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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2504.03050 |
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| _version_ | 1866915961512132608 |
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| author | Greenlees, J. P. C. |
| author_facet | Greenlees, J. P. C. |
| contents | If G is a finite group, some aspects of the modular representation theory depend on the cochains C^*(BG; k), viewed as a commutative ring spectrum. We consider its singularity category (in the sense of the author and Stevenson arxiv 1702.07957) and show that it is the bounded derived category of the Ω-Tate ring spectrum (k-nullification of the Koszul dual, C_*(ΩBG_p)). We establish a form of Gorenstein duality for C_*(ΩBG_p) and a form of Tate duality for the Ω-Tate homology. If C^*(BG; k) is a homotopical complete intersection in a strong sense there is a stable Koszul complex construction of the Ω-Tate spectrum.
[v3: (1) role of ci condition clarified.(2) Novel statements flagged, Ω-Tate named and highlighted.(3) Study of the norm map expanded.] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_03050 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The singularity category and duality for complete intersection groups Greenlees, J. P. C. Algebraic Topology Group Theory 20J06, 55P43 If G is a finite group, some aspects of the modular representation theory depend on the cochains C^*(BG; k), viewed as a commutative ring spectrum. We consider its singularity category (in the sense of the author and Stevenson arxiv 1702.07957) and show that it is the bounded derived category of the Ω-Tate ring spectrum (k-nullification of the Koszul dual, C_*(ΩBG_p)). We establish a form of Gorenstein duality for C_*(ΩBG_p) and a form of Tate duality for the Ω-Tate homology. If C^*(BG; k) is a homotopical complete intersection in a strong sense there is a stable Koszul complex construction of the Ω-Tate spectrum. [v3: (1) role of ci condition clarified.(2) Novel statements flagged, Ω-Tate named and highlighted.(3) Study of the norm map expanded.] |
| title | The singularity category and duality for complete intersection groups |
| topic | Algebraic Topology Group Theory 20J06, 55P43 |
| url | https://arxiv.org/abs/2504.03050 |