The motivic tt-geometry of real quadrics

TL;DR

This paper explores the tensor-triangular geometry of motives generated by real quadrics, determining the structure of their spectrum at prime 2 and relating it to real Artin-Tate spectra.

Contribution

It provides the first detailed description of the Balmer spectrum for motives of real quadrics, including its non-Noetherian nature and connections to real algebraic geometry.

Findings

The spectrum at prime 2 is a countably infinite, non-Noetherian space of Krull dimension 2.

Established relationships between the spectrum, real Artin-Tate spectrum, and Vishik's isotropic points.

Full description of the spectrum of integral motives of quadrics over real algebraic numbers.

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.