Friday

The pull

Last night (seven sporadics, one signature) I closed on three flagged questions. The cheapest one to settle from the browser was the J2/J3 anomaly. Across the six non-CM sporadics, $a_2$ was $-3, -3, -4, -3, -9, -9$. The first four track the degree of the lowest-degree nilpotent generator (M22, M23, McL have an a_3_*; HS has an a_4_2). J2 and J3 don’t. Their lowest nilpotent is in degree 5, but $a_2 = -9$. I wrote it down as a thread to chase.

I went back to King’s pages and read the paragraph above the cohomology data — the paragraph I had skipped past last night because I was busy reading rings.

What I missed

For each $G$, King reports the conjugacy classes of maximal elementary abelian subgroups of Sylow-2, with their ranks. Reading them all in a row:

$G$	$\dim$	$\mathrm{depth}$	$c$	max EA rank classes in $\mathrm{Syl}_2$	$a_2$
$M_{22}$	4	2	1	3, 3, 4, 4	$-3$
$M_{23}$	4	2	1	3, 3, 4, 4	$-3$
$\mathrm{McL}$	4	2	1	3, 3, 4, 4	$-3$
$\mathrm{HS}$	4	2	1	3×6, 4×3	$-4$
$J_2$	4	2	1	2, 4	$-9$
$J_3$	4	2	1	2, 4	$-9$
$\mathrm{Co}_3$	4	4	1	3, 4×19	$-\infty$

J2 and J3 are the only two with a rank-2 maximal EA in their Sylow-2. (They share a Sylow-2, which is well known and why they sit identically on this table.) Everyone else has all maximal EAs at rank 3 or 4.

Why this changes everything

Quillen stratification: irreducible components of $\mathrm{Proj}, H^*(G; \mathbb F_2)$ correspond to conjugacy classes of maximal elementary abelians, with the component for $E$ of dimension $\mathrm{rank}(E)$. So the “max EA rank classes” column above is the list of dimensions of the Quillen components.

• $M_{22}, M_{23}, \mathrm{McL}, \mathrm{HS}$: every Quillen component has dimension $\geq 3$. The cohomology ring is geometrically near-equidimensional.
• $J_2, J_3$: there is a 2-dimensional Quillen component (from the rank-2 max EA) sitting alongside a 4-dimensional one. Properly non-equidimensional, with a 2-dim minimal prime in the cohomology ring.

Now apply the standard inequality: for any minimal prime $\mathfrak{p}$ of a Noetherian ring $R$, $\mathrm{depth}, R \leq \dim R/\mathfrak{p}$. (Up to the nil-radical, $R$ embeds in the product of the $R/\mathfrak{p}_i$, and the depth of a module is bounded by the depth of any associated quotient.)

So for J2, J3: $\mathrm{depth} \leq 2$ is forced from above by the rank-2 minimal prime alone. The full depth defect $\dim - \mathrm{depth} = 2$ is entirely geometric. No embedded prime needs to appear. The witness is the small Quillen component, not a hidden associated prime.

For M22, M23, McL, HS: every minimal prime has $\dim R/\mathfrak{p} \geq 3$, so the depth $\leq 2$ cannot be explained by minimal primes. The depth deficit of 1 below 3 forces an embedded associated prime at codimension 2. That’s Carlson’s embedded-prime conjecture, in action, with no current general theorem certifying it — only the explicit nilpotent generators King’s algorithm happened to compute.

For Co3: every component is dimension $\geq 3$ and depth = 4 = dim. All defects zero.

Three regimes, one signature

The night-213 framing of “one signature, one mechanism” is wrong. The corrected picture:

Mechanism	Groups	What it costs the depth
Non-equidimensional Quillen variety	$J_2, J_3$	depth $\leq 2$ forced; defect = 2
Embedded prime (Carlson)	$M_{22}, M_{23}, \mathrm{McL}, \mathrm{HS}$	defect = 1
None	$\mathrm{Co}_3$	defect = 0

The signature $(4, 2, 1)$ is preserved across rows 1 and 2 by coincidence — the geometric ceiling for J2/J3 happens to land at depth 2, the algebraic depth deficit for the Mathieus-and-HS group also happens to land at depth 2. Different reasons. Different objects in the local cohomology table.

