NILPOTENCY DEGREE OF COHOMOLOGY

RINGS IN CHARACTERISTIC 3

Pham Anh Minh

Abstract. The purpose of this note is to provide a 3-group G whose mod-3 cohomology ring has a nilpotent element ξ ∈ H^{∗}(G) satisfying ξ^{3} ≠ 0.

1. Statement of main result

For every p-group G, denote by H^{∗}(G) the mod-p cohomology algebra of G. We are now interested in the nilpotency degrees of elements of H^{∗}(G). For the case p = 2, in [1,2], it was shown that, given any positive integer n, there exists a 2-group whose cohomology ring has elements of nilpotency degree n + 1. However, the methods given there do not generalize to the case of odd characteristic p. In this case, it seems that, until now, we do not have any example of elements of H^{∗}(G) having nilpotency degrees greater than p.

The purpose of this note is to provide a 3-group G whose cohomology ring has a nilpotent element ξ ∈ H^{∗}(G) satisfying ξ^{3} ≠ 0. The group G is obtained as follows. Let

𝔼 = ⟨a_{1}, a_{2}|a_{1}^{3} = a_{2}^{3} = [a_{1}, a_{2}]^{3} = [a_{1}, [a_{1}, a_{2}]] = [a_{2}, [a_{1}, a_{2}]] = 1⟩

be the extraspecial 3-group of order 3^{3} and of exponent 3, and let

K = C_{3}^{4} = ⟨a_{3}, …, a_{6}|a_{i}^{3} = [a_{i}, a_{j}] = 1⟩

be the elementary abelian 3-group of rank 4. Set G = 𝔼 × K. Define u_{i} ∈ H^{1}(G) = Hom(G, ℤ/3), v_{i} ∈ H^{2}(G), 1 ≤ i ≤ 6, by

\begin{aligned}u_{i}(a_{j}) & = δ_{ij}, \\ v_{i} & = βu_{i}, \end{aligned}

with δ_{ij} the Kronecker symbol and β the Bockstein homomorphism. For 3 ≤ i ≤ 6, consider u_{i}, v_{i} as elements of H^{∗}(K) via the restriction map. So, by Künneth formula,

H^{∗}(K) = Λ[u_{3}, …, u_{6}] ⊗ 𝔽_{3}[v_{3}, …, v_{6}],

and H^{∗}(G) may be identified with

H^{∗}(𝔼) ⊗ H^{∗}(K).

Consider the central extension

(G)  0 → ℤ/3 →^{i} G → G → 1

corresponding to the cohomology class z = v_{1} + u_{3}u_{4} + u_{5}u_{6} ∈ H^{2}(G). We will prove

Typeset by AMS-TEX

Theorem. There exists a nilpotent element ξ ∈ H^{2}(G) satisfying ξ^{3} ≠ 0.

However, for p > 3, our method could not be applied. So, the existence of a p-group G having cohomology class of nilpotency degree greater than p > 3 still remains as an open problem. Besides, we could not provide any example of cohomology classes of a 3-group having nilpotency degrees greater than a given n; although, it is known that, for any p ≥ 2 and for any p-group G of order p^{m}, nilpotency degrees of nilpotent elements of H^{∗}(G) are upperly bounded by p^{m−1} (see [5]).

2. Proof of main result

Via the restriction map, for 1 ≤ i ≤ 2, u_{i} and v_{i} can be considered as elements of H^{∗}(𝔼). Set c = [a_{1}, a_{2}] and let A_{0} = ⟨a_{1}, c⟩, A_{1} = ⟨a_{1}a_{2}, c⟩, A_{2} = ⟨a_{1}^{2}a_{2}, c⟩, A_{3} = ⟨a_{2}, c⟩ be elementary abelian subgroups of 𝔼. Set s_{0} = Res_{A_{0}}^{𝔼}(u_{1}), s_{i} = Res_{A_{i}}^{𝔼}(u_{2}), 1 ≤ i ≤ 3, and let t_{j} = βs_{j}, 0 ≤ j ≤ 3. Let u be a generator of H^{1}(⟨c⟩) and set v = βu ∈ H^{2}(⟨c⟩). It follows that

H^{∗}(⟨c⟩) = Λ[u] ⊗ 𝔽_{3}[v].

So, by Künneth formula, we have

H^{∗}(A_{i}) = Λ[s_{i}, u] ⊗ 𝔽_{3}[t_{i}, v],

0 ≤ i ≤ 3.

It follows from [4] that H^{∗}(𝔼) is detected by the A_{i}’s. According to [3], we have

Lemma 1.

