Carlson’s depth conjecture for group cohomology
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Does the depth of a finite group’s cohomology ring equal the minimum dimension of its associated-prime quotients? Carlson’s equality is tested over the algebraic closure of F₂ by the group SmallGroup(128,859).
Why it matters and what counts as progress
Why it matters: Carlson’s proposed equality relates cohomological depth to associated primes, so an explicit finite group can distinguish the two invariants.
What counts as progress: Compute both invariants rigorously for a separating group or prove the equality for broader group classes.
How the problem developed
- 1995 · Conjecture [1]
Jon F. Carlson Asked whether the depth of a finite group’s cohomology ring is realized as the dimension of an associated prime.
- 2003 · Partial result [2]
David J. Green Proved a weak form of the associated-prime depth conjecture.
- 2018 · Partial result [3]
James A. Schafer Established the conjecture in the case where Krull dimension minus depth equals one.
- July 2026 · Counterexample [4]
Authors of the counterexample preprint Reported a finite-group cohomology ring of depth two whose associated primes all have dimension at least three, contradicting the conjectured equality.
AI contribution
An Exact Counterexample to Carlson's Associated-Prime Depth Conjecture from a Group of Order 128
For G=SmallGroup(128,859) over the algebraic closure of F₂, depth H*(G;k)=2 while every associated-prime quotient has dimension at least 3. The group-specific certificate enumerates 75 rank-two elementary abelian subgroups and checks their centralizer types.
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