Bousfield--Kan 完备化在次可缩表示中的应用

中文摘要 AI 辅助

我们通过自由单纯群的消解、其过滤谱序列以及一个非交换算术方图之间的相互作用来研究 Bousfield--Kan 完备化。对于每个有限型的自由离散单纯群，我们将其整式幂零完备化表示为有理幂幺完备化与所有 pro-$p$ 完备化的乘积在一个显式 adelic 单纯群上的同伦拉回。adelic 项通过在有限幂零阶段取限制积，然后取其逆向极限而构成；对原始分量群不要求幂零性假设。可缩表示的有限次表示为此构造提供了一个显式应用。指定关系子的独立性使得有理和 mod-$p$ 过滤谱序列的正次数项消失，并在商塔上验证了收敛性。自由单纯消解的连续比较在特征零下通过该范畴中的态射和同伦实现了与常值自由幂幺群的等价。对于相应的表示复形 $K$，我们得到 $R_\infty K\simeq K(F_R(Z),1)$，其中 $R=\mathbb Q,\mathbb F_p,\mathbb Z$，$Z$ 索引一个互补基，$F_R(Z)$ 分别表示自由幂幺群的有理点、自由 pro-$p$ 群或自由幂零群。在幂零阶段的相容收缩在此情形下识别了算术方图的所有四个条目。

英文摘要

We study Bousfield--Kan completions through the interaction of free simplicial resolutions, their filtration spectral sequences, and a noncommutative arithmetic square. For every free discrete simplicial group of finite type, we express its integral pronilpotent completion as the homotopy pullback of its rational prounipotent completion and the product of its pro-$p$ completions over an explicit adelic simplicial group. The adelic entry is formed by taking restricted products at finite nilpotent stages and then their inverse limit; no nilpotency assumption on the original group of components is required. Finite subpresentations of contractible presentations provide an explicit application of this construction. Independence of the specified relators makes the positive-degree terms of the rational and mod-$p$ filtration spectral sequences vanish, with convergence verified on the quotient towers. Continuous comparison of free simplicial resolutions then realizes, in characteristic zero, the equivalence with a constant free prounipotent group by morphisms and homotopies in that category. For the corresponding presentation complex $K$ we obtain $R_\infty K\simeq K(F_R(Z),1)$ for $R=\mathbb Q,\mathbb F_p,\mathbb Z$, where $Z$ indexes a complementary basis and $F_R(Z)$ denotes, respectively, the rational points of a free prounipotent group, a free pro-$p$ group, or a free pronilpotent group. Compatible contractions at the nilpotent stages identify all four entries of the arithmetic square in this case.

发表机构

• Moscow Center for Fundamental and Applied Mathematics, Lomonosov Moscow State University（莫斯科基础与应用数学中心，莫斯科国立大学）

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