Permutation twisted cohomology, remixed

For each endotrivial complex arising from Bredon homology of a representation sphere, we construct $p$-local quasi-isomorphisms, called forerunners, enabling us to extend Balmer–Gallauer’s results in arXiv:2307.04398 Part II 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 line bundles, that is, every endotrivial is locally isomorphic to a shifted 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 all $p$-groups of order at most $p^3$, and conjecture Noetherianity holds for all finite $p$-groups.