Tensor Triangular Geometry of Module Categories over Group Rings
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| Natura: | Recurso digital |
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2025
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| author | SÉRGIO DE ANDRADE, PAULO |
| author_facet | SÉRGIO DE ANDRADE, PAULO |
| contents | This paper explores the intricate structure of module categories over group rings through the lens of tensor triangular geometry. Tensor triangular geometry, pioneered by Paul Balmer, provides a powerful framework for classifying the thick subcategories of certain triangulated categories endowed with a compatible tensor product. We apply this sophisticated machinery to the stable module category of a finite group G over a field k, a fundamental object in modular representation theory. Our investigation aims to delineate the Balmer spectrum of these categories, demonstrating its profound connection to classical invariants such as support varieties arising from the group cohomology ring. We detail the theoretical underpinnings, including the definition of tensor triangulated categories, thick subcategories, and the Balmer spectrum, and then illustrate how these concepts manifest in the context of group rings. The paper establishes a geometric realization of the stable module category's thick subcategories, offering a unified perspective on their classification and revealing deep algebraic-geometric structures. We discuss the implications of these findings for understanding the stable module category and its modules, drawing parallels with other areas of mathematics where tensor triangular geometry has been successfully applied. The research culminates in a comprehensive understanding of how the geometric insights offered by Balmer's theory illuminate the complex world of group representations. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17689507 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Tensor Triangular Geometry of Module Categories over Group Rings SÉRGIO DE ANDRADE, PAULO This paper explores the intricate structure of module categories over group rings through the lens of tensor triangular geometry. Tensor triangular geometry, pioneered by Paul Balmer, provides a powerful framework for classifying the thick subcategories of certain triangulated categories endowed with a compatible tensor product. We apply this sophisticated machinery to the stable module category of a finite group G over a field k, a fundamental object in modular representation theory. Our investigation aims to delineate the Balmer spectrum of these categories, demonstrating its profound connection to classical invariants such as support varieties arising from the group cohomology ring. We detail the theoretical underpinnings, including the definition of tensor triangulated categories, thick subcategories, and the Balmer spectrum, and then illustrate how these concepts manifest in the context of group rings. The paper establishes a geometric realization of the stable module category's thick subcategories, offering a unified perspective on their classification and revealing deep algebraic-geometric structures. We discuss the implications of these findings for understanding the stable module category and its modules, drawing parallels with other areas of mathematics where tensor triangular geometry has been successfully applied. The research culminates in a comprehensive understanding of how the geometric insights offered by Balmer's theory illuminate the complex world of group representations. |
| title | Tensor Triangular Geometry of Module Categories over Group Rings |
| url | https://doi.org/10.5281/zenodo.17689507 |