AI 中文总结
本文证明Ash-Miller-Patzt与Brown-Chan-Galatius-Payne独立构造的Steinberg同调上的双分次交换Hopf代数结构相同，二者均由Steinberg模上的join与抛物限制运算诱导。
AI 中文摘要
两个近期独立构造在具有有理Steinberg系数的$\mathrm{GL}(\mathbb{Z})$同调上给出了双分次交换Hopf代数结构。其一由Ash--Miller--Patzt提出，使用sharbly分解；另一由Brown--Chan--Galatius--Payne提出，使用Quillen谱序列及$BK(\mathbb{Z})$上的秩过滤映射。我们证明这两个结构一致。两者均由Steinberg模上的join与抛物限制运算诱导。
英文摘要
Two recent constructions independently give bigraded commutative Hopf algebra structures on the homology of $\mathrm{GL}(\mathbb{Z})$ with rational Steinberg coefficients. One, by Ash--Miller--Patzt, uses the sharbly resolution. The other, by Brown--Chan--Galatius--Payne, uses the Quillen spectral sequence and rank-filtered maps on $BK(\mathbb{Z})$. We prove that the two structures coincide. Both are induced by join and parabolic restriction operations on Steinberg modules.
Comments17 pages
