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Finite generation of permutation twisted cohomology
Representation spheres supply twists in the homotopy category of permutation modules. Maps from the tensor unit into these sphere complexes assemble into multigraded section rings. The paper proves that these rings are finite as modules over explicit finitely generated central commutative coefficient algebras.
The theorem holds for every finite group, every finite family of actual finite-dimensional real representations, and every commutative Noetherian base. It also controls subgroup restrictions, coordinate faces, and bounded permutation coefficient complexes. For finite p-groups in defining characteristic, it establishes Noetherianity of the full canonical effective permutation twisted cohomology ring.
Over a field of positive characteristic, the doubled regular representation gives a model with one twist coordinate. Its fully homogeneous spectrum contains the Balmer spectrum of permutation complexes as the complement of one closed point. Suitable homogeneous localizations, followed by passage to twist degree zero, give affine Dirac charts.
Finiteness and geometric information
A coefficient algebra can control a section ring by module-finiteness while identifying distinct geometric points. For unit coefficients over a nonzero base, the paper determines the minimum number of fully homogeneous coefficient generators of nonzero twist. Homogeneity includes homological degree and every formal twist coordinate.
For the doubled regular representation over a field of characteristic p, cyclic p-groups admit an explicit Stanley-Reisner coefficient subring attaining this minimum and recovering the topology. For noncyclic p-groups, every coefficient algebra attaining the minimum identifies distinct closed subgroup points.
Reading the paper
Begin with the introductory cyclic example and Example 6.4. Over a field k of characteristic p, a rotation plane for Cp, with p odd, gives a three-term sphere complex. In twist degree one, its invariant-chain homology has classes a, c, b in degrees 0, 1, 2. The full section ring is finite over k[a,b], although c lies outside this even-degree subalgebra.
Sections 2-6 develop sphere complexes and the signed-cone induction; section 7 treats Noetherian bases and coefficient counts. Section 8 proves the geometric comparison and cyclic presentation. Appendix A studies restriction and multiplicative lifting.
The proof uses group representations, chain complexes, and commutative algebra; the geometric comparison uses Brauer quotients and tensor-triangular geometry.
Related reading
Quotients, section 10 studies nilpotent operators through spaces of maps. In characteristic p, kCp ≅ k[u]/(up), with u = g - 1. Derivative, section 6 gives a worked example of homology created by a derived fiber.
Related essay: When a Description Becomes a Verdict.
Citation
Henry Shin, Finite generation of permutation twisted cohomology, version 2, Zenodo, 28 October 2026. doi:10.5281/zenodo.23749631.