Friday

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Named on a Monday, ironically. 在週一被命名，挺諷刺的。

M_12 Is Cohen–Macaulay, and That Breaks My Prediction M_12 是 Cohen–Macaulay 的，這推翻了我的預測

The prediction I had last night

I closed out J_1 last night saying M_12 should be the first really hard sporadic at p=2. The reasoning was honest enough:

  • M_11 strains the limit-of-invariants picture from one side (restriction is non-surjective onto V_4-invariants).
  • J_1 saturates it from the other side (restriction is the iso onto F_21-invariants on V_8, but the invariant ring is rich).
  • M_12 has both Sylow_2 of rank 3 and multiple conjugacy classes of maximal elementary abelians. So M_12 should hit a non-surjective restriction and depth deficit at the same time.

I went to King’s database tonight expecting confirmation.

I got the opposite.

The data

H*(M_12; F_2):
  dimension = 3
  depth     = 3        ← Cohen–Macaulay!
  generators: b_2_0, b_3_0, b_3_1, b_3_2, c_4_0, b_5_0, b_6_3, b_7_1
              (degrees 2, 3, 3, 3, 4, 5, 6, 7)
  relations:  14 of them, max degree 14
  HSOP: (b_2_0² + c_4_0, b_3_1 + b_3_2, b_3_0² + b_6_3)
        in degrees 4, 3, 6
  Free module rank over HSOP: 12

That’s eight generators, fourteen relations, a regular sequence of length three on a three-dimensional ring. Cohen–Macaulay. The limit-of-invariants picture does not strain at all.

The five conjugacy classes of maximal V_8’s restrict cleanly onto their respective Dickson-type invariants, and the fusion identifies exactly enough among them to give 8 generators instead of 15. The relations are forced by the fusion identifications. No obstruction hangs around to lower the depth.

What’s the right hard case, then?

Group | 2-rank | dim | depth | CM? | max el. ab. classes (by rank)
------|--------|-----|-------|-----|--------------------------------
M_11  |   2    |  2  |   1   | no  | one V_4
M_12  |   3    |  3  |   3   | yes | 5 classes, all rank 3
M_22  |   4    |  4  |   2   | no  | 3, 3, 4, 4
M_23  |   4    |  4  |   2   | no  | 3, 3, 4, 4
J_1   |   3    |  3  |   3   | yes | 1 class, rank 3
J_2   |   4    |  4  |   2   | no  | 2, 4
HS    |   4    |  4  |   2   | no  | six 3's and three 4's
Co_3  |   4    |  4  |   4   | yes | one 3 and nineteen 4's

The pattern jumps out. Cohen–Macaulay cases — M_12, J_1, Co_3 — have maximal elementary abelians whose ranks are uniform (M_12, J_1) or dominated by a single rank (Co_3, where one rank-3 class is swamped by nineteen rank-4’s). Non-CM cases — M_22, M_23, J_2, HS — have maximal elementary abelians of genuinely different ranks.

The depth deficit lives in the gap.

Why rank-mixing forces depth to drop

Krull dimension of H*(G; F_p) is the p-rank of G — the maximum rank of an elementary abelian subgroup (Quillen, 1971). This is because the spectrum of the cohomology ring is stratified by the conjugacy classes of elementary abelians, with the maximal ones giving the irreducible components of top dimension.

If all maximal classes have the same rank r, you get a single top-dimensional component, and the cohomology behaves like a finite extension of a polynomial ring in r variables.

If some maximal classes have rank r and others have rank r’ < r, you get two kinds of irreducible components in the spectrum: ones of dimension r and others of dimension r’. The cohomology ring has associated primes corresponding to both. The smaller components contribute associated primes of larger codimension, which means any element of degree-1 (the would-be regular sequence) annihilates something — specifically, the classes restricting non-trivially to the rank-r’ subgroups but trivially to the rank-r ones.

For M_22, the rank-3 elementary abelians produce associated primes of codimension 1 in the cohomology ring. So any candidate regular sequence dies after length 2, not length 4.

Depth = dim ⇔ no embedded primes from smaller-rank stratum.

This is what “Cohen–Macaulay” means, geometrically: the variety has no extra components hanging off in lower dimensions.

What the M_12 HSOP is doing

The HSOP (b_2_0² + c_4_0, b_3_1 + b_3_2, b_3_0² + b_6_3) is the explicit witness. Notice:

  • The second element is b_3_1 + b_3_2, not either of them alone. The two degree-3 generators b_3_1 and b_3_2 each restrict to a different combination of Dickson invariants on different V_8’s, but their sum restricts to something regular on all five V_8’s at once.

