Nilpotence and descent in equivariant stable homotopy theory

被引:61
作者
Mathew, Akhil [1 ]
Naumann, Niko [2 ]
Noel, Justin [2 ]
机构
[1] Harvard Univ, Cambridge, MA 02138 USA
[2] Univ Regensburg, NWF Math 1, Regensburg, Germany
关键词
Stable equivariant homotopy theory; Stable homotopy theory; Localization; Completion; Nilpotence; Koszul duality; Eilenberg-Moore spectral sequence; Infinity categories; Tensor triangulated categories; Spectral sequences; Descent; Unipotence; Equivariant topological K-theory; SPECTRAL SEQUENCE; COHOMOLOGY; CATEGORIES; MODULES; INDEX; MODEL;
D O I
10.1016/j.aim.2016.09.027
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Let G be a finite group and let F be a family of subgroups of G. We introduce a class of G-equivariant spectra that we call F-nilpotent. This definition fits into the general theory of torsion, complete, and nilpotent objects in a symmetric monoidal stable infinity-category, with which we begin. We then develop some of the basic properties of 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 infinity-categories of module spectra over objects such as equivariant real and complex K-theory and Borelequivariant 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. (C)2016 Elsevier Inc. All rights reserved.
引用
收藏
页码:994 / 1084
页数:91
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