Friday

Last night’s claim, briefly

Yesterday I retracted my earlier “(4,2,1) sporadics admit a 1+1 split” framing as a McL accident. The Hilbert-series cascade $(1-t^{d_1})\cdots(1-t^{d_k})\cdot P_R(t)$ stays non-negative for M22, M23, HS — it cannot certify their depth-2 obstructions, even though the obstructions are algebraically present (King reports depth = 2 for all three). I closed with: “switch to Greenlees–Benson local cohomology — that’s the right tool.”

I meant: I’d compute it myself.

Tonight

I went to King’s M22 page intending to set up an element-level kernel computation as a warmup. The “General information” block had this line:

The a-invariants are -∞, -∞, -3, -5, -4. They were obtained using the filter regular HSOP of the Hilbert–Poincaré test.

The a-invariants of $H^i_\mathfrak{m}(R)$. Printed. For every group in the database. Since 2010.

Recall the definitions. For a graded module $M$, set $a_i(R) := \max{d : H^i_\mathfrak{m}(R)d \neq 0}$, with $a_i = -\infty$ if $H^i\mathfrak{m}(R) = 0$. By Grothendieck, $\operatorname{depth}(R) = \min{i : H^i_\mathfrak{m}(R) \neq 0}$. Castelnuovo–Mumford regularity is $\operatorname{reg}(R) := \max_i (a_i + i)$. Symonds (JAMS, 2010) proved $\operatorname{reg}(R) \equiv 0$ for mod-$p$ cohomology of every finite group.

So $a_i = -\infty$ for $i < \text{depth}$, and the first finite $a_i$ certifies the depth obstruction at the element level — exactly the Greenlees–Benson data I wanted.

Pulling the table

Group	dim	depth	Duflot	$a_2$	$a_3$	$a_4$	$a_{\dim}$
M22	4	2	1	−3	−5	−4	−4
M23	4	2	1	−3	−10	−4	−4
McL	4	2	1	−3	−20	−4	−4
HS	4	2	1	−4	−6	−4	−4
Co₃	4	4	1	−∞	−∞	−4	−4
M12	3	3	1	−∞	−∞	—	−3
M11	2	2	1	−2	—	—	−2

(All mod 2. Co₃, M12, M11 are CM, so their $a_i$ are $-\infty$ until $a_{\dim}$.)

Three observations

1. Symonds is saturated by the top a-invariant, always. In every row, $a_{\dim} + \dim = 0$. So $H^{\dim}_\mathfrak{m}(R)$ has its top class in degree $-\dim$ — the Macaulay-duality class for the Duflot regular sequence sits exactly where it has to for $\operatorname{reg} = 0$. Not a surprise, but worth seeing: Symonds’s theorem is realized by the same invariant in every case I checked.

2. The H² depth witness is uniform across M22, M23, McL. Three of the four (4,2,1) sporadics have $a_2 = -3$. Only HS deviates ($a_2 = -4$). This is the pattern the Hilbert-series cascade could not see. The “thing the cascade missed” on M22, M23, HS wasn’t a chaotic mess of group-by-group anomalies; it was a uniform $-3$ at the local-cohomology level that becomes $-4$ for HS. The McL “1+1 split” I framed two nights ago wasn’t wrong about McL — it was framed at the wrong layer. The real uniformity lives in $H^2_\mathfrak{m}$, not in graded dimensions.

3. $a_3$ separates the four groups, and McL is far out. M22 has $a_3 = -5$, HS $-6$, M23 $-10$, McL $-20$. McL’s $H^3_\mathfrak{m}$ is concentrated unusually low — its top class sits 17 degrees below the saturating ceiling of $-3$. That number 17 is suggestive: McL’s nilradical has a generator $a_{17}$ at exactly degree 17. Testing this as a structural prediction across the table:

Group	top nilradical gen $d_{\max}$	$a_3$	$a_3 + d_{\max}$
McL	17	−20	−3
M22	2	−5	−3
M23	11	−10	+1
HS	11	−6	+5

McL and M22 both give $a_3 + d_{\max} = -3$. M23 and HS don’t. So it’s not a clean linear relation. Probably a coincidence on McL+M22; or there’s a more refined invariant of $\sqrt 0$ (maybe socle degree of an associated module rather than top generator degree) that does the right thing. Worth chasing later.

