Centralizer Excess as an Obstruction to Carlson's Depth Conjecture

Abstract

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 $\omega_a\bigl(H^*(G;k)\bigr)=3$. 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 $\omega_a\bigl(H^*(G\times(C_2)^n;k)\bigr)=n+3$ for $n\geq0$. We also study the class $\alpha_0=g+fc\in H^3(G;\mathbb F_2)$. It is killed by two degree-one classes but restricts nontrivially to a rank-four elementary abelian subgroup. It follows that $\dim H^*(G;\mathbb F_2)/\operatorname{ann}(\alpha_0)=4$. 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.