Lazard 关于 N-序列的实现问题：Bar 障碍与有限商下降
Lazard's Realization Problem for N-Series: Bar Obstructions and Finite-Quotient Descent
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中文总结 AI 辅助
本文研究 Lazard 关于群 N-序列由整数群环滤过实现的问题，通过 bar 群余核刻画核，给出有限群的可判定性，并构造反例说明有限商下降可能失败。
中文摘要 AI 辅助
Lazard 提出了一个问题：一个群的哪些 N-序列可由整数群环的乘法滤过产生。给定群 G 的一个 N-序列 H_•，设 A_• 为增广理想的滤过，由元素 x-1（x∈H_a）的乘积张成，并按 a 加权。每个实现滤过都包含 A_•，因此 H_• 可实现当且仅当映射 H_n/H_{n+1}→A_n/A_{n+1}，h↦h-1，是单射。我们证明每个核都是某个显式映射的规范化 bar 群之余核；对于有限群，这使得可实现性可通过整数线性代数判定。对于具有共尾有限商塔的 profinite 群，若元素 h∈H_n 在每个有限商中满足 h-1∈A_{n+1}，则其在离散群环中满足 h-1∈A_{n+1} 当且仅当其 Losey 表达式的宽度有界。此下降可能失败。利用 Tahara 的三类 2-群以及 Hartl-Mikhailov-Passi 对第四维子群的描述，我们构造了一个有限 2-群的可数乘积 P 和一个元素 h∈γ_3(P)，该元素位于 P 的每个有限子乘积的第四维子群中，但不在 D_4(P) 中。
英文摘要
Lazard asked which $N$-series of a group arise from multiplicative filtrations of the integral group ring. Given an $N$-series $H_\bullet$ of a group $G$, let $A_\bullet$ be the filtration of the augmentation ideal spanned by products of elements $x-1$, $x\in H_a$, weighted by $a$. Every realizing filtration contains $A_\bullet$, so $H_\bullet$ is realizable exactly when the maps $H_n/H_{n+1}\to A_n/A_{n+1}$, $h\mapsto h-1$, are injective. We show that each kernel is the cokernel of an explicit map of normalized bar groups; for a finite group this makes realizability decidable by integer linear algebra. For a profinite group with a cofinal tower of finite quotients and an $N$-series induced from it, suppose $h\in H_n$ and $h-1\in A_{n+1}$ in every finite quotient. Then $h-1\in A_{n+1}$ in the discrete group ring exactly when the widths of its finite-level Losey expressions are bounded. This descent can fail. Using Tahara's class-three $2$-groups and the Hartl-Mikhailov-Passi description of the fourth dimension subgroup, we construct a countable product $P$ of finite $2$-groups. It contains an element of $γ_3(P)$ that lies in the fourth dimension subgroup of every finite subproduct of $P$ but not in $D_4(P)$.
