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

This paper refutes Carlson's 1995 conjecture that the depth of a finite group's cohomology ring is always realized by an associated prime, by constructing an explicit counterexample using a specific group of order 128 where the depth is 2 but no associated prime of dimension 2 exists.

Original authors: Xinan Dai, Wenhao Deng, Yingdong Shi, Tailin Wu, Yuchen Yang

Published 2026-07-28

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CC BY 4.0

Original authors: Xinan Dai, Wenhao Deng, Yingdong Shi, Tailin Wu, Yuchen Yang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Detective Work of Mathematical Shapes

Imagine you are a detective trying to solve a mystery inside a giant, invisible city made entirely of numbers and shapes. This city is called "Group Cohomology," and it's a place where mathematicians study the hidden patterns of symmetry in finite groups—think of a group as a set of rules for how objects can be shuffled or rotated without breaking them. In this city, there are two very important clues that help detectives understand the layout: "depth" and "associated primes."

Think of depth as the height of a sturdy ladder you can build inside the city. To build a rung on your ladder, you need to find a special number that doesn't cause the whole structure to collapse (a "non-zero-divisor"). The deeper the ladder, the more stable and complex the city is. On the other hand, think of associated primes as the specific addresses of the city's weakest spots or "dead ends." These are the locations where certain numbers get stuck and can't move forward.

For a long time, mathematicians had a hunch, a rule of thumb called Carlson's Conjecture. They believed that the height of your ladder (the depth) would always match the size of the smallest dead end (the dimension of an associated prime). It seemed logical: if you can build a ladder of height 2, there should be a dead end exactly 2 units wide. But in the world of math, hunches are just guesses until someone proves them true or finds a single exception that breaks the rule.

The Great Counterexample

This paper is the story of a team of mathematicians who decided to test that hunch by building a very specific, tricky city and seeing if the rule held up. They chose a group called SmallGroup(128, 859), which is a collection of 128 elements, and they looked at it through the lens of a field called F2 (a number system with only two values, 0 and 1).

First, they did the hard work of measuring the "ladder height" of this group. Using precise algebraic tools, they proved that the depth of this group's cohomology ring is exactly 2. This means you can build a ladder with two rungs, but no more.

Next, they had to check the "dead ends." According to the old rule (Carlson's Conjecture), there should be a dead end that is exactly 2 units wide. To find this, they used a clever mathematical bridge discovered by a researcher named Okuyama. This bridge said: "If there is a dead end of width 2, there must be a specific type of smaller group inside our city—a 'rank-two elementary abelian subgroup'—whose own internal structure has a depth of exactly 2."

So, the team went on a hunt. They listed every single one of these special subgroups inside their city. They found exactly 75 of them. Then, they checked the "depth" of the centralizer (the neighborhood) for each of these 75 subgroups.

Here is where the surprise happened.

For most of these subgroups, a known theorem (Duflot's theorem) showed that their neighborhoods were very deep—at least 3 units deep.
For the remaining two tricky types, the team did a massive, exact calculation using computer-assisted algebra. They found that even these neighborhoods had a depth of at least 3.

The result was a perfect contradiction. The main city had a depth of 2, but every single possible "dead end" they could find was associated with a neighborhood that was at least 3 units deep. There was no dead end of width 2 anywhere to be found.

The Verdict

Because they found a city where the ladder height (2) is strictly smaller than the smallest dead end (3), the team has proven that Carlson's Conjecture is false. They didn't just guess; they provided an "exact certificate," a step-by-step algebraic proof that can be checked by anyone. They showed that for the group SmallGroup(128, 859), the depth is 2, but the smallest associated prime has a dimension of at least 3.

In simple terms, they found a place in the mathematical universe where the rules of symmetry are more complex and "deep" than the simple rule of thumb suggested. The ladder is shorter than the smallest gap, proving that the old map was missing a crucial detail. This isn't just a small correction; it's a complete rewrite of a long-held belief about how these mathematical cities are built.

Technical Summary: An Exact Counterexample to Carlson's Conjecture on Depth and Associated Primes

1. Problem Statement

The paper addresses Question 3.1 posed by J. F. Carlson in his 1995 work on depth and transfer maps. The question asks whether, for a finite group $G$ and a field $k$ of characteristic $p$ , the depth of the cohomology ring $H^*(G; k)$ is always realized by the dimension of one of its associated primes.

Formally, let $A = H^*(G; k)$ . The depth of $A$ , denoted $\text{depth } A$ , is the maximal length of a regular sequence in the ideal of positive-degree elements. The associated primes of $A$ are prime ideals $\mathfrak{p} = \text{ann}_A(a)$ for some $a \in A$ . The conjecture posits the equality:
$\text{depth } H^*(G; k) = \min_{\mathfrak{p} \in \text{Ass } H^*(G; k)} \dim H^*(G; k)/\mathfrak{p}$
While the inequality $\text{depth } A \le \dim A/\mathfrak{p}$ is standard, the conjecture asserts that the lower bound provided by the depth is always attained by some associated prime. Previous work by Green, Schafer, and Garaialde Ocaña et al. confirmed this equality for specific classes of $p$ -groups, but the general case remained open as of late 2025.