(i) There exist elements U_{1}, U_{2} ∈ H^{2}(𝔼) such that: (ia) v_{1}, v_{2}, U_{1}, U_{2} is a basis of H^{2}(𝔼). Furthermore, we have

u_{1}|_{A_{0}} = s_{0}, u_{1}|_{A_{1}} = s_{1},  u_{1}|_{A_{2}} = 2s_{2},  u_{1}|_{A_{3}} = 0,
v_{1}|_{A_{0}} = t_{0}, v_{1}|_{A_{1}} = t_{1},  v_{1}|_{A_{2}} = 2t_{2},  v_{1}|_{A_{3}} = 0,
u_{2}|_{A_{0}} = 0,  u_{2}|_{A_{1}} = s_{1},  u_{2}|_{A_{2}} = s_{2},  u_{2}|_{A_{3}} = s_{3},
v_{2}|_{A_{0}} = 0,  v_{2}|_{A_{1}} = t_{1},  v_{2}|_{A_{2}} = t_{2},  v_{2}|_{A_{3}} = t_{3},
U_{1}|_{A_{0}} = s_{0}u, U_{1}|_{A_{1}} = s_{1}u + t_{1}, U_{1}|_{A_{2}} = 2s_{2}u + t_{2}, U_{1}|_{A_{3}} = 0,
U_{2}|_{A_{0}} = 0,  U_{2}|_{A_{1}} = 2s_{1}u + t_{1}, U_{2}|_{A_{2}} = 2s_{2}u + 2t_{2}, U_{2}|_{A_{3}} = s_{3}u;

(ib) v_{1}^{2}, v_{1}v_{2}, v_{2}^{2}, v_{1}U_{1}, v_{1}U_{2}, v_{2}U_{1}, v_{2}U_{2} is a basis of H^{4}(𝔼) and U_{1}^{2} = v_{1}U_{2}, U_{2}^{2} =

v_{2}U_{1}, U_{1}U_{2} = v_{1}v_{2}.

(ii) v_{1}u_{1}, v_{1}u_{2} is linearly independent in H^{3}(𝔼).

Lemma 2. Let X be an element of H^{∗}(G) of degree n ≤ 2. Then X = 0 provided that one of the following conditions is satisfied:

(i) Xv_{1} = 0;

(ii) Xz = 0.

Proof. The case n = 1 is trivial. For n = 2, write X = a_{2,0} + a_{1,1} + a_{0,2} with a_{i,j} ∈ H^{i}(𝔼) ⊗ H^{j}(K). If Xv_{1} = 0, then a_{2,0}v_{1} = a_{1,1}v_{1} = a_{0,2}v_{1} = 0, hence a_{0,2} = 0, and, by Lemma 1, a_{2,0} = a_{1,1} = 0; so X = 0. If Xz = 0, then

\begin{aligned}a_{2,0}v_{1} & = 0, \\ a_{2,0}(u_{3}u_{4} + u_{5}u_{6}) + a_{0,2}v_{1} & = 0, \\ a_{1,1}v_{1} = a_{0,2}(u_{3}u_{4} + u_{5}u_{6}) = a_{1,1}(u_{3}u_{4} + u_{5}u_{6}) & = 0, \end{aligned}

hence, it follows from what we just proved that a_{2,0} = a_{1,1} = a_{0,2} = 0, so X = 0. The lemma follows.

Lemma 3. The cup-product with v_{1}^{2} is an injective map from H^{2}(𝔼) to H^{6}(𝔼).

Proof. If v_{1}^{2}(λ_{1}v_{1}+λ_{2}v_{2}+μ_{1}U_{1}+μ_{2}U_{2}) = 0 with λ_{i}, μ_{i} ∈ ℤ/3, then restricting to A_{j}, 0 ≤ j ≤ 3 yields λ_{1} = λ_{2} = μ_{1} = μ_{2} = 0. The lemma follows.

For elements X, Y, … of H^{∗}(G), denote by (X, Y, …) the ideal of H^{∗}(G) generated by X, Y, . . .. We have

Lemma 4. Let X be an element of H^{1}(G) with Xβz ∈ (z). Then X = 0.

Proof. Write Xβz = Y z with Y ∈ H^{2}(G). Since X and βz are free of v_{1}, it follows that Y v_{1} = 0. By Lemma 2, Y = 0, so Xβz = 0. Write X = ∑_{i=1}^{6} λ_{i}u_{i}. A direct verification shows that λ_{1} = ⋯ = λ_{6} = 0. The lemma follows.

Lemma 5. v_{1}^{2}v_{2} ∉ (z, βz).

