Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The motivic tt-geometry of real quadrics

Jean Paul Schemeil

Abstract

We study the tensor-triangular geometry of the category of Voevodsky motives generated by real quadrics. At the prime 2, we determine its Balmer spectrum, and find that it is a countably infinite, non-Noetherian space of Krull dimension 2. We detail the relationship between this space, the real Artin-Tate spectrum computed by Balmer-Gallauer, and Vishik's isotropic points. We conclude by combining our computation with Balmer-Gallauer's results on Artin-Tate motives to obtain a full description of the spectrum of integral motives of quadrics over real algebraic numbers.

The motivic tt-geometry of real quadrics

Abstract

We study the tensor-triangular geometry of the category of Voevodsky motives generated by real quadrics. At the prime 2, we determine its Balmer spectrum, and find that it is a countably infinite, non-Noetherian space of Krull dimension 2. We detail the relationship between this space, the real Artin-Tate spectrum computed by Balmer-Gallauer, and Vishik's isotropic points. We conclude by combining our computation with Balmer-Gallauer's results on Artin-Tate motives to obtain a full description of the spectrum of integral motives of quadrics over real algebraic numbers.

Paper Structure

This paper contains 22 sections, 32 theorems, 180 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. tt-geometric recollections
  4. Motivic recollections
  5. Mixed motives of quadrics
  6. Mixed motives of quadrics over real closed fields
  7. Function fields of real anisotropic quadrics
  8. Real isotropic category of quadrics
  9. A jointly conservative family of tt-functors
  10. The Balmer spectrum of some cellular motivic categories
  11. Computation of $\operatorname{Spc}(\mathscr{K}^c_2)$
  12. Computation of $\operatorname{Spc}(\mathscr{K}^c_n)$ for $3\leq n < \infty$
  13. Isotropic case
  14. The prime tt-ideals
  15. Generators and specialisation relations
  16. ...and 7 more sections

Key Result

Theorem 2.1

If a family of geometric functors as above is jointly conservative, then the induced map on spectra is surjective.

Theorems & Definitions (77)

  • Theorem 2.1: barthel2024surjectivity
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3: Definition of $\pi_1$
  • proof
  • Lemma 3.4: Definition of $\pi_2$
  • proof
  • ...and 67 more