Nilpotence and descent in equivariant stable homotopy theory

TL;DR

This paper introduces the concept of -nilpotent G-spectra in equivariant stable homotopy theory, establishing their properties and showing many equivariant cohomology theories are nilpotent for abelian subgroups.

Contribution

It defines -nilpotent spectra, develops their basic properties, and proves that key equivariant cohomology theories are -nilpotent for abelian subgroups.

Findings

-nilpotent spectra generalize torsion, complete, and nilpotent objects.

Equivariant K-theories are -nilpotent for abelian subgroups.

Structural theorems for module spectra over equivariant cohomology theories.

Abstract

Let $G$ be a finite group and let $\mathscr{F}$ be a family of subgroups of $G$. We introduce a class of $G$-equivariant spectra that we call $\mathscr{F}$-nilpotent. This definition fits into the general theory of torsion, complete, and nilpotent objects in a symmetric monoidal stable $\infty$-category, with which we begin. We then develop some of the basic properties of $\mathscr{F}$-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 $\infty$-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.