AI 中文总结
该研究以SmallGroup(128,859)为反例，揭示卡尔森深度猜想失效的机制，提出中心化子过剩准则，证明该反例的直积仍为反例，并分析了相关上同调类的低次表现。
AI 中文摘要
设$G=\text{SmallGroup}(128,859)$，$k=\boldsymbol{\bar{\text{F}}}_2$。上同调环$H^*(G;k)$的深度为2，尽管每个相伴素理想的商维数至少为3。我们分离出卡尔森深度猜想这一失效背后的机制：利用奥克耶玛定理的双向结论，我们将相伴素维数谱与秩$r$对应，其中秩为$r$的初等阿贝尔子群$E$满足$\text{depth}\thinspace H^*(C_G(E);k)=r$。由此得到中心化子过剩准则：当$d=\text{depth}\thinspace H^*(K;k)$时，卡尔森等式成立当且仅当秩为$d$的过剩量消失。对$G$而言，每个秩2中心化子的深度至少为3，因此正秩2过剩量正是确切的局部阻碍。正深度级过剩量在与初等阿贝尔2群的直积下保持不变，故$G\times (C_2)^n$对所有$n\thinspace\thinspace0$都是反例。我们还考察了该现象的低次表现：上同调类$\boldsymbol{\text{α}}_0=g+fc\thinspace\thinspace H^3(G;\text{F}_2)$被两个一次类零化，但非平凡限制到秩4初等阿贝尔子群，因此$\boldsymbol{\text{α}}_0$生成的循环模具有全克鲁尔维数4，可见低次零化未必产生卡尔森所预测的维数2素零化子，全维支撑并不等于整个上同调谱。
英文摘要
Let $G=\operatorname{SmallGroup}(128,859)$ and $k=\overline{k}$. The cohomology ring $H^*(G;k)$ has depth two, and we prove that the minimum quotient dimension of an associated prime is exactly three. Okuyama's theorem shows that an integer $r$ occurs as such a dimension exactly when there is an elementary abelian subgroup $E\leq G$ of rank $r$ with $\operatorname{depth} H^*(C_G(E);k)=r$. We use this equivalence to define the centralizer excess. If $d=\operatorname{depth} H^*(K;k)$, Carlson's equality holds precisely when some rank-$d$ subgroup has zero excess. For $G$, all rank-two centralizers have positive excess. A complete enumeration of the thirty-one actual rank-three elementary abelian subgroups finds six zero-excess witnesses. Hence $ω_a\bigl(H^*(G;k)\bigr)=3$. Since $H^*(G\times(C_2)^n;k)\cong H^*(G;k)[u_1,\dots,u_n]$, the standard behavior of associated primes under polynomial extension gives $ω_a\bigl(H^*(G\times(C_2)^n;k)\bigr)=n+3$ for $n\geq0$. We also study the class $α_0=g+fc\in H^3(G;\mathbb F_2)$. It is killed by two degree-one classes but restricts nontrivially to a rank-four elementary abelian subgroup. It follows that $\dim H^*(G;\mathbb F_2)/\operatorname{ann}(α_0)=4$. Thus two explicit linear annihilators do not force a two-dimensional cyclic support. The assertion is about Krull dimension; it does not say that the support is the whole spectrum.