2. Methodology

The authors employ a three-layered strategy combining group-theoretic enumeration, commutative algebra, and exact computer-assisted verification to construct a counterexample.

A. Theoretical Reduction (Okuyama's Bridge)
The proof relies on Okuyama's Theorem (Theorem 3.1), which establishes a correspondence between associated primes and elementary abelian subgroups. Specifically, if an associated prime $\mathfrak{p}$ has quotient dimension $s$ , there exists an elementary abelian $p$ -subgroup $E \le G$ of rank $s$ such that:
$\text{depth } H^*(C_G(E); k) = s$
Consequently, to disprove the conjecture for a group with $\text{depth } H^*(G; k) = 2$ , it suffices to show that for every rank-two elementary abelian subgroup $E \le G$ , the depth of its centralizer $C_G(E)$ is strictly greater than 2 (i.e., $\ge 3$ ).

B. Group-Theoretic Enumeration
The authors select the finite group $G = \text{SmallGroup}(128, 859)$ over the field $k = \mathbb{F}_2$ . This group was identified from the catalogue of Green and King, which records its Krull dimension as 4 and its depth as 2.

The authors perform an exhaustive enumeration of all rank-two elementary abelian subgroups (isomorphic to $C_2 \times C_2$ ) in $G$ .
The procedure lists all non-identity involutions, forms commuting pairs, and deduplicates subgroups.
This yields exactly 75 such subgroups, which fall into 6 distinct isomorphism types of centralizers.

C. Exact Algebraic Certificates
For each of the 6 centralizer types, the authors determine the depth of the cohomology ring $H^*(C_G(E); \mathbb{F}_2)$ :

Four Types (Centers of Rank $\ge 3$ ): For centralizers isomorphic to $\text{SmallGroup}(16, 10)$ , $(16, 14)$ , $(32, 22)$ , and $(32, 46)$ , the authors apply Duflot's Theorem. Since the rank of the center of these groups is at least 3, Duflot's theorem immediately implies $\text{depth } H^*(C_G(E); \mathbb{F}_2) \ge 3$ .
Two Exceptional Types (Centers of Rank 2): For centralizers isomorphic to $\text{SmallGroup}(64, 90)$ $SmallGroup (64, 90)$ and $(64, 216)$ $(64, 216)$ , Duflot's theorem only guarantees depth $\ge 2$ $\geq 2$ . To rule out depth 2, the authors construct exact regular sequences of length 3 using explicit polynomial presentations of the cohomology rings.
- They utilize colon ideal computations: $(I : f) = I$ certifies that $f$ is a non-zero-divisor modulo $I$ .
- They verify three successive colon equalities for specific homogeneous elements in the presentation of each ring, proving the existence of a regular sequence of length 3.
- These computations are performed over $\mathbb{F}_2$ using Gröbner basis techniques. By Lemma 5.3 (base change invariance), these results hold for any algebraically closed field of characteristic 2.

3. Key Results

The paper establishes the following theorem:

Theorem 2.1: Let $G = \text{SmallGroup}(128, 859)$ and $k = \mathbb{F}_2$ . Then:

$\text{depth } H^*(G; k) = 2$ .
$\omega_a(H^*(G; k)) \ge 3$ , where $\omega_a$ is the minimum dimension of an associated prime quotient.
Consequently, $\text{depth } H^*(G; k) < \min_{\mathfrak{p} \in \text{Ass } H^*(G; k)} \dim H^*(G; k)/\mathfrak{p}$ .

The proof of $\text{depth } H^*(G; k) = 2$ is also provided via exact certificates:

A regular sequence of length 2 ( $f_1, f_2$ ) is constructed.
A non-zero socle element $w$ is exhibited in the quotient $A/(f_1, f_2)$ , proving that no further regular elements exist, thus fixing the depth at exactly 2.

4. Significance and Claims

The paper claims to provide the first exact counterexample to Carlson's conjecture regarding the equality of depth and the dimension of associated primes.

Resolution of an Open Problem: The work definitively answers Question 3.1 in the negative, demonstrating that the lower bound supplied by the depth of a cohomology ring is not always attained by an associated prime.
Methodological Rigor: The authors emphasize that the result is not based on heuristic approximations or numerical estimates. Instead, it relies on exact algebraic certificates (colon ideal equalities and Gröbner basis normal forms) and exhaustive group enumeration.
Reproducibility: The paper includes the finite group presentation, the cohomology ring presentations, the enumeration summary, and the specific algebraic certificates required for independent verification.
Scope: The counterexample is specific to the group $\text{SmallGroup}(128, 859)$ in characteristic 2. The paper does not claim to characterize all groups for which the conjecture fails, nor does it propose a new general formula for the gap between depth and associated prime dimensions.

The authors note that the construction relies on pre-existing results (Green and King's catalogue and Okuyama's theorem) but that the "load-bearing" new content is the complete enumeration of the rank-two subgroups and the subsequent exact depth calculations for their centralizers.