Carlson’s associated-prime depth conjecture

Carlson posed the question in 1995: whether the depth of every finite-group cohomology ring equals the minimum dimension among its associated primes. The claimed example has

$$\operatorname{depth}H^{*}(G;\overline{\mathbb{F}}_{2})=2<\omega_{\mathrm{a}}\bigl(H^{*}(G;\overline{\mathbb{F}}_{2})\bigr),$$

with the latter quantity at least $3$.

Known results

• Green proved the equality for finite $p$-groups attaining Duflot’s lower bound.
• Carlson established the case $\dim H^{*}(G,k)=2$.
• Schäfer proved it when the Cohen–Macaulay defect is at most one.
• Garaialde Ocaña, González-Sánchez, and Guerrero Sánchez proved it for an infinite family of finite $p$-groups.

2026 counterexample claim

The paper reports an exhaustive calculation involving $75$ rank-two elementary abelian subgroups and six centralizer types; every relevant centralizer has depth at least $3$, ruling out associated primes of quotient dimension $2$. A related paper reports the same obstruction. The counterexample is currently an unverified claim rather than a settled resolution.

Current status (as of 2026): The general equality is claimed false by a finite-group example with depth $2$ and associated-prime minimum at least $3$; independent verification and the broader mathematical status remain open.