Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Periods in equivariant and motivic contexts

Martin Gallauer

TL;DR

The paper defines and develops the period as a multiplicative invariant for stably symmetric monoidal ∞-categories, establishing its basic properties and showing how it stratifies tt-spectra similarly to the characteristic stratification in rings. It then builds a broad toolkit (dévisage, amplification, Zariski descent, and functoriality) to transfer period information across tt-categories and their spectra, and applies these tools to a spectrum of examples. In particular, the work analyzes periods in modular representation theory, derived permutation modules, and motivic tt-geometry, including isotropic points and their mappings between motivic spectra, and demonstrates how periods organize complex phenomena in equivariant and motivic settings. The results illuminate how periodicity interacts with t-structures, weight structures, and orbit-type categories, offering a unifying perspective on the structure of tt-schemes and their points. Overall, the paper provides both foundational theory and concrete computations that advance the understanding of periods as organizing principles in stable homotopy theory and motives.

Abstract

We define the period as a multiplicative characteristic of stably symmetric monoidal $\infty$-categories, develop its basic properties, and study many examples, with a focus on `ordinary' equivariant and motivic homotopy theory. We apply the findings to isotropic points in motivic tt-geometry. (Includes an appendix by Ivo Dell'Ambrogio on generalized comparison maps in tt-geometry.)

Periods in equivariant and motivic contexts

TL;DR

The paper defines and develops the period as a multiplicative invariant for stably symmetric monoidal ∞-categories, establishing its basic properties and showing how it stratifies tt-spectra similarly to the characteristic stratification in rings. It then builds a broad toolkit (dévisage, amplification, Zariski descent, and functoriality) to transfer period information across tt-categories and their spectra, and applies these tools to a spectrum of examples. In particular, the work analyzes periods in modular representation theory, derived permutation modules, and motivic tt-geometry, including isotropic points and their mappings between motivic spectra, and demonstrates how periods organize complex phenomena in equivariant and motivic settings. The results illuminate how periodicity interacts with t-structures, weight structures, and orbit-type categories, offering a unifying perspective on the structure of tt-schemes and their points. Overall, the paper provides both foundational theory and concrete computations that advance the understanding of periods as organizing principles in stable homotopy theory and motives.

Abstract

We define the period as a multiplicative characteristic of stably symmetric monoidal -categories, develop its basic properties, and study many examples, with a focus on `ordinary' equivariant and motivic homotopy theory. We apply the findings to isotropic points in motivic tt-geometry. (Includes an appendix by Ivo Dell'Ambrogio on generalized comparison maps in tt-geometry.)

Paper Structure

This paper contains 24 sections, 43 theorems, 101 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Characteristics
  3. Periods
  4. Motivic tt-geometry
  5. Contents
  6. Acknowledgments
  7. Conventions
  8. Periods of rings…
  9. …and categories
  10. Period stratification
  11. Ample families
  12. Dévissage
  13. Zariski descent
  14. Comparison map
  15. Fully faithful functors
  16. ...and 9 more sections

Key Result

Lemma 2.8

Let $R$ be a graded ring of odd period. Then the commutativity constraint is $\epsilon=1$. In particular, if $R$ is Dirac then $2=0$ in $R$.

Figures (3)

  • Figure 1: The period stratification for $\mathop{\mathrm{stmod}}\nolimits(D_8;k)$. In blue the stratum of period $1$; in red a single point with period $2$.
  • Figure 2: The periodic locus for $\mathop{\mathrm{DPerm}}\nolimits(Q_8;k)^{\omega}$. In blue the stratum of period $1$: a $\mathbb{P}^1_k$ with the $\mathbb{Z}/2$-rational points doubled; in red a single point with period $4$.
  • Figure 3: The periodic locus for $\mathop{\mathrm{DPerm}}\nolimits(D_8;k)^{\omega}$. Blue: period $1$, red: period $2$.

Theorems & Definitions (179)

  • Definition 1.2
  • Example 2.2
  • Definition 2.3
  • Remark 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Lemma 2.8
  • proof
  • Remark 2.9
  • ...and 169 more