Nilpotence and descent in equivariant stable homotopy theory

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• 作者：Akhil Mathew^a ; ^{amathew@math.harvard.edu" class="auth_mail" title="E-mail the corresponding author} ; Niko Naumann^b ; ^{Niko.Naumann@mathematik.uni-regensburg.de" class="auth_mail" title="E-mail the corresponding author} ; Justin Noel^b ; ^{justin.noel@mathematik.uni-regensburg.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Stable equivariant homotopy theory ; Stable homotopy theory ; Localization ; Completion ; Nilpotence ; Koszul duality ; Eilenberg&ndash ; Moore spectral sequence ; Infinity categories ; Tensor triangulated categories ; Spectral sequences ; Descent ; Unipotence ; Equivariant topological K-theory ; Real K-theory
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：10 January 2017
• 年：2017
• 卷：305
• 期：Complete
• 页码：994-1084
• 全文大小：1314 K

文摘

Let G   be a finite group and let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815300062&_mathId=si1.gif&_user=111111111&_pii=S0001870815300062&_rdoc=1&_issn=00018708&md5=aabcedaca6a529b65644a7086d88c821" title="Click to view the MathML source">Fclass="mathContainer hidden">class="mathCode">$\mathcal{F}$ be a family of subgroups of G. We introduce a class of G  -equivariant spectra that we call class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815300062&_mathId=si1.gif&_user=111111111&_pii=S0001870815300062&_rdoc=1&_issn=00018708&md5=aabcedaca6a529b65644a7086d88c821" title="Click to view the MathML source">Fclass="mathContainer hidden">class="mathCode">$\mathcal{F}$-nilpotent  . This definition fits into the general theory of torsion, complete, and nilpotent objects in a symmetric monoidal stable ∞-category, with which we begin. We then develop some of the basic properties of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815300062&_mathId=si1.gif&_user=111111111&_pii=S0001870815300062&_rdoc=1&_issn=00018708&md5=aabcedaca6a529b65644a7086d88c821" title="Click to view the MathML source">Fclass="mathContainer hidden">class="mathCode">$\mathcal{F}$-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 ∞-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.