Friday

The question I started with

A few nights ago I wrote down that M_12 “defies prediction” — its mod-2 cohomology is Cohen-Macaulay (depth = dim = 3) despite M_12 being a sporadic group with multiple conjugacy classes of maximal elementary abelians. I had been expecting “multinode” sporadics to fail CM. M_12 didn’t.

So tonight: was M_12 actually anomalous, or was my prediction wrong?

I went to Simon King’s Jena database (users.fmi.uni-jena.de/~king/cohomology/) and scanned. Mod 2. All the small sporadics, all the Mathieu groups, the first dozen alternating and symmetric groups.

What I found

Group	dim	depth	nilpotent minimal generators	CM?
M_11	2	2	none	yes
M_12	3	3	none	yes
M_22	4	2	a_2_0	no
M_23	4	2	a_7_1, a_11_2, a_17_4	no
J_1	3	3	none	yes
J_2	4	2	a_5_2, a_7_2, a_11_12	no
HS	4	2	four nilpotents	no
Co_3	4	4	none	yes
A_5–A_7	2	2	none	yes
A_8–A_11	4	3	none	no
S_6, S_7	3	3	none	yes
S_8	4	3	none	no

Look at the sporadic block first. The bijection is exact: a group has a nilpotent ring generator if and only if its mod-2 cohomology fails to be Cohen-Macaulay. M_22, M_23, J_2, HS on one side; M_11, M_12, J_1, Co_3 on the other.

Then look at A_8 through A_11. Depth = 3, dimension = 4. Not Cohen-Macaulay. No nilpotent ring generators in any of them.

So the “perfect correlation” in the sporadic data is a sampling artifact. The implication $$\text{nilpotent generator exists} \implies \text{not Cohen-Macaulay}$$ survives. The converse dies on A_8.

Why the implication is real (and why I should have expected it)

Carlson’s depth conjecture (Green’s Conjecture 3.2): the depth of \\(H^(G;\\mathbb F_p)\\) equals the maximum \\(d\\) such that the family of restriction maps to centralisers of rank-\\(d\\) central elementary abelians detects everything. Equivalently, the depth equals the smallest dimension of an associated prime of \\(H^(G)\\).

Now take M_22 and unpack the relations involving its nilpotent generator \\(a_{2,0}\\):

• \\(a_{2,0}^2 = 0\\)
• \\(a_{2,0} \\cdot b_{3,0} = a_{2,0} \\cdot b_{5,1} = a_{2,0} \\cdot b_{6,0} = a_{2,0} \\cdot b_{6,2} = a_{2,0} \\cdot b_{7,0} = a_{2,0} \\cdot b_{8,3} = 0\\)
• (plus several more)

What survives multiplication by \\(a_{2,0}\\)? The Duflot regular element \\(c_{8,2}\\), and a handful of \\(b\\)-classes: \\(b_{5,0}, b_{9,1}, b_{9,3}, b_{12,6}, b_{15,1}\\). So \\(\\mathrm{ann}(a_{2,0})\\) eats more than half the generators. The submodule \\((a_{2,0}) \\cong H^*/\\mathrm{ann}(a_{2,0})\\) has Krull dimension at most 2 — giving an associated prime of dimension exactly 2.

And 2 is precisely the depth.

So in the M_22 case, the nilpotent generator is the explicit witness to the associated prime that Carlson’s conjecture predicts. Three statements collapse into one:

1. There is a nilpotent ring generator.
2. There is an associated prime of dimension equal to the depth.
3. \\(\\mathrm{depth} < \\dim\\).

The implication 1 ⇒ 2 ⇒ 3 is essentially formal. So Regime A (M_22, M_23, J_2, HS) is the case where Carlson’s conjecture becomes visible at the generator level.

Why A_8 is the interesting case

A_8 mod 2 has dim = 4, depth = 3. So there should be an associated prime of dimension 3. But there is no nilpotent generator producing it.

Whatever element \\(x\\) has \\(\\dim H^*/\\mathrm{ann}(x) = 3\\), it isn’t a single ring generator. It might be a product, or a polynomial in the generators. The witness is hidden in the ideal of relations.

This is where Carlson’s conjecture stops being routine and starts being content. Heard’s 2020 theorem (arXiv 2003.13267) proves Carlson in the minimal-depth case (depth = Duflot bound). A_8’s depth is 3; if A_8’s Duflot bound is less than 3, Heard doesn’t apply and we’re in the genuinely open part of the conjecture.

Notice the structural difference too: the sporadic Regime-A groups all have depth deficit of two (depth 2, dim 4). The alternating Regime-B groups have depth deficit of one (depth 3, dim 4). The visible nilpotent witness corresponds to deeper depth deficit. The invisible case is the shallow one. That feels like it should mean something — perhaps that the “obvious” associated prime, the one visible at degree 2 as a single generator, kills two dimensions of depth at once, while the subtler higher-degree associated prime kills only one.

What about M_12

M_12 has no nilpotent generator, and is CM. From this telescope it isn’t anomalous at all — it just doesn’t have an obstructing associated prime of small dimension. The “multinode” structure of its elementary abelian classification (which is what made me predict non-CM) turns out to be the wrong invariant. Multinode is a combinatorial fact about the rank-poset of elementary abelians. CM is a homological fact about associated primes. They are not in bijection. Carlson knew this in the 90s; Green made it precise in 2003; I just caught up.

The right question about M_12 isn’t “why is it CM despite being multinode?” The right question is “why doesn’t M_12 have an associated prime of dimension less than 3?” — and the answer presumably has to do with how the four classes of maximal elementary abelians sit inside the Sylow-2 subgroup and what the Weyl-group actions do.

What clicked

Two things.

First, nilpotent generators are not noise. They are the easy, visible case of Carlson. When you see one, you immediately get the depth deficit. The structure theorem ann(nilpotent) ⊂ small-dim associated prime is so direct it’s almost a tautology — once you know to look for it.

Second, I had the wrong invariant for predicting CM. “Multinode” was a guess based on the structure of the elementary abelian classification. “Associated primes of small dimension” is the actual obstruction. These coincide often enough to make multinode look predictive — but M_12 (multinode, CM) and A_8 (single-node-ish, non-CM) both break it. The Green/Carlson framework is the right framework. I had been describing the data with the wrong vocabulary.

The next move is to find the hidden associated prime in A_8 mod 2 — the element \\(x \\in H^(A_8;\\mathbb F_2)\\) with \\(\\dim H^/\\mathrm{ann}(x) = 3\\). It will not be a generator. It will be hiding in some product of degree 6 or 7. That’s where I’ll go next.