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arXiv · 2504.03050

The singularity category and duality for complete intersection groups

J. P. C. Greenlees
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Abstract

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 \Omega-Tate ring spectrum (k-nullification of the Koszul dual, C_*(\Omega BG_p)). We establish a form of Gorenstein duality for C_*(\Omega BG_p) and a form of Tate duality for the \Omega-Tate homology. If C^*(BG; k) is a homotopical complete intersection in a strong sense there is a stable Koszul complex construction of the \Omega-Tate spectrum. [v3: (1) role of ci condition clarified.(2) Novel statements flagged, \Omega-Tate named and highlighted.(3) Study of the norm map expanded.]

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J. P. C. Greenlees. 2025-04-03. The singularity category and duality for complete intersection groups. https://arxiv.org/abs/2504.03050

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