AI 中文总结
本文证明忠实平坦Hopf Galois扩张的Ştefan谱序列具有乘法性，引入相关形式体系构造对应谱序列，结合比较定理完成等同，在多类代数结构上得到新乘法性结果。
AI 中文摘要
设H是域k上具有双射反极的Hopf代数，B/A是平坦右H-Galois扩张。Ştefan构造了一个收敛于Hochschild上同调HH^{p+q}(B, N)的谱序列，其E₂^{p,q}=H^p(H, HH^q(A, N))。本文证明当B/A是忠实平坦时，该Ştefan谱序列是乘法的。更准确地说，本文引入了通过幺半范畴上的lax monoidal函子构造函子性乘法谱序列的新形式体系，为任意忠实平坦Hopf Galois扩张构造了函子性乘法谱序列，结合Künzer的比较定理可将其与Ştefan谱序列从E₂页起等同。应用包括强分次代数、smash积、交叉积、群与李代数扩张，在若干情形得到了新的乘法性结果。
英文摘要
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. Ştefan 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 Ştefan 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 Ştefan's spectral sequence from the $E_2$-page follows from Künzer's comparison theorems. Applications include strongly graded algebras, smash products, crossed products, group and Lie algebra extensions, with new multiplicativity results in several cases.
