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Permutation twisted cohomology, reduced and immersed

  • Juan Omar Gómez
  • Sam K. Miller
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Abstract

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.

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Cite

BibTeXRIS
BibTeX
@misc{gomez2026permutation,
  title = {{Permutation twisted cohomology, reduced and immersed}},
  author = {Juan Omar Gómez and Sam K. Miller},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40602},
  eprint = {2610.40602},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40602},
}
RIS
TY  - UNPB
TI  - Permutation twisted cohomology, reduced and immersed
AU  - Juan Omar Gómez
AU  - Sam K. Miller
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40602
UR  - https://arxiv.org/abs/2610.40602
AB  - 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.
ER  -