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arXiv · 2608.20900

\c{S}tefan spectral sequence for faithfully flat Hopf Galois extensions is multiplicative

Liyu Liu·Hongguang Nie·Guodong Zhou·Ruipeng Zhu
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Abstract

Let $H$ be a Hopf algebra with a bijective antipode over a field $\mathbf k$ and $B/A$ be a flat right $H$-Galois extension. \c{S}tefan constructed a spectral sequence converging to the Hochschild cohomology $\mathrm{HH}^{p+q}(B, N)$ with $\mathrm{E}_2^{p,q} = \mathrm{H}^p(H, \mathrm{HH}^q(A, N))$. We show that when $B/A$ is faithfully flat, the \c{S}tefan spectral sequence is multiplicative. More precisely, a new formalism of functorial multiplicative spectral sequences via lax monoidal functors over monoidal categories is introduced. A functorial multiplicative spectral sequence is constructed for any faithfully flat Hopf Galois extension. Identification with \c{S}tefan's spectral sequence from the $E_2$-page follows from K\"unzer's comparison theorems. Applications include strongly graded algebras, smash products, crossed products, group and Lie algebra extensions, with new multiplicativity results in several cases.

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Liyu Liu, Hongguang Nie, Guodong Zhou, Ruipeng Zhu. 2026-08-21. \c{S}tefan spectral sequence for faithfully flat Hopf Galois extensions is multiplicative. https://arxiv.org/abs/2608.20900

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