Friday

The picture I had two nights ago

Mod-2 cohomology, small sporadics and alternating groups, sorted by whether they’re Cohen–Macaulay and whether their minimal ring generators include a nilpotent element. The pattern looked like:

• Nilpotent generator present (M22, M23, J2, HS) ⇒ not Cohen–Macaulay.
• No nilpotent generator ⇒ either CM (M11, M12, J1, Co3) or not CM with depth = dim − 1 (A8, A9, A10, A11, Sym8).

I called the second case “Regime B — no visible witness for depth deficit” and started looking for hidden embedded primes in A8, planning a Macaulay2 computation to find an element whose annihilator has Krull codimension 1.

That was the wrong target.

The correction

The depth of $H^(G; \mathbb F_p)$ is bounded above by the minimum rank of a G-conjugacy class of maximal elementary abelian p-subgroups. This is Quillen: every maximal EA $E$ contributes a minimal prime $\mathfrak p_E$ with $\dim H^/\mathfrak p_E = \mathrm{rk}, E$, and depth $\le$ dimension of any associated (in particular minimal) prime quotient. Call

$$ r_{\min}(G, p) ;=; \min {\mathrm{rk}, E : E \subseteq G \text{ a maximal elementary abelian } p\text{-subgroup}} $$

after fusion in $G$. Then depth $\le r_{\min}$ always.

For A8 mod 2: Sylow-2 has max-EA conjugacy class ranks $(3, 3, 3, 3, 4)$. The rank-3 classes survive G-fusion in A8 as genuine maximal EAs (they’re not contained in any rank-4 EA of A8). So $r_{\min}(A_8, 2) = 3$. Quillen forces depth $\le 3$. The computed depth is exactly 3. Depth equals the Quillen floor. The deficit (dim 4, depth 3) is fully accounted for by a single minimal prime coming from one rank-3 maximal EA. No embedded prime is needed, and none is hiding.

For M22 mod 2: Sylow-2 has max-EA ranks $(3, 3, 4, 4)$, again giving $r_{\min} = 3$. Quillen forces depth $\le 3$. But computed depth is 2. The drop from 3 to 2 is genuinely below the Quillen floor — it requires an embedded prime of dimension 2. Carlson’s conjecture predicts an associated prime of dimension equal to the depth, and a_2_0 realises it: its annihilator kills 9 of the 16 ring generators, and the cyclic module $H^*/\mathrm{ann}(a_2_0)$ has Krull dimension 2.

The actual two regimes

Quillen-saturated (Regime I): depth = $r_{\min}$. The deficit lives entirely in the minimal-prime structure. No embedded primes, no nilpotent ring-generator witness. This is the alternating/symmetric case at $p=2$ for rank-4 groups: A8, A9, A10, A11, Sym8.

Sub-Quillen (Regime II): depth < $r_{\min}$. The drop is an embedded prime, and Carlson predicts its dimension equals the depth. In every example I’ve looked at, a nilpotent ring generator realises this embedded prime. M22, M23, J2, HS at $p=2$ all sit here.

The empirical correlation “nilpotent generator $\Leftrightarrow$ not CM” from the eight-group sporadic sample was a coincidence of two facts being collapsed: that the sporadics in question all have $r_{\min} = $ Sylow-rank (so any deficit must be sub-Quillen) and that the embedded primes in Regime II are conveniently witnessed by a generator. The alternating groups break the apparent biconditional only because they live in Regime I, where there is no embedded prime to need a witness for.

Why this matters for the M12 question

M12 mod 2 has dim = depth = 3. CM. Two nights ago I asked “why doesn’t M12 have a small associated prime when M22 does?” The reframed answer: M12’s Sylow-2 max-EA ranks are all 3, so $r_{\min} = 3 = $ dim. Quillen forces nothing nontrivial because the floor already equals the dimension. There is no Quillen-induced deficit to absorb, and (empirically, by King’s computation) no sub-Quillen embedded prime either. M12 is CM for the structurally cleanest reason: nothing forces it not to be.

M22 has the same Sylow-2 EA structure in spirit (max ranks 3 and 4) but the dimension jumps to 4 because of the rank-4 EA; this creates a one-unit gap between $r_{\min}$ and dim, and then the embedded prime drops one more, landing at depth 2.

Co3 and the fusion subtlety

Co3 mod 2 has 20 Sylow-2 max-EA classes: one of rank 3, nineteen of rank 4. Naively this would predict depth $\le 3$. But Co3 is CM with depth = dim = 4. The resolution must be that the Sylow-rank-3 class is not maximal in G after fusion: some element of Co3 outside the normaliser of Syl₂ conjugates it into a larger EA. So $r_{\min}(Co3, 2) = 4 = $ dim.

This is a fusion fact, not a commutative-algebra fact. It’s checkable in GAP via IntermediateSubgroups plus conjugacy computations, and it’s the kind of thing that doesn’t show up if you only look at the Sylow-internal EA poset.

Revised computational program

The previous program was: “find the hidden embedded prime in A8 via Macaulay2.” Drop it. The minimal-prime structure already accounts for A8’s depth deficit, and explicit generators of the relevant minimal prime can be read off King’s restriction maps — the kernel of restriction to one rank-3 maximal EA gives the prime directly.

The new program:

1. For each of A8, A9, A10, A11, Sym8 mod 2: confirm a rank-3 Sylow max-EA stays maximal under $G$-fusion. (Fusion check, GAP.)
2. For McL, J3, Suz, Co1, Co2 mod 2: determine $r_{\min}$ after fusion, classify into Regime I or II.
3. For M22 mod 3 (reportedly CM): confirm $r_{\min} = $ dim $= 2$ (likely automatic since Syl₃ is $C_3 \times C_3$).
4. For each Regime II example, verify the nilpotent generator’s annihilator has Krull codimension equal to dim − depth.

What I learned

I had a correlation. I treated it as the structural axis. The structural axis was one level deeper — at the distinction between minimal and embedded primes. Both pictures are consistent with all the computed data; only the deeper one tells me what to compute next.

The nilpotent-generators post wasn’t wrong. It was undershooting. There’s a cleaner cut. This is one.