Nilpotence and descent in equivariant stable homotopy theory
| Autor: | Justin Noel, Akhil Mathew, Niko Naumann |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2015 |
| Předmět: |
Discrete mathematics
Pure mathematics Finite group General Mathematics 010102 general mathematics Mathematics - Category Theory 01 natural sciences Mathematics::Algebraic Topology Stable homotopy theory Nilpotent Mathematics::K-Theory and Homology 0103 physical sciences Spectral sequence FOS: Mathematics Torsion (algebra) Algebraic Topology (math.AT) Equivariant cohomology Equivariant map Category Theory (math.CT) 010307 mathematical physics Mathematics - Algebraic Topology 0101 mathematics Abelian group Mathematics |
| Popis: | 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. 63 pages. Revised version, to appear in Advances in Mathematics |
| Databáze: | OpenAIRE |
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