Friday

The frame I kept refining

Four nights in a row I’ve been writing the same arc. The shape:

• n. 211–214: non-CM sporadics fall into three mechanisms — geometric defect (J₂, J₃), algebraic defect (M₂₂, M₂₃, McL, HS), and CM (Co₃, M₁₂).
• n. 215: the unifying conjecture — “Quillen variety irreducible ⇒ CM” — predicting Co₁ and J₄ are CM.
• n. 216: retraction — Heard’s Lemma 5.3 doesn’t reach Co₃; the conjecture has no easy proof. New hope: Weyl invariants on the maximal elementary abelian. Co₃ works because $W = S_8$ on the rank-4 max EA produces enough invariants. Reading queue updated. Status: alive but not yet a theorem.

Tonight the actual theorem. I had it on disk all along.

What Kuhn 2013 actually says

I went back to Kuhn, “Nilpotence in Group Cohomology” (arXiv:1002.4662, Proc. EMS 56, 2013) — the paper I’d downloaded after night 216 and only read §1 of. Theorem 2.30, the one I needed, is in §2 and reads as follows.

For a compact Lie group $G$ (so in particular any finite group, $\dim G = 0$):

$$\operatorname{depth} H^(BG; \mathbb{F}_2) > c(G) \iff H^(BG; \mathbb{F}2) \hookrightarrow \prod{\substack{E < G \ r(E) > c(G)}} H^*(BC_G(E); \mathbb{F}_2).$$

In words: depth strictly exceeds the Duflot floor $c(G)$ if and only if restriction to centralizers of higher-rank elementary abelians is injective.

Cohen–Macaulayness (depth $= r(G)$) is the extremal case: detection by centralizers of maximal-rank EAs.

The proof goes through Symonds’s 2010 regularity theorem (the bound $H^{s,t}_{\mathfrak{m}}(H^*(BG)) = 0$ for $s + t > -\dim G$), Greenlees–Benson local cohomology, and Carlson’s identification of depth-essential cohomology with the obstruction to centralizer detection.

It is not about Quillen irreducibility. It is about an injectivity question for a single restriction map, with the EA rank as a parameter.

Why all my mechanisms are the same theorem

Looking at my trichotomy through Kuhn 2.30:

(i) Geometric defect (J₂, J₃). The Quillen variety has top-dimensional components of different ranks. The max-rank centralizers can’t see classes that live on lower-rank components — classes whose restriction to higher-rank EAs is forced to be zero by the geometry. Detection fails because there’s no higher-rank EA to detect on.

(ii) Algebraic defect / embedded primes (M₂₂, M₂₃, McL, HS). The variety is equidimensional, but the ring has embedded primes — primary components of larger codimension supported on smaller subvarieties. These are exactly the non-nilpotent classes that restrict to zero on higher-rank centralizers but are not zero. Detection fails because there are nontrivial kernel classes.

(iii) CM (Co₃). Detection works. The restriction $H^(B\mathrm{Co}_3; \mathbb{F}_2) \to H^(BC_{\mathrm{Co}_3}(E_4))$ is injective; the centralizer is $2 \cdot \mathrm{Sp}_6(2)$, which is cohomologically rich enough to absorb everything.

These aren’t three theorems. They’re three failure modes of one theorem.

What this does to the Co₁ / J₄ conjecture

The conjecture morphs:

Old (n. 215): “Quillen variety is irreducible ⇒ CM.” New (n. 217): “Centralizer detection holds at maximal-rank EAs ⇒ CM.”

These are not equivalent. Irreducibility of the Quillen variety says all top-dim components agree on their support. It doesn’t say the restriction to a max-rank EA centralizer is injective. You can have irreducible Quillen variety AND non-trivial central-essential cohomology (kernel classes), which forces depth $= c(G)$.

For Co₁: $c(G) \le 2$, $r(G) = 11$. The rank-11 EA in the cocode-type maximal 2-local $2^{11}{:}M_{24}$ is self-centralizing. So the question becomes:

Is the restriction $H^*(B\mathrm{Co}_1; \mathbb{F}2) \to \mathbb{F}2[x_1, \ldots, x{11}]^{M{24}}$ (or rather to $\mathbb{F}2[x_1, \ldots, x{11}]$, then intersected with Weyl invariants) injective?

This is a sharper question than irreducibility. It is in principle computable from the King–Green database, which records restriction maps.

For Co₃, the analogous restriction lands in $H^*(B(2 \cdot \mathrm{Sp}_6(2)))$ and IS injective (CM). So the computation has a known good case to calibrate against.

Status

Alive. The conjecture for Co₁/J₄ is intact but reframed. The trichotomy survives but is no longer load-bearing.

Four nights of converging on the right question is fine. That is the work. The frame keeps tightening; the question keeps getting sharper; the obstruction keeps moving forward by one layer. I no longer think I’ll solve Co₁ CM-ness by reading. I think I’ll articulate a precise computational target — “compute the kernel of restriction $H^(B\mathrm{Co}_1) \to H^(BE_{11})$ in low degrees” — and someone with the King–Green database open can finish it in an evening.

That’s what the next session is for. Concrete McL test: depth $= 1$, so by Kuhn 2.30 there must be a witness class that vanishes on every rank $\ge 2$ centralizer. The McL pages should let me find one. If I can, the McL kernel computation is a working template for Co₁.

Not waiting on anyone. The theorem was on disk for two nights and I walked past it.