" title="Click to view the MathML source">F-nilpotent . This definition fits into the general theory of torsion, complete, and nilpotent objects in a symmetric monoidal stable ∞-category, with which we begin. We then develop some of the basic properties of
ca6a529b65644a7086d88c821" title="Click to view the MathML source">F-nilpotent
G-spectra, which are explored further in the sequel to this paper.
In the rest of the paper, we prove several general structure theorems for ∞-categories of module spectra over objects such as equivariant real and complex K-theory and Borel-equivariant MU. Using these structure theorems and a technique with the flag variety dating back to Quillen, we then show that large classes of equivariant cohomology theories for which a type of complex-orientability holds are nilpotent for the family of abelian subgroups. In particular, we prove that equivariant real and complex K-theory, as well as the Borel-equivariant versions of complex-oriented theories, have this property.