# An Exact Counterexample to Carlson's Associated-Prime Depth Conjecture from a Group of Order 128

arXiv: https://arxiv.org/abs/2607.23732 · HTML: https://mlsanity.com/paper/2607.23732/

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- First posted on arXiv 26 Jul 2026.

## Abstract (arXiv)

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=SG128859,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≤ G satisfying depth H^*(CG(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.