arXiv:2610.37693v1 Announce Type: new Abstract: Let $A$ be a finite Hopf algebra over a Noetherian commutative ring $R$. For any map $f : R \to k$ where $k$ is a field, there is a natural map on the cohomology of $A$ under this change of coefficients $H^\ast(f): H^\ast(A, R) \otimes_R k \to H^\ast(A_k, k)$. In this paper we examine this map for $R = \mathbb Z[\zeta]$ the cyclotomic integers with $\zeta$ an $\ell$th 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 $\mathbb Z[\zeta]$ and show that, with mild restrictions on $\ell$, every fiber map whose residue field has a good prime characteristic induces an isomorphism on cohomology.
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