The Spectrum of the Derived Category of Artin Motives: A tt-Geometric Analysis via Modular Representation Theory

Core Concepts

This paper explores the tensor-triangular geometry of the derived category of Artin motives, leveraging modular representation theory of profinite groups to classify its thick tensor-ideals and understand its spectral structure.

Abstract

• Bibliographic Information: Balmer, P., & Gallauer, M. (2024). THE SPECTRUM OF ARTIN MOTIVES. arXiv preprint arXiv:2401.02722.
• Research Objective: This paper aims to classify the thick tensor-ideals of the derived category of Artin motives, denoted DAMgm(F; k), where F is a base field and k is a field of coefficients. The authors approach this problem through the lens of tensor-triangular geometry and utilize tools from modular representation theory of profinite groups.
• Methodology: The authors leverage the equivalence between DAMgm(F; k) and the bounded homotopy category of finitely generated permutation kG-modules, denoted K(G; k), where G is the absolute Galois group of F. They employ the continuity result of [Gal18] to express the spectrum of K(G; k) as a limit of spectra of K(G/N; k) for open normal subgroups N of G. They then utilize the modular H-fixed points functor, ΨH, to analyze the spectrum and classify the thick tensor-ideals.
• Key Findings:
  • The paper provides a complete classification of the tt-primes in K(G; k) for any profinite group G.
  • It establishes that when the characteristic of k is 0, the spectrum of K(G; k) consists of a single point.
  • When the characteristic of k is p > 0, every tt-prime is uniquely determined (up to G-conjugation) by a closed pro-p-subgroup H of G and a homogeneous prime in the cohomology of the Weyl group of H.
  • The paper explicitly computes the spectrum of DAMgm(F; k) when F is a finite field, demonstrating that it is the Alexandroff extension of a specific space W∞.
  • Notably, the authors prove that the big derived category of Artin motives, DAM(F; k), is stratified in the sense of Barthel-Heard-Sanders when F is a finite field.
• Main Conclusions: The study reveals a rich and intricate structure within the derived category of Artin motives. The classification of thick tensor-ideals provides a deeper understanding of this category and its spectral properties. The stratification result for finite base fields highlights the well-behaved nature of Artin motives in this setting.
• Significance: This research significantly contributes to the field of motivic homotopy theory by providing a comprehensive analysis of the tensor-triangular geometry of Artin motives. The explicit computations and classification results offer valuable insights into the structure and properties of these motives.
• Limitations and Future Research: While the paper focuses on Artin motives, it acknowledges the broader challenge of understanding the motivic derived category DM(F) and the motivic stable homotopy category SH(F) from a tensor-triangular perspective. Further research could explore the generalization of these results to these larger categories. Additionally, investigating the behavior of stratification under colimits in more general settings could provide further insights into the structure of derived categories.

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The spectrum of Artin motives

by Paul Balmer,... at arxiv.org 11-06-2024

https://arxiv.org/pdf/2401.02722.pdf

Deeper Inquiries

How do the techniques used in this paper for analyzing Artin motives extend to understanding the tt-geometry of more general motivic categories like DM(F) or SH(F)?

While the techniques in the paper provide a powerful lens for analyzing Artin motives, extending them to more general motivic categories like DM(F) or SH(F) presents significant challenges. Here's why:

Increased Complexity: DM(F) and SH(F) are vastly more complex than the category of Artin motives. They encompass a broader range of objects and possess richer structures. For instance, DM(F) includes motives of all varieties, not just zero-dimensional ones, and SH(F) incorporates the intricate world of motivic homotopy theory.

Lack of Direct Galois Action: A key ingredient in the analysis of Artin motives is their close connection to representations of the absolute Galois group. This connection is less direct for DM(F) and SH(F). While there are Galois actions on certain realizations of motives, these actions don't translate straightforwardly to the level of the triangulated categories themselves.

Absence of Explicit Generators: The paper heavily relies on the explicit generators of the category of Artin motives, namely the permutation modules. Finding analogous generators for DM(F) or SH(F) is a major open problem. Without such generators, directly applying the techniques of restriction and modular fixed points becomes difficult.
However, despite these challenges, the paper offers valuable insights that could guide future research:

Inspiration for New Techniques: The success of restriction and modular fixed points in the context of Artin motives suggests exploring analogous techniques tailored to the structures of DM(F) and SH(F). For example, one might investigate restrictions to suitable subcategories or develop modular fixed points functors for motivic actions of pro-p-groups.

