insight - Equivariant Homotopy Theory - # Separable Algebras

Classifying Separable Commutative Algebras in Equivariant Stable Homotopy Theory

Q: How do the findings of this paper extend to more general equivariant settings beyond the specific categories and conditions considered?

While the paper focuses on specific categories like compact G-spectra and derived G-Mackey functors, with G primarily a p-group, the techniques and ideas hint at potential extensions to broader equivariant contexts. Here's a breakdown: Beyond p-groups: The reliance on p-group structure is prominent in utilizing tools like the Segal conjecture and properties of p-completions. Extending to general finite groups necessitates new approaches. One potential avenue is exploring generalizations of the Segal conjecture for families of subgroups, potentially leveraging tools from chromatic homotopy theory. Generalized Geometric Fixed Points: The paper heavily utilizes geometric fixed points. Generalizing the results might involve considering other families of subgroups and their associated fixed point functors. This could lead to modified versions of the indecomposability and retraction conditions tailored to these families. Weakening Conditions: The IC and RC conditions are quite strong. Investigating whether they can be weakened while still yielding a classification is a natural direction. For instance, could one relax the indecomposability requirement to a condition on the Picard group of the category? Beyond Stable Homotopy: The core concepts of separable algebras and their classification translate to other equivariant settings. Exploring analogous questions in equivariant derived algebraic geometry, for instance, could be fruitful. This might involve studying categories of equivariant sheaves or studying equivariant versions of tt-geometry.

Q: Could there be alternative conditions or characterizations that guarantee all separable commutative algebras are standard, even for groups beyond p-groups?

The existence of non-standard separable algebras for groups like C6 suggests that directly extending the p-group results to all finite groups is unlikely without alternative conditions. Here are some potential avenues for exploration: Conditions on the Group: Instead of focusing solely on the algebra, imposing restrictions on the group itself might be fruitful. For instance, could one classify groups where all separable algebras are standard? This might involve properties related to fusion systems, representation theory, or group cohomology. Chromatic Information: The counterexamples for C6 hint that chromatic homotopy theory might play a role. Perhaps conditions involving the Morava K-theories or other chromatic data of the algebra could help distinguish standard and non-standard cases. Categorical Characterizations: Instead of explicit conditions, exploring categorical properties of Perf(A)hG or Perf(A)tG that guarantee the standard classification could be insightful. This might involve studying the Balmer spectrum of these categories, their Picard groups, or their relationship with equivariant derived categories. Connections to Group Actions: The standard separable algebras arise from finite G-sets, which are closely tied to how G acts on objects. Investigating conditions on the action of G on A itself, or on related categories, might lead to a characterization.

Q: What are the implications of the existence of non-standard separable commutative algebras for the broader study of equivariant stable homotopy theory and its applications?

The presence of non-standard separable algebras in equivariant stable homotopy theory has several interesting implications: Richer Étale Topology: In tt-geometry, separable algebras are closely related to the étale topology. The existence of non-standard examples suggests a richer and potentially more intricate étale topology in the equivariant setting, going beyond what is captured by subgroups and their fixed points. Subtleties of Equivariant Descent: Separable algebras are central to descent theory, which studies how to recover global information from local data. The existence of non-standard algebras indicates that equivariant descent is more subtle than the non-equivariant case. It suggests that understanding the "local-to-global" principle in equivariant settings requires a deeper understanding of these exotic separable algebras. Complexity of Mackey Functors: Derived Mackey functors, which are closely related to the categories studied in the paper, play a crucial role in equivariant algebraic topology and representation theory. The existence of non-standard separable algebras in this context suggests a greater complexity in the structure and classification of Mackey functors than previously anticipated. New Invariants and Phenomena: The discovery of non-standard separable algebras hints at the existence of new invariants and phenomena in equivariant stable homotopy theory. Understanding these invariants could provide deeper insights into the structure of equivariant spectra and their relationship with group actions. Overall, the existence of non-standard separable commutative algebras highlights the richness and complexity of equivariant stable homotopy theory. It suggests that the equivariant setting harbors a wealth of new structures and phenomena waiting to be uncovered, with potentially far-reaching consequences for related areas of mathematics.

Core Concepts

This research paper investigates and classifies separable commutative algebras within the framework of equivariant stable homotopy theory, focusing on conditions that guarantee these algebras arise from finite G-sets.