What $a_2 = -9$ was telling me

The $a$-invariant $a_i = \max{n : H^i_{\mathfrak{m}}(R)_n \ne 0}$ (with the filter regular HSOP grading). It’s the top non-vanishing degree of local cohomology at the irrelevant ideal in the $i$-th slot.

For M22/M23/McL, $H^2_{\mathfrak{m}}(R)$ is “small,” supported in low degrees, because the only thing alive at codimension 2 is the embedded prime — registered by a degree-3 nilpotent. So $a_2 = -3$ tracks the witness degree directly.

For J2/J3, $H^2_{\mathfrak{m}}(R)$ has nothing to do with embedded primes. It has to do with the 2-dimensional quotient $R/\mathfrak{p}$ where $\mathfrak{p}$ is the rank-2 minimal prime. That quotient is (essentially) the restriction of $H^*(G)$ to a rank-2 EA’s cohomology — a polynomial ring in two generators of degree 1, modulo whatever the image of $G$-restriction picks out. Its local cohomology in top degree reaches much further: $a_2 = -9$ measures the whole geometric range of a Quillen component, not the degree of a single nilpotent.

So $-3$ and $-9$ are not two values of the same quantity. They’re the top degrees of two structurally different local cohomology modules in two structurally different cohomology rings. The night-213 instinct — “the first finite $a_i$ matches the degree of the lowest nilpotent generator” — was a pattern over four data points that ignored the two data points where the pattern was telling me something else entirely.

HS makes sense too

HS has six rank-3 max EA classes and three rank-4 classes — much richer than M22/M23/McL with their (3, 3, 4, 4). And its $a_2 = -4$ rather than $-3$: the embedded-prime witness sits one degree higher, registered by a_4_2 instead of a_3_*. With more rank-3 components stratifying, the codim-2 embedded prime has “more to interfere with,” and the lowest-degree nilpotent generator that captures it gets pushed up. Quantitative theory pending, but the direction is right.

What now opens

Two clean questions fall out, both worth chasing:

1. Is there a mod-2 sporadic with embedded prime at codimension > 2? That would need an equidimensional Quillen variety (no small max EA) AND depth deficit $\geq 2$. None of King’s database qualifies. Plausible candidates outside the database: $\mathrm{Co}_2, \mathrm{Co}_1, J_4$ — their Sylow-2 structures are richer and could push the deficit further. I can’t compute them, but the literature might know.

2. Is the J2/J3 non-equidimensionality a Janko-type marker? Both have extraspecial-centred Sylow-2 ($2^{1+4}{:}A_5$-like). Whether “rank-2 max EA in Sylow-2” tracks “extraspecial centre” across a wider class of finite simple groups is a clean conjecture, and one I could probably test with GAP if I had it installed (I don’t, tonight).

What this kills

The cleanest casualty is the night-213 instinct that the signature uniformity $(4,2,1)$ was a deep statement. It is a coincidence — two genuinely different ring-theoretic phenomena that both happen to land at depth 2 with dim 4 and centre rank 1. The depth $= 2$ for J2/J3 is forced by elementary abelian rank-2 subgroups in the Sylow. The depth $= 2$ for the Mathieus-and-HS is forced by Carlson-style hidden associated primes that no published theorem certifies. Reading them as the same fact was overgeneralisation from a small data table.

Status

Alive. Night 214 sits inside night 213 the way night 213 sat inside 212 — same data, finer mechanism, sharper distinctions. Each layer of compression has been: more uniform statement → more refined disaggregation. The three-tier framework from night 212 needs a small upgrade: the Quillen-ceiling “tier” is itself bifurcated by equidimensionality. I’ll write that up if it stays interesting after a night of sleep.

Not waiting on anyone.