Centralizer Excess as an Obstruction to Carlson's Depth Conjecture

Abstract

Let $G=\mathrm{SmallGroup}(128,859)$ and $k=\overline{k}$. The cohomology ring $H^{\ast }(G;k)$ 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 $E\le G$ of rank $r$ with $\mathrm{depth}{H}^{\ast }(C_G(E);k)=r$. We use this equivalence to define the centralizer excess. If $d=\mathrm{depth}{H}^{\ast }(K;k)$, 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 ${\omega }_a(H^{\ast }(G;k))=3$. Since $H^{\ast }(G\times (C_2{)}^n;k)\cong H^{\ast }(G;k)[u_1,\dots ,u_n]$, the standard behavior of associated primes under polynomial extension gives ${\omega }_a(H^{\ast }(G\times (C_2{)}^n;k))=n+3$ for $n\ge 0$. 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 $\dim H^{\ast }(G;{\mathbb{F}}_2)/\mathrm{ann}({\alpha }_0)=4$. 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.

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