Nilpotence and descent in equivariant stable homotopy theory

• Published in: arXiv.org 2016-09
• Main Authors: Mathew, Akhil, Naumann, Niko, Noel, Justin
• Format: Article
• Language: English
• Subjects: Homology; Homotopy theory; Spectra; Subgroups; Theorems
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1507.06869

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.