发表机构
Fudan University; Westlake University; ShanghaiTech University（复旦大学; 西湖大学; 上海科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究卡尔森关于有限群上同调环深度与关联素数维数关系的猜想，通过列举特定128阶群的二阶初等阿贝尔子群及相关计算，给出该猜想的否定答案，证明该群上同调环无二维关联素数。
AI 中文摘要
1995年卡尔森在关于深度和转移的论文的问题3.1中，询问有限群上同调环的深度是否总是由其某个关联素数的维数实现。我们给出了否定答案。设\(G=\SG{128}{859}\)，\(k=\kbar\)。一个精确的表示证书证明\(\depth H^*(G;k)=2\)。奥久山的关联素数定理会将一个二维关联素数转化为一个二阶初等阿贝尔子群\(E\leq G\)，使得\(\depth H^*(C_G(E);k)=2\)。我们列举了\(G\)的所有75个二阶初等阿贝尔子群并得到六种中心化子类型。杜弗洛定理表明其中四种类型的上同调深度至少为三，而精确的理想商证书显示其余两种类型有长度为三的正则序列。因此每个二阶中心化子的上同调深度至少为三，所以\(H^*(G;k)\)没有二维关联素数。文中还包含有限群表示、三个上同调环表示、枚举总结以及精确代数证书以供独立验证。
英文摘要
In Question~3.1 of his 1995 paper on depth and transfer, Carlson asked whether the depth of a finite-group cohomology ring is always realized by the dimension of one of its associated primes. We give a negative answer. Let \[ G=\SG{128}{859},\qquad k=\kbar. \] An exact presentation certificate proves that $\depth H^*(G;k)=2$. Okuyama's associated-prime theorem would convert an associated prime of dimension two into a rank-two elementary abelian subgroup $E\leq G$ satisfying $\depth H^*(C_G(E);k)=2$. We enumerate all $75$ rank-two elementary abelian subgroups of $G$ and obtain six centralizer types. Duflot's theorem gives depth at least three for four types, while exact ideal-quotient certificates exhibit regular sequences of length three for the remaining two. Hence every rank-two centralizer has cohomological depth at least three, so $H^*(G;k)$ has no associated prime of dimension two. The finite group presentation, the three cohomology-ring presentations, the enumeration summary, and the exact algebraic certificates are included for independent verification.
