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Cohomological change of coefficients for noncocommutative Hopf algebras
Abstract
Let A be a finite Hopf algebra over a Noetherian commutative ring R. For any map f : R → k where k is a field, there is a natural map on the cohomology of A under this change of coefficients H∗(f): H∗(A, R) ⊗R k → H∗(Ak, k). In this paper we examine this map for R = ℤ[ζ] the cyclotomic integers with ζ an ℓth root of unity. We show that under a finite generation assumption, the induced map gives a homeomorphism on the spectra of the two cohomology rings. Finally, we turn to small quantum groups defined over ℤ[ζ] and show that, with mild restrictions on ℓ, every fiber map whose residue field has a good prime characteristic induces an isomorphism on cohomology.
Related
- Cohomological Hall algebras and Quot schemes of curves
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- A Lie Correspondence in the Setting of Hopf and Frobenius Algebras
BibTeX
@misc{magland2026cohomological,
title = {{Cohomological change of coefficients for noncocommutative Hopf algebras}},
author = {Charlie Magland},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37693},
eprint = {2610.37693},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37693},
}
RIS
TY - UNPB
TI - Cohomological change of coefficients for noncocommutative Hopf algebras
AU - Charlie Magland
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.37693
UR - https://arxiv.org/abs/2610.37693
AB - Let A be a finite Hopf algebra over a Noetherian commutative ring R. For any map f : R → k where k is a field, there is a natural map on the cohomology of A under this change of coefficients H^∗(f): H^∗(A, R) ⊗_R k → H^∗(A_k, k). In this paper we examine this map for R = ℤ[ζ] the cyclotomic integers with ζ an ℓth root of unity. We show that under a finite generation assumption, the induced map gives a homeomorphism on the spectra of the two cohomology rings. Finally, we turn to small quantum groups defined over ℤ[ζ] and show that, with mild restrictions on ℓ, every fiber map whose residue field has a good prime characteristic induces an isomorphism on cohomology.
ER -