On the tensor-triangular geometry of isotropic motives

TL;DR

The paper analyzes the tensor-triangular geometry of isotropic motives over flexible fields with $\mathbb{F}_2$-coefficients, proving that the Balmer spectra of isotropic Tate, Artin, and Artin-- Tate motives are singleton, and that their homological spectra likewise reduce to a single prime via the weight-structure framework. It validates Balmer's Nerves of Steel conjecture in these cases and characterizes the corresponding homological primes, while also establishing non-stratification in the big isotropic categories and detailing generation properties and infinite Rouquier dimensions. The methodology hinges on strongly weight-nilpotent generators and explicit control of isotropic motivic cohomology (notably the exterior algebra on Milnor-type generators) to constrain tt-ideals. The results illuminate how isotropisation simplifies tt-geometry and provide insights relevant to the broader aim of understanding Voevodsky's motives, with implications for generation and stratification beyond the isotropic setting. Overall, the work yields a clear, tractable picture of the tt-geometry in these isotropic settings and connects it to classical problems in motivic triangulated categories.

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.

Figures (1)

• Figure 1: The coordinates of the associated graded pieces of $X^{\otimes n }$ lie above $Chow_n$ and have $(a)$-coordinate bounded between $-Mn$ and $Mn$ for some fixed $M$. The negation of the coordinates of the isotropic motivic cohomology groups of the point $\operatorname{Spec} (k)$ are bounded above by the green dashed line. For large enough $n$, these regions are disjoint.

Theorems & Definitions (91)

• Theorem 1.2: \ref{['cor:spectrumofDTM']}
• Theorem 1.3: \ref{['theorem:spectraofDAM/DATM']}
• Theorem 1.4: \ref{['cor:NoSholds']}
• Theorem 1.5: \ref{['prop:stratification']}
• Theorem 1.6: \ref{['prop:DTMnotfinitelygenerated']}
• Example 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5