Derived induction and restriction theory

Abstract

Let G be a finite group. To any family F of subgroups of G, we associate a thick ⊗-ideal F Nil of the category of G-spectra with the property that every G-spectrum in F Nil (which we call F-nilpotent) can be reconstructed from its underlying H-spectra as H varies over F. A similar result holds for calculating G-equivariant homotopy classes of maps into such spectra via an appropriate homotopy limit spectral sequence. In general, the condition E ∈ F Nil implies strong collapse results for this spectral sequence as well as its dual homotopy colimit spectral sequence. As applications, we obtain Artin and Brauer type induction theorems for G-equivariant E-homology and cohomology, and generalizations of Quillen's Fp-isomorphism theorem when E is a homotopy commutative G-ring spectrum.We show that the subcategory F Nil contains many G-spectra of interest for relatively small families F. These include G-equivariant real and complex K-theory as well as the Borel-equivariant cohomology theories associated to complex oriented ring spectra, the Lnlocal sphere, the classical bordism theories, connective real K-theory, and any of the standard variants of topological modular forms. In each of these cases we identify the minimal family such that these results hold.

Examples of Descent up to Nilpotence

“…We have the following two basic analogs over other bases, all proved in [MNN15]. The first is a basic "complex-oriented" form of Theorem 4.26, and can be proved in many ways, such as the use of the flag variety.…”

“…This uses some algebraic facts about Mackey functors. We refer to [MNN15] for details. The statement after inverting |G| is nontrivial (in general, calculating E * (BG)[1/|G|]) for E a spectrum is a difficult problem) and appears in [HKR00] for R = E n and related ring spectra, and follows from the spectral sequence as well.…”

“…In this paper, our goal is to give an introduction to these ideas and an overview of several examples, emphasizing the F -nilpotence of [MNN17,MNN15]. This paper is mostly intended as an exposition, although some of the results are improvements or variants of older ones.…”

“…I would like to thank Bhargav Bhatt, Srikanth Iyengar, and Jacob Lurie for helpful discussions. I would especially like to thank my collaborators Niko Naumann and Justin Noel; much of this material is drawn from [MNN17,MNN15]. Most of all, I would like to thank Mike Hopkins: most of these ideas originated in his work.…”

The Balmer spectrum of the equivariant homotopy category of a finite abelian group

“…Remark 3.6.i) In the language of[MNN15], the vanishing criterion in Theorem 3.5 for Φ F (E) is equivalent to the determination of the derived defect base of E: Since EF ⊗ E is a ring spectrum, its A-fixed points, i.e., ΦF (E), vanish if and only if it is itself equivariantly contractible, i.e., EF ⊗ E = 0. By definition, this is equivalent to F containing the derived defect base of E which consists of those subgroups of A of p-rank at most n by [MNN15, Prop.…”