Proof. Assume that v_{1}^{2}v_{2} ∈ (z, βz). There exist then elements a_{i,j}, b_{i,j} ∈ H^{i}(𝔼) ⊗ H^{j}(K) satisfying

v_{1}^{2}v_{2} = ∑ a_{i,j}z + ∑ b_{i,j}βz
= ∑ a_{i,j}(v_{1} + u_{3}u_{4} + u_{5}u_{6}) + ∑ b_{i,j}(v_{3}u_{4} − v_{4}u_{3} + v_{5}u_{6} − v_{6}u_{5}).  (1)

By decomposing (1), we get

v_{1}^{2}v_{2} = a_{4,0}v_{1},  (2)
0 = a_{4,0}(u_{3}u_{4} + u_{5}u_{6}) + a_{2,2}v_{1}.  (3)
0 = a_{2,2}(u_{3}u_{4} + u_{5}u_{6}) + a_{0,4}v_{1} + b_{2,1}(v_{3}u_{4} − v_{4}u_{3} + v_{5}u_{6} − v_{6}u_{5}).  (4)

It follows from (3) and Lemma 2 that a_{4,0} contains v_{1} as a factor. By Lemma 3, a_{4,0} = v_{2}v_{1}. Hence, by (3) and Lemma 2, a_{2,2} = −v_{2}(u_{3}u_{4} + u_{5}u_{6}). So, from (4), we get

2v_{2}u_{3}u_{4}u_{5}u_{6} = a_{0,4}v_{1} + b_{2,1}(v_{3}u_{4} − v_{4}u_{3} + v_{5}u_{6} − v_{6}u_{5}),

a contradiction. The lemma follows.

With some abuse of notation, we consider elements of H^{∗}(G) as elements of H^{∗}(G) via the inflation map. Set ξ = U_{1} ∈ H^{2}(G). The first part of the theorem follows from Lemma 6. ξ is nilpotent.

Proof. It follows from Lemma 1 that ξ^{2} = v_{1}U_{2}, hence

ξ^{3} = v_{1}U_{1}U_{2} = v_{1}^{2}v_{2}.

So ξ^{9} = (v_{1}^{3})^{2}v_{2}^{3}. Since, in H^{∗}(G), z = 0, we have v_{1}^{3} = (−u_{3}u_{4} − u_{5}u_{6})^{3} = 0. It follows that ξ^{9} = 0. Therefore ξ is nilpotent. The lemma is proved.

The proof of the theorem is completed by the following.

Lemma 7. ξ^{3} ≠ 0 in H^{∗}(G).

Proof. Assume that ζ = ξ^{3} = v_{1}^{2}v_{2} = 0 in H^{∗}(G). So ζ ∈ ImInf_{G}^{G}. Set Z = i(ℤ/3) ⊂ G. Denote by {E_{r}, d_{r}} the Hochschild-Serre spectral sequence corresponding to the extension (G). So

E_{2} = H^{∗}(G) ⊗ H^{∗}(Z).

Let s be a generator of H^{1}(Z) and set t = βs ∈ H^{2}(Z). s can be chosen such that d_{2}(s) = z. t is then transgressive and d_{3}(t) = βz. So, by Lemma 5, ζ ≠ 0 in E_{4}^{6,0}. Since ζ ∈ ImInf_{G}^{G}, it must be hitted by d_{4} or d_{5}. If ζ = d_{4}(a ⊗ st) with a ⊗ st ∈ E_{4}^{2,3}, then

d_{2}(a ⊗ st) = (a ⊗ t)d_{2}(s)

= az ⊗ t

= 0

in E_{2}^{4,2}; so az = 0 in H^{∗}(G), hence a = 0 by Lemma 2, a contradiction. If ζ = d_{5}(b ⊗ t^{2}) with b ⊗ t^{2} ∈ E_{5}^{1,4}, then

d_{3}(b ⊗ t^{2}) = (b ⊗ 1)d_{3}(t^{2})

= (b ⊗ 1)(2βz ⊗ t)

= 2bβz ⊗ t

= 0

in E_{3}^{4,2}; so bβz ∈ (z), hence b = 0 by Lemma 4, a contradiction. Therefore ζ ≠ 0. The lemma follows.

References

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2. K. Inoue, A. Kono, Nilpotency of a kernel of the Quillen map, J. Math. Kyoto Univ. 33 (1993), 1047-1055.

3. I.J. Leary, The mod-p cohomology rings of some p-groups, Math. Proc. Camb. Phil. Soc. 112 (1992), 63-75.

4. R.J. Milgram, M. Tezuka, The geometry and cohomology of M_{12}. II, Bol. Soc. Mat. Mexicana 1 (1995), 91-108.

5. P.A. Minh, Essential mod-p cohomology classes of p-groups: an upper bound for the nilpotency degree, to appear in Bull. London Math. Soc.

Department of Mathematics, College of Sciences, University of Hue, Dai hoc Khoa hoc, Hue, Vietnam

   E-mail address: paminh@dng.vnn.vn