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3 results

  • Part Two - Connecting the Dots: Resources, Tools, and Representations

  • Edited by Tommaso Caselli, Eduard Hovy, Carnegie Mellon University, Pennsylvania, Martha Palmer, University of Colorado Boulder, Piek Vossen, Vrije Universiteit, Amsterdam
  • Book:
    Computational Analysis of Storylines
    Published online:
    06 November 2021
    Print publication:
    25 November 2021, pp 143-144

  • Summary

    For an equivariant commutative ring spectrum R, ?0R has algebraic structure reflecting the presence of both additive transfers and multiplicative norms. The additive structure gives rise to a Mackey functor and the multiplicative structure yields the additional structure of a Tambara functor. If R is an N? ring spectrum in the category of genuine G-spectra, then all possible additive transfers are present and ?0R has the structure of an incomplete Tambara functor. However, if R is an N? ring spectrum in a category of incomplete G-spectra, the situation is more subtle. In this chapter, we study the algebraic theory of Tambara structures on incomplete Mackey functors, which we call bi-incomplete Tambara functors. Just as incomplete Tambara functors have compatibility conditions that control which systems of norms are possible, bi-incomplete Tambara functors have algebraic constraints arising from the possible interactions of transfers and norms. We give a complete description of the possible interactions between the additive and multiplicative structures.

  • 7 - Bi-incomplete Tambara Functors

  • Edited by Scott Balchin, David Barnes, Queen's University Belfast, Magdalena Kędziorek, Radboud Universiteit Nijmegen, Markus Szymik, Norwegian University of Science and Technology, Trondheim
  • Book:
    Equivariant Topology and Derived Algebra
    Published online:
    29 October 2021
    Print publication:
    18 November 2021, pp 276-313

  • Summary

    For an equivariant commutative ring spectrum R, ?0R has algebraic structure reflecting the presence of both additive transfers and multiplicative norms. The additive structure gives rise to a Mackey functor and the multiplicative structure yields the additional structure of a Tambara functor. If R is an N? ring spectrum in the category of genuine G-spectra, then all possible additive transfers are present and ?0R has the structure of an incomplete Tambara functor. However, if R is an N? ring spectrum in a category of incomplete G-spectra, the situation is more subtle. In this chapter, we study the algebraic theory of Tambara structures on incomplete Mackey functors, which we call bi-incomplete Tambara functors. Just as incomplete Tambara functors have compatibility conditions that control which systems of norms are possible, bi-incomplete Tambara functors have algebraic constraints arising from the possible interactions of transfers and norms. We give a complete description of the possible interactions between the additive and multiplicative structures.

  • On the Balmer spectrum for compact Lie groups

  • Part of
  • Tobias Barthel, J. P. C. Greenlees, Markus Hausmann
  • Journal:
    Compositio Mathematica / Volume 156 / Issue 1 / January 2020
    Published online by Cambridge University Press:
    14 November 2019, pp. 39-76
    Print publication:
    January 2020

  • We study the Balmer spectrum of the category of finite $G$-spectra for a compact Lie group $G$, extending the work for finite $G$ by Strickland, Balmer–Sanders, Barthel–Hausmann–Naumann–Nikolaus–Noel–Stapleton and others. We give a description of the underlying set of the spectrum and show that the Balmer topology is completely determined by the inclusions between the prime ideals and the topology on the space of closed subgroups of $G$. Using this, we obtain a complete description of this topology for all abelian compact Lie groups and consequently a complete classification of thick tensor ideals. For general compact Lie groups we obtain such a classification away from a finite set of primes $p$.