Permutation twisted cohomology, reduced and immersed

In this paper, we introduce a `reduced' form of permutation twisted cohomology as introduced by Balmer--Gallauer and further developed by Miller. This ring is bigraded by a well-chosen choice of endotrivial complex, rather than multigraded by a family of endotrivial complexes. The Balmer spectrum of the small derived category of permutation modules openly immerses into the homogeneous spectrum of this ring. We also show that for $p$-groups, Miller's `remixed' twisted cohomology ring also admits the Balmer spectrum as an open subspace, and confirm in both cases it can be upgraded to have Dirac scheme structure; furthermore, we prove that the two open covers witnessing this are in general different.