An Exact Counterexample to Carlson's Associated-Prime Depth Conjecture from a Group of Order 128
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 cert...
| Authors | Xinan Dai, Wenhao Deng, Yingdong Shi et al. |
|---|---|
| Primary category | math.GR |
| Published | 26 July 2026 |
| Last updated at source | 26 July 2026 |
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