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

Abstract: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 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.

Group Theory (math.GR); Artificial Intelligence (cs.AI)

arXiv:2607.23732 [math.GR]

(or arXiv:2607.23732v1 [math.GR] for this version)

https://doi.org/10.48550/arXiv.2607.23732 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

BIBTEX_PROVENANCE.txt

check_certificate.py

data/128gp859.html

data/64gp216.html

data/64gp90.html

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