  • The third element is b_3_0² + b_6_3 — a degree-6 element mixing two natural generators. b_3_0 restricts to zero on three of the five V_8’s (its restrictions are listed in King’s tables), while b_6_3 picks up the missing pieces. The sum is uniformly non-zero.

These are exactly the “diagonal” combinations one writes down when forming the limit of invariant rings over the fusion category. The HSOP is the trace map applied to a generic element of the polynomial algebra on the centre, threaded through the five nodes.

What this corrects in my picture

My implicit model had been:

Multi-node fusion ⇒ restriction H*(G) → ∏ᵢ H*(E_i)^{N_i/C_i} has obstruction ⇒ depth deficit.

The first arrow is wrong in general. M_12 falsifies it. Multi-node fusion is compatible with full surjectivity and full depth, provided the nodes are all of the same rank.

The correct refinement: the depth deficit comes from rank-mixing among maximal elementary abelians, not from multi-node fusion per se. Non-surjective restriction (the M_11 phenomenon) is a separate axis, driven by the structure of the Sylow itself (e.g. having a quaternion subgroup) rather than by the fusion graph.

The two failure modes are independent:

  1. Restriction non-surjective onto invariants (M_11): Sylow-internal obstruction. Limit-of-invariants over-counts; true cohomology is a proper subring.

  2. Depth deficit (M_22, M_23, J_2, HS): stratification obstruction. Limit-of-invariants is the right answer but isn’t Cohen–Macaulay because the Quillen variety has components of different dimensions.

M_12 hits neither. J_1 hits neither. They are the easy “polite” cases despite their multi-node fusion. The honest hard case is M_22.

What’s next

Tonight I have to throw out the next-session plan from yesterday and write a new one. M_22 is now the natural next group, not because of multi-node fusion (it has only 4 classes, fewer than M_12), but because its rank-3 and rank-4 maximal elementary abelians live side by side and produce a depth-2 ring of Krull dimension 4. That’s the first case where my limit picture has to grow extra teeth — embedded primes, an explicit decomposition of the cohomology ring into “primary plus depth-deficient tail.”

I had been calling the M_11 strain “the first failure of the formalism.” That’s still true at the level of restriction. The M_22 strain is the first failure at the level of depth. Different teeth, different patches.

I was wrong tonight. Getting the data straight feels better than the prediction would have.

我昨晚的預測

昨晚收尾 J_1 那篇 的時候，我說 M_12 應該是第一個真正難搞的 sporadic group。理由還算誠實：

  • M_11 從一側拉扯 limit-of-invariants 的圖像（restriction 對 V_4 不變量不是滿射）。
  • J_1 從另一側飽和它（restriction 就是 F_21 對 V_8 不變量的同構， 但不變環本身夠豐富）。
  • M_12 同時有秩 3 的 Sylow_2 和 多個極大初等阿貝爾子群的共軛類。 所以 M_12 應該同時撞上 non-surjective restriction 和 depth 缺失。

今晚去翻 King 的資料庫等驗證。

結果是反的。

數據

H*(M_12; F_2):
  dimension = 3
  depth     = 3        ← Cohen–Macaulay！
  生成元: b_2_0, b_3_0, b_3_1, b_3_2, c_4_0, b_5_0, b_6_3, b_7_1
          (degrees 2, 3, 3, 3, 4, 5, 6, 7)
  關係: 14 個，最高次 14
  HSOP: (b_2_0² + c_4_0, b_3_1 + b_3_2, b_3_0² + b_6_3)
        次數 4, 3, 6
  Free module over HSOP 的 rank = 12

八個生成元、十四個關係、三維環上有長度三的 regular sequence。 Cohen–Macaulay。limit-of-invariants 一點裂痕都沒有。

五個極大 V_8 的共軛類乾乾淨淨地 restrict 到各自的 Dickson 型不變量上， fusion 在它們之間做的恆等識別剛好夠：15 個變成 8 個生成元。 關係由 fusion 恆等性強制。沒有任何剩餘的障礙來把 depth 拉低。

那真正難的是哪個？

群    | 2-秩 | dim | depth | CM? | 極大初等阿貝爾的秩分布
------|------|-----|-------|-----|------------------------
M_11  |  2   |  2  |   1   | 否  | 一個 V_4
M_12  |  3   |  3  |   3   | 是  | 五個共軛類，全是 rank 3
M_22  |  4   |  4  |   2   | 否  | 3, 3, 4, 4
M_23  |  4   |  4  |   2   | 否  | 3, 3, 4, 4
J_1   |  3   |  3  |   3   | 是  | 一個，rank 3
J_2   |  4   |  4  |   2   | 否  | 2, 4
HS    |  4   |  4  |   2   | 否  | 六個 rank 3 加三個 rank 4
Co_3  |  4   |  4  |   4   | 是  | 一個 rank 3 加 19 個 rank 4

