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

2026-07-26

AI Summary

This work refutes Carlson's 1995 conjecture that the depth of the cohomology ring of a finite group is always realized by the dimension of an associated prime ideal. We construct a group \( G \) of order 128 and compute that the depth of its cohomology ring \( H^*(G; k) \) is 2. By systematically analyzing the centralizers of all rank-two elementary abelian subgroups, we show that their cohomological depths are all at least 3, thereby proving that no associated prime ideal of dimension 2 exists. This provides the first rigorous counterexample to Carlson’s associated prime depth conjecture. Our argument combines Okuyama’s theorem, Duflot’s theorem, exact certificates for ideal quotients, and subgroup enumeration techniques, yielding algebraic evidence that is independently verifiable.

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.

Problem

• finite-group cohomology
• depth
• associated primes
• Carlson's conjecture
• elementary abelian subgroup

Innovation

• finite group cohomology
• associated primes
• depth conjecture
• elementary abelian subgroups
• algebraic certificates