发表机构
Eötvös Loránd University; UAEU; NTNU（厄特沃什·罗兰大学; 阿联酋大学; 挪威科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究引入伽罗瓦上链代数的弱A_n-形式性，刻画有限群的A_n-形式性，证明数域在特定素数条件下有限A_3-嵌入问题可解但无相容解族，揭示真正A_3-形式性可检测全局相容性阻碍。
AI 中文摘要
我们引入伽罗瓦上链代数的弱A_n-形式性，对于模p上同调集中在次数不超过2的有限群，通过有限嵌入问题及其构成的射影系统刻画弱与真正的A_n-形式性。随后我们研究数域的A_n-形式性：设K为数域，p为奇素数且μ_p⊄K，当p≥5时，以及在自然局部分圆条件下p=3时，我们证明每个相关的有限A_3-嵌入问题均可解；与之相对，对每个满足μ_p⊄K的奇素数p，我们证明不存在有限阶解的相容族，因此C^•(G_K,𝔽_p)不是真正的A_3-形式。该证明利用切博塔廖夫密度、支配域及Gras-Munnier准则来阻碍有限阶解的相容性，且基于Maire-Mináč-Ramakrishna-Tân的前期工作。这表明，真正的A_3-形式性可检测数域的全局相容性阻碍，该阻碍对每个单独的有限嵌入问题不可见，也对单个强Massey消失问题产生的阻碍不可见。
英文摘要
We introduce weak $A_n$-formality for Galois cochain algebras and, for profinite groups whose mod-$p$ cohomology is concentrated in degrees at most two, characterise weak and genuine $A_n$-formality through finite embedding problems and the projective system they form. We then study $A_n$-formality for number fields. Let $K$ be a number field and let $p$ be an odd prime with $μ_p\not\subset K$. For $p\geq 5$, and for $p=3$ under a natural local cyclotomic condition, we prove that every associated finite $A_3$-embedding problem is solvable. In contrast, for every odd $p$ with $μ_p\not\subset K$, we show that no compatible family of finite-stage solutions exists, and hence $C^\bullet(G_K,\mathbb F_p)$ is not genuinely $A_3$-formal. The proof uses Chebotarev density, governing fields, and the Gras-Munnier criterion to obstruct the compatibility of finite-stage solutions, and it builds on previous work of Maire-Mináč-Ramakrishna-Tân. This shows that genuine $A_3$-formality detects a global compatibility obstruction for number fields invisible to every individual finite embedding problem and invisible to obstructions arising from individual strong Massey vanishing problems.
Comments30 pages, comments very welcome
