推翻了关于群上同调深度的长期猜想，给出一个128阶群的反例。
An Exact Counterexample to Carlson's Associated-Prime Depth Conjecture from a Group of Order 128
- 构造128阶群G=SG(128,859)，证明其上同调深度为2。
- 验证所有秩2初等阿贝尔子群的中心化子深度均≥3，无维度为2的关联素理想。
- 提供完整代数证明与可验证证书，适合代数拓扑与群表示研究者。
在1995年关于深度与传递的论文中，Carlson提出问题：有限群上同调环的深度是否总由某个关联素的理想维度实现？本文给出否定回答。令 $ G = ext{SG}(128,859) $，$ k = \overline{k} $，精确证明 $ \depth H^*(G;k) = 2 $。Okuyama的关联素定理表明，若存在维度为2的关联素，则应存在秩2初等阿贝尔子群 $ E \leq G $，使得 $ \depth H^*(C_G(E);k) = 2 $。我们枚举了G中全部75个秩2初等阿贝尔子群，得到六类中心化子；其中四类由Duflot定理保证深度≥3，其余两类通过精确理想商证书构造出长度为3的正则序列。因此所有秩2中心化子深度均≥3，说明 $ H^*(G;k) $ 不存在维度为2的关联素。论文包含群的精确表示、上同调环的三份呈现、枚举汇总及精确代数证书，可供独立验证。
原文摘要 · Abstract (English)
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.
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