Permutation twisted cohomology, remixed

Abstract

For each endotrivial complex for a $p$-group arising from Bredon homology of a representation sphere, we construct $p$-local quasi-isomorphisms, called forerunners. These enable us to extend Balmer--Gallauer's results in arXiv:2307.04398 concerning the tensor-triangular geometry of permutation modules for elementary abelian $p$-groups to all $p$-groups. We construct an open cover of the Balmer spectrum under which all endotrivials are tt-line bundles, that is, every endotrivial is locally isomorphic to a shift of the tensor unit. We define a remixed permutation twisted cohomology ring for which the canonical comparison map from the Balmer spectrum to the homogeneous spectrum of the twisted cohomology ring is injective. If the twisted cohomology ring is Noetherian, the comparison map is an open immersion, and the open cover endows the Balmer spectrum with Dirac scheme structure. We prove Noetherianity holds for Dedekind groups and the dihedral group of order 8, and conjecture it holds for all $p$-groups.