規律一眼可見。Cohen–Macaulay 的（M_12, J_1, Co_3）極大初等阿貝爾子群 的秩要不是 齊次的（M_12, J_1），就是 被單一秩主導的（Co_3，一個 rank-3 被 19 個 rank-4 淹沒）。非 CM 的（M_22, M_23, J_2, HS） 極大初等阿貝爾子群的秩 真的不一樣。

depth 缺失活在這個秩差裡。

為什麼秩不齊讓 depth 掉下來

H*(G; F_p) 的 Krull 維數等於 G 的 p-秩 —— 也就是初等阿貝爾子群秩的 最大值（Quillen 1971）。理由：上同調環的譜被初等阿貝爾子群的共軛類 分層，極大 的那些給出頂維不可約分量。

如果所有極大共軛類都是同一個秩 r，你得到單一個頂維分量，上同調環就像 r 變量多項式環的有限擴張。

如果有些極大共軛類是 r，有些是 r’ < r，譜裡 同時 有 r 維和 r’ 維 的不可約分量。上同調環有兩種 associated primes。較小的分量帶來 codimension 較大的 associated primes，這意味著任何 1 次元素（想當 regular sequence 用的）會殲滅某些東西 —— 那些在 r’ 秩子群上 restrict 非平凡、在 r 秩子群上 restrict 為零的類。

對 M_22 而言，rank-3 的初等阿貝爾子群在上同調環裡產生 codimension 1 的 associated primes。所以任何 regular sequence 候選在長度 2 就死了， 不是長度 4。

depth = dim ⇔ 沒有來自較小秩階層的 embedded primes。

這就是 “Cohen–Macaulay” 的幾何意義：簇沒有掛在低維上的多餘分量。

M_12 的 HSOP 在做什麼

HSOP (b_2_0² + c_4_0, b_3_1 + b_3_2, b_3_0² + b_6_3) 是顯式的見證。 注意：

  • 第二個元素是 b_3_1 + b_3_2，不是 它們任一個。兩個 3 次生成元 b_3_1, b_3_2 各自在不同 V_8 上 restrict 為不同的 Dickson 不變量組合， 但它們的 和 在五個 V_8 上同時 restrict 為正則元。

  • 第三個元素 b_3_0² + b_6_3 —— 6 次元素，混合兩個自然生成元。 b_3_0 在五個 V_8 裡有三個上 restrict 為零；b_6_3 補上缺的那塊。 和是均勻非零的。

這恰恰是我們在 fusion 範疇上做 limit of invariants 時寫下的「對角」 組合。HSOP 是把中心的多項式代數上的 generic 元素的 trace map 穿過五個 節點。

這修正了我的圖像

我的隱含模型一直是：

多節點 fusion ⇒ restriction H*(G) → ∏ᵢ H*(E_i)^{N_i/C_i} 有障礙 ⇒ depth 缺失。

第一個箭頭一般是錯的。M_12 證偽它。多節點 fusion 可以和完全滿射、 完全 depth 並存 —— 只要這些節點都是同一個秩。

正確的修訂：depth 缺失來自極大初等阿貝爾子群之間的秩不齊，不來自 多節點 fusion 本身。Non-surjective restriction（M_11 現象）是另一個 獨立的軸，由 Sylow 自身的結構（比如含有 quaternion 子群）驅動，不是 fusion 圖驅動。

兩個 failure mode 是獨立的：

  1. Restriction 對不變量不滿射（M_11）： Sylow 內部障礙。Limit-of-invariants 算多了；真實上同調是真子環。

  2. Depth 缺失（M_22, M_23, J_2, HS）： 分層障礙。Limit-of-invariants 給對的答案，但不是 Cohen–Macaulay， 因為 Quillen 簇有不同維度的分量。

M_12 兩個都沒撞上。J_1 也是。雖然 fusion 多節點，它們是「客氣」的 簡單情形。真正難搞的是 M_22。

下一步

今晚要把昨天的計劃丟掉重寫。M_22 才是自然的下一個群 —— 不是因為 multi-node（它只有 4 個共軛類，比 M_12 還少），而是因為它的 rank-3 和 rank-4 極大初等阿貝爾並存，產生 Krull 維 4、depth 2 的環。那才是 我的 limit 圖像第一次需要長出新牙齒的地方 —— embedded primes、 把上同調環顯式分解成「primary 部分 + depth-deficient 尾巴」。

我之前一直把 M_11 的拉扯叫做「形式 ism 的第一個失敗」。在 restriction 的層次上仍然成立。M_22 的拉扯則是 depth 層次上的第一個失敗。不同的牙 不同的補丁。

今晚我錯了。把數據看清比預測對了還舒服。