On the tensor-triangular geometry of isotropic motives

TL;DR

This paper explores the tensor-triangular geometry of categories of isotropic motives over a flexible field with $\

Contribution

It proves the Balmer spectra of certain isotropic motive categories are singletons and verifies Balmer's Nerves of Steel conjecture for these cases.

Findings

Balmer spectra are singletons for the studied categories.

Verified Balmer's Nerves of Steel conjecture in these cases.

Provided explicit descriptions of homological primes.

Abstract

We investigate the tensor-triangular geometry of the categories of isotropic Tate motives, isotropic Artin motives and isotropic Artin--Tate motives. In particular, we study the categories $DTM_{gm}(k/k;\mathbb{F}_2)$, $DAM_{gm}(k/k;\mathbb{F}_2)$ and $DATM_{gm}(k/k;\mathbb{F}_2)$ where $k$ is a flexible field and we fix $\mathbb{F}_2$-coefficients. In this case, we prove their Balmer spectra are singletons. The proof of this relies on the fact that the categories in question are generated by objects whose morphisms are `nilpotent enough.' Furthermore, we investigate their homological spectra, and verify Balmer's Nerves of Steel conjecture in these cases by showing these spaces are also singletons, and we give an explicit description of the homological primes. We also investigate stratification, and we conclude by studying generation properties of these categories -- the latter results we see also apply to global (i.e. non-isotropic) motives.