Understanding Building Blocks: Artin motives can be viewed as fundamental building blocks within DM(F) and SH(F). A thorough understanding of their tt-geometry could provide a foundation for studying the tt-geometry of the larger categories.

Stratification as a Guiding Principle: The stratification result for Artin motives over finite fields raises the question of whether similar stratification phenomena occur in DM(F) or SH(F). While proving such results would be highly non-trivial, the possibility of stratification could serve as a guiding principle for investigating the spectrum and support theory of these categories.

Could there be alternative characterizations of the spectrum of K(G; k) that avoid the reliance on pro-elementary abelian groups or twisted cohomology, potentially leading to a more streamlined description?

Finding alternative characterizations of Spc(K(G; k)) that bypass pro-elementary abelian groups or twisted cohomology is an intriguing open question. While the paper provides a concrete description, a more streamlined approach would be desirable. Here are some potential avenues for exploration:

Exploiting Categorical Properties: Instead of relying on external tools like group cohomology, one could try to characterize the spectrum directly from the categorical properties of K(G; k). This might involve identifying specific objects or morphisms that detect the tt-primes or leveraging the structure of the tensor product and internal Hom.

Connections to Representation Theory: Deeper connections between the spectrum and the representation theory of G could lead to alternative descriptions. For instance, exploring the relationship between tt-primes and specific classes of indecomposable or irreducible representations might offer new perspectives.

Geometric Interpretations: In the case of Artin motives, the spectrum has a geometric interpretation in terms of points of the scheme Spec(Z) and the behavior of motives over different finite fields. Seeking analogous geometric interpretations for general profinite groups could provide more intuitive characterizations.

Alternative Cohomology Theories: While the paper focuses on ordinary group cohomology, exploring other cohomology theories for profinite groups, such as continuous cohomology or profinite group homology, might offer different perspectives on the spectrum.
Finding such alternative characterizations would not only provide a more elegant description of Spc(K(G; k)) but could also shed new light on the interplay between representation theory, tt-geometry, and the structure of profinite groups.

What are the implications of the stratification result for DAM(F; k) when F is a finite field for other aspects of motivic homotopy theory, such as the study of motivic cohomology or motivic Steenrod operations?

The stratification result for DAM(F; k) when F is a finite field has potentially significant implications for other aspects of motivic homotopy theory:

Understanding Motivic Cohomology: Stratification provides a powerful tool for studying the structure of categories through their localizing subcategories. This could lead to a deeper understanding of motivic cohomology, which is representable in the motivic stable homotopy category SH(F). By analyzing the localizations of DAM(F; k), one might gain insights into the behavior of motivic cohomology groups and their relations to the tt-geometry of Artin motives.

Analyzing Motivic Steenrod Operations: Motivic Steenrod operations are cohomology operations acting on motivic cohomology groups. These operations play a crucial role in understanding motivic phenomena. The stratification of DAM(F; k) could provide a framework for studying the action of motivic Steenrod operations on the motivic cohomology of Artin motives. This could potentially lead to new insights into the structure of these operations and their interplay with the tt-geometry of motives.

Connections to Other Motivic Categories: The category DAM(F; k) embeds into larger motivic categories like DM(F) and SH(F). The stratification result raises the question of whether and how this stratification interacts with the structure of these larger categories. For instance, one might investigate whether the localizations of DAM(F; k) induce meaningful localizations of DM(F) or SH(F), potentially providing a way to decompose these complex categories into more manageable pieces.

Computational Tools: Stratification often leads to powerful computational tools. The explicit description of the spectrum and the classification of localizing subcategories in DAM(F; k) could facilitate computations of motivic cohomology groups, motivic Steenrod operations, and other motivic invariants for Artin motives.
In summary, the stratification result for DAM(F; k) provides a new lens through which to view and analyze various aspects of motivic homotopy theory. It has the potential to deepen our understanding of motivic cohomology, motivic Steenrod operations, and the relationships between different motivic categories, as well as to provide new computational tools for studying these objects.