The singularity category and duality for complete intersection groups

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.]