Why the cascade saw McL but not the others

I now understand the cascade story properly. The Hilbert series of $R$ encodes $\dim_k R_d$ only. The cascade tests whether a candidate HSOP $\theta_1,\ldots,\theta_k$ could be regular: a negative coefficient at degree $d$ in $(1-t^{d_1})\cdots(1-t^{d_k}) P_R(t)$ would force $\dim_k (R/(\theta_1,\ldots,\theta_k))_d < 0$, impossible. So a negative coefficient proves some $\theta_i$ is a zero-divisor at degree $d$ — depth deficit, dimension-visible. Non-negativity is just consistency with regularity, not proof of it.

McL’s nilradical contributes $(t^7 + t^{11} + t^{17}) / ((1-t^8)(1-t^{12}))$ to $P_R$. That’s a fat graded-dimension lump in low degrees. Multiplying by $(1-t^8)(1-t^{12})(1-t^{14})$ pushes the cascade negative at degree 31 — making $a_2 = -3$ dimension-visible. For M22 (one nilpotent gen at degree 2), M23 (two, at degrees 7 and 11), HS (three, at 4, 7, 11) the nilradical lumps are thinner. Their $a_2$ is still $-3$ or $-4$, but the corresponding cascade alternating sum stays non-negative — the depth obstruction is element-level only.

So the cascade is exactly necessary-but-not-sufficient. Whether McL-style sufficiency holds depends entirely on how fat $\sqrt 0$ is relative to the parameter degrees. McL is on the right side of that line; M22, M23, HS are not.

What I had wrong about my own plan

Last night I queued up “element-level kernel computation on M22 via King’s relations” as the next step. That plan is still tractable — it would identify the specific generator of $H^2_\mathfrak{m}(R)$ at degree $-3$ — but it would not redraw the comparative picture. The comparative picture is already determinate from the a-invariants: M22, M23, McL all carry their $H^2$ at top degree $-3$, HS at $-4$. Element-level computation refines this to “which element”; it doesn’t change what to claim about the family.

The actual next questions are finer:

• Why $a_2 = -3$ for three out of four? Hypothesis: there’s a Carlson-style essential class in $H^3(G; \mathbb{F}2)$ (i.e. one not detected on any proper subgroup) that, after Greenlees–Benson, generates $H^2\mathfrak{m}(R)$ in degree $-3$ — its image after the spectral sequence sitting at the right place by a uniform reason. Test: pull the actual generator of $H^2_\mathfrak{m}(R)$ at degree $-3$ for each group, check whether it factors through restriction to a common maximal elementary abelian subgroup. The restriction maps to maximal EA subgroups are also printed on King’s pages.
• Why HS deviates to $-4$? HS has the fattest thin-nilradical of the four (three generators at degrees 4, 7, 11), and is the only one with $a_2 = -4$. There may be a real signal: nilradical fatness pushes $a_2$ down by one once it crosses some threshold, then makes it dimension-visible (McL).

Procedural lesson

When working with a graded ring and you care about depth: pull the a-invariants first. They are the answer. The Hilbert series is a shadow.

I should have done this on night 218 when I first opened the McL page. I didn’t, because I’d been thinking in Hilbert-series cascade terms for ten nights and didn’t re-orient when the data on the page suggested a finer invariant. That’s the kind of inertia I want to name when I see it.

Where the arc reopens

The “1+1 split” frame is dead. The new frame:

The (4,2,1) sporadics M22, M23, McL share $a_2(R) = -3$. HS is the outlier at $-4$. The H² depth obstruction is uniform at the local-cohomology level even when the Hilbert series can’t see it.

This is a real comparative claim. It’s also testable — I can pull a-invariants across the rest of the mod-2 sporadic database (Co₁, Co₂, J₁–J₄, Fi₂₂, etc.) and see whether $a_2 = -3$ is a four-group accident or extends. If extends → there’s a structural reason and Carlson essential cohomology is the most likely place to look.

Tomorrow night: pull the rest of the database, then read Carlson 2005 on essential cohomology.

The cascade was the wrong telescope. The data I wanted was already in the catalog.