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Examples of Descent up to Nilpotence

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Geometric and Topological Aspects of the Representation Theory of Finite Groups (PSSW 2016)

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Abstract

We give a survey of the ideas of descent and nilpotence, beginning with the theory of thick subcategories. We focus on examples arising from chromatic homotopy theory (such as Rognes’ Galois extensions) and from group actions, as well as a few examples in algebra. These ideas provide tools for studying certain invariants of tensor-triangulated categories.

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Notes

  1. 1.

    We note that the descent-theoretic approach to Picard groups uses \(\infty \)-categorical technology in an essential manner.

  2. 2.

    We remind the reader that taking retracts, in the \(\infty \)-categorical setting, is not a finite homotopy limit.

  3. 3.

    In fact, \(\eta ^3 = 0\) in \(\pi _*(KO)\).

  4. 4.

    Stated another way, if \(BGL_1(R)\) denotes the classifying space of rank 1 R-modules, then the composite \( BC_p \rightarrow BGL_1(R) \) classifying the representation sphere is nullhomotopic. We refer to [2] for a detailed treatment.

  5. 5.

    For the results below, it is best not only to restrict to \(\mathbb {E}_{\infty }\)-rings as many natural examples are not (or not known to be) \(\mathbb {E}_{\infty }\). If R is \(\mathbb {E}_{\infty }\), the two statements in the question are equivalent.

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Acknowledgements

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 [54, 55]. Most of all, I would like to thank Mike Hopkins: most of these ideas originated in his work. I am grateful to the referee and to Niko Naumann for several corrections. While this article was written, I was supported by the NSF Graduate Fellowship under grant DGE-114415.

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  1. University of Chicago, Chicago, IL, USA

    Akhil Mathew

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  1. Department of Mathematics, University of Georgia, Athens, GA, USA

    Jon F. Carlson

  2. Department of Mathematics, University of Utah, Salt Lake City, UT, USA

    Srikanth B. Iyengar

  3. Department of Mathematics, University of Washington, Seattle, WA, USA

    Julia Pevtsova

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Mathew, A. (2018). Examples of Descent up to Nilpotence. In: Carlson, J., Iyengar, S., Pevtsova, J. (eds) Geometric and Topological Aspects of the Representation Theory of Finite Groups. PSSW 2016. Springer Proceedings in Mathematics & Statistics, vol 242. Springer, Cham. https://doi.org/10.1007/978-3-319-94033-5_11

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