Nilpotence and descent in equivariant stable homotopy theory
By A. Mathew, N. Axel Naumann, J. Noel. 2015
152 citations recorded by the source corpus.
Abstract 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 ∞-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 ∞-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.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/fb054dc38abef735d24817ce64bd2018ef3d42b6
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In Discrete Mathematics and Combinatorics
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Citations in the corpus
- Synthetic spectra and the cellular motivic category (2018, methodology): “ formal enough to yield a spectral sequence whenever X,A are objects of a presentable, symmetric monoidal stable ∞-category, the Adams spectral sequence is developed in this generality for example in [MNN17]. In particular, we can work with SynE and take as our objects the synthetic analogues νX, νA”
- Local duality in algebra and topology (2015, background): “n functors for modules by Dwyer, Iyengar, and Greenlees [DG02, DGI06, Gre01]. Similar ideas have appeared in derived algebraic geometry [Lur11] and very recently in equivariant stable homotopy theory [MNN17]. The difference between our approach and the classical one becomes most transparent in the ca”