Abstract

Bibliographic Information: Naumann, N., Pol, L., & Ramzi, M. (2024). Separable commutative algebras in equivariant homotopy theory. arXiv preprint arXiv:2411.06845v1.
Research Objective: This paper aims to classify separable commutative algebras in the context of equivariant stable homotopy theory, addressing the question of whether these algebras always arise from finite G-sets (referred to as "standard" algebras).
Methodology: The authors employ techniques from equivariant stable homotopy theory, particularly focusing on geometric fixed points, homotopy fixed points, Tate constructions, and pullback decompositions of equivariant spectra. They introduce three key conditions: the indecomposable condition (IC), the retraction condition (RC), and the separably closed condition.
Key Findings: The paper establishes that if a commutative ring G-spectrum R satisfies the IC, RC, and separably closed conditions for all subgroups of G, then every separable commutative algebra in the category of compact R-modules is standard. This result is particularly significant for p-groups, where it implies that all separable commutative algebras are standard in both compact G-spectra and compact derived G-Mackey functors.
Main Conclusions: The authors conclude that the classification of separable commutative algebras in equivariant stable homotopy theory is intricately linked to the properties of geometric fixed points and the behavior of certain key conditions. They demonstrate that while all separable algebras are standard for p-groups under specific conditions, this does not hold true in general, as illustrated by counterexamples for the group C6.
Significance: This research significantly advances the understanding of separable commutative algebras in equivariant stable homotopy theory, providing a framework for their classification and highlighting the crucial role of geometric fixed points. The results have implications for the study of Galois groups and the interplay between algebra and topology in equivariant contexts.
Limitations and Future Research: The study primarily focuses on specific conditions and categories within equivariant stable homotopy theory. Further research could explore the classification of separable commutative algebras in more general settings, investigate the implications of these findings for other equivariant contexts, and delve deeper into the counterexamples presented to understand the limitations of the standard classification.

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Separable commutative algebras in equivariant homotopy theory

by Niko Naumann... at arxiv.org 11-12-2024

https://arxiv.org/pdf/2411.06845.pdf

Separable commutative algebras in equivariant homotopy theory

Deeper Inquiries

How do the findings of this paper extend to more general equivariant settings beyond the specific categories and conditions considered?

While the paper focuses on specific categories like compact G-spectra and derived G-Mackey functors, with G primarily a p-group, the techniques and ideas hint at potential extensions to broader equivariant contexts. Here's a breakdown:

Beyond p-groups: The reliance on p-group structure is prominent in utilizing tools like the Segal conjecture and properties of p-completions. Extending to general finite groups necessitates new approaches. One potential avenue is exploring generalizations of the Segal conjecture for families of subgroups, potentially leveraging tools from chromatic homotopy theory.

Generalized Geometric Fixed Points: The paper heavily utilizes geometric fixed points.  Generalizing the results might involve considering other families of subgroups and their associated fixed point functors. This could lead to modified versions of the indecomposability and retraction conditions tailored to these families.

Weakening Conditions: The IC and RC conditions are quite strong. Investigating whether they can be weakened while still yielding a classification is a natural direction. For instance, could one relax the indecomposability requirement to a condition on the Picard group of the category?

Beyond Stable Homotopy: The core concepts of separable algebras and their classification translate to other equivariant settings. Exploring analogous questions in equivariant derived algebraic geometry, for instance, could be fruitful. This might involve studying categories of equivariant sheaves or studying equivariant versions of tt-geometry.

Could there be alternative conditions or characterizations that guarantee all separable commutative algebras are standard, even for groups beyond p-groups?

The existence of non-standard separable algebras for groups like C6 suggests that directly extending the p-group results to all finite groups is unlikely without alternative conditions. Here are some potential avenues for exploration:

Conditions on the Group:  Instead of focusing solely on the algebra, imposing restrictions on the group itself might be fruitful. For instance, could one classify groups where all separable algebras are standard? This might involve properties related to fusion systems, representation theory, or group cohomology.

Chromatic Information: The counterexamples for C6 hint that chromatic homotopy theory might play a role. Perhaps conditions involving the Morava K-theories or other chromatic data of the algebra could help distinguish standard and non-standard cases.

Categorical Characterizations:  Instead of explicit conditions, exploring categorical properties of Perf(A)hG or Perf(A)tG that guarantee the standard classification could be insightful. This might involve studying the Balmer spectrum of these categories, their Picard groups, or their relationship with equivariant derived categories.

Connections to Group Actions:  The standard separable algebras arise from finite G-sets, which are closely tied to how G acts on objects.  Investigating conditions on the action of G on A itself, or on related categories, might lead to a characterization.

What are the implications of the existence of non-standard separable commutative algebras for the broader study of equivariant stable homotopy theory and its applications?

The presence of non-standard separable algebras in equivariant stable homotopy theory has several interesting implications:

Richer Étale Topology: In tt-geometry, separable algebras are closely related to the étale topology. The existence of non-standard examples suggests a richer and potentially more intricate étale topology in the equivariant setting, going beyond what is captured by subgroups and their fixed points.

Subtleties of Equivariant Descent: Separable algebras are central to descent theory, which studies how to recover global information from local data. The existence of non-standard algebras indicates that equivariant descent is more subtle than the non-equivariant case. It suggests that understanding the "local-to-global" principle in equivariant settings requires a deeper understanding of these exotic separable algebras.

Complexity of Mackey Functors: Derived Mackey functors, which are closely related to the categories studied in the paper, play a crucial role in equivariant algebraic topology and representation theory. The existence of non-standard separable algebras in this context suggests a greater complexity in the structure and classification of Mackey functors than previously anticipated.

New Invariants and Phenomena:  The discovery of non-standard separable algebras hints at the existence of new invariants and phenomena in equivariant stable homotopy theory. Understanding these invariants could provide deeper insights into the structure of equivariant spectra and their relationship with group actions.
Overall, the existence of non-standard separable commutative algebras highlights the richness and complexity of equivariant stable homotopy theory. It suggests that the equivariant setting harbors a wealth of new structures and phenomena waiting to be uncovered, with potentially far-reaching consequences for related areas of mathematics.