Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Profinite equivariant spectra and their tensor-triangular geometry

Scott Balchin, David Barnes, Tobias Barthel

Abstract

We study the tensor-triangular geometry of the category of equivariant $G$-spectra for $G$ a profinite group, $\mathsf{Sp}_G$. Our starting point is the construction of a ``continuous'' model for this category, which we show agrees with all other models in the literature. We describe the Balmer spectrum of finite $G$-spectra up to the ambiguity that is present in the finite group case; in particular, we obtain a thick subcategory theorem when $G$ is abelian. By verifying the bijectivity hypothesis for $\mathsf{Sp}_G$, we prove a nilpotence theorem for all profinite groups. Our study then moves to the realm of rational $G$-equivariant spectra. By exploiting the continuity of our model, we construct an equivalence between the category of rational $G$-spectra and the algebraic model of the second author and Sugrue, which improves their result to the symmetric monoidal and $\infty$-categorical level. Furthermore, we prove that the telescope conjecture holds in this category. Finally, we characterize when the category of rational $G$-spectra is stratified, resulting in a classification of the localizing ideals in terms of conjugacy classes of subgroups. To facilitate these results, we develop some foundational aspects of pro-tt-geometry. For instance, we establish and use the continuity of the homological spectrum and introduce a notion of von Neumann regular tt-categories, of which rational $G$-spectra is an example.

Profinite equivariant spectra and their tensor-triangular geometry

Abstract

We study the tensor-triangular geometry of the category of equivariant -spectra for a profinite group, . Our starting point is the construction of a ``continuous'' model for this category, which we show agrees with all other models in the literature. We describe the Balmer spectrum of finite -spectra up to the ambiguity that is present in the finite group case; in particular, we obtain a thick subcategory theorem when is abelian. By verifying the bijectivity hypothesis for , we prove a nilpotence theorem for all profinite groups. Our study then moves to the realm of rational -equivariant spectra. By exploiting the continuity of our model, we construct an equivalence between the category of rational -spectra and the algebraic model of the second author and Sugrue, which improves their result to the symmetric monoidal and -categorical level. Furthermore, we prove that the telescope conjecture holds in this category. Finally, we characterize when the category of rational -spectra is stratified, resulting in a classification of the localizing ideals in terms of conjugacy classes of subgroups. To facilitate these results, we develop some foundational aspects of pro-tt-geometry. For instance, we establish and use the continuity of the homological spectrum and introduce a notion of von Neumann regular tt-categories, of which rational -spectra is an example.
Paper Structure (53 sections, 119 theorems, 181 equations, 4 figures)

This paper contains 53 sections, 119 theorems, 181 equations, 4 figures.

Table of Contents

  1. Pro-tensor-triangular geometry
  2. Tensor-triangular geometry and the theory of stratification
  3. The spectrum
  4. Stratification and costratification in tensor-triangular geometry
  5. Presheaves of tensor-triangulated categories and their stalks
  6. Continuity of the homological spectrum
  7. Limits and colimits of $\infty$-categories
  8. Filtered colimits of tt-categories and their spectrum
  9. Recollections on the homological spectrum and its support
  10. Continuity for the homological spectrum
  11. The geometry of tensor-triangular stalks
  12. Basic properties of tt-stalks
  13. Continuity of finite localizations and stalks
  14. Zero-dimensional tensor-triangular geometry
  15. Profinite spaces
  16. ...and 38 more sections

Key Result

Theorem A

Let $G$ be a profinite group. The category $\mathsf{Sp}_{G}$ is equivalent to Fausk's model of equivariant $G$-spectra for any profinite group $G$. Moreover, if $G = \mathop{\mathrm{lim}}\nolimits_{i \in I}G_i$ with $G_i$ finite, then the Balmer spectrum is the limit of the spectra for the finite qu

Figures (4)

  • Figure 1: A schematic for $\mathop{\mathrm{Spc}}\nolimits(\mathsf{Sp}_{\mathbb{Z}_p}^\omega)$ with a comparison to the Zariski spectrum of the Burnside ring of $\mathbb{Z}_p$. There is a copy of the contents of the dotted box for each prime $q \neq p$, and these do not interact with one another. The dots in the upper half of the schematic correspond to tt-primes, and lines correspond to inclusions, with smaller primes going up and to the right on the page.
  • Figure 2: A schematic illustrating of the nature of blueshift.
  • Figure 3: The first 3 steps of the construction of Pełczyński space via a middle-thirds method.
  • Figure : A Peasant Woman Binding Sheaves (after Millet), van Gogh, 1889.

Theorems & Definitions (311)

  • Theorem A: \ref{['thm:gsp_contfausk']}
  • Theorem B: \ref{['cor:nilpotence']}
  • Theorem C: \ref{['thm:prism_abelian']}
  • Theorem D: \ref{['cor:sheavescompgen']}, \ref{['thm:eqsheaves_continuity']}
  • Theorem E: \ref{['thm:sheafalgebraicmodel']}
  • Theorem F: \ref{['cor:ttstalkformula']}, \ref{['cor:continuousttstalk']}
  • Theorem G: \ref{['prop:qgsp_ttstalk']}
  • Theorem H: \ref{['thm:scatterediffcountable']}
  • Theorem I: \ref{['thm:qgsp_tc']}
  • Remark 1.2
  • ...and 301 more