A COUNTEREXAMPLE TO THE mod mod mod-ppp VERSION

OF THE \overline{Ch Ch Ch}-CONJECTURE FOR ANY ODD PRIME ppp

PHAM ANH MINH

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

Abstract. The \overline{Ch}-conjecture states that, for every p-group G (p odd), H^{∗}(G, 𝔽_{p})/\sqrt{0} is generated by Chern classes of unitary representations of G and images under transfers from proper subgroups of G. It was known that, for p = 3, the extraspecial p-group of order 27 and exponent 3 was a counterexample to that conjecture. We now provide counterexample to that conjecture for any odd prime p.

Let p be an odd prime number. For every p-group G, denote by H^{∗}(G) = H^{∗}(G, 𝔽_{p}) the mod-p cohomology algebra of G. The Chern subring, Ch(G) ⊂ H^{∗}(G), is the subring generated by Chern classes of unitary representations of G. Define \overline{Ch}(G) recursively as the subring of H^{∗}(G) generated by Ch(G) and the image under transfer of \overline{Ch}(H) for proper subgroups H of G. The mod-p version of the \overline{Ch}-conjecture was stated in [9] that \overline{Ch}(G)/\sqrt{0} = H^{∗}(G)/\sqrt{0}. However, this conjecture does not hold, at least for p = 3. In fact, by considering the extraspecial p-group 𝔼 of order p^{3} and of exponent p, one can prove that, for p = 3, \overline{Ch}(𝔼)/\sqrt{0} ⫋ H^{∗}(𝔼)/\sqrt{0} (see [4, Prop. 9.4]).

It is reasonable to ask whether the conjecture holds for p > 3 (note that, in this case, \overline{Ch}(𝔼)/\sqrt{0} = H^{∗}(𝔼)/\sqrt{0}, see e.g. [5]). The purpose of this note is to construct a counterexample to this conjecture for any odd prime p, as follows. Given an odd prime p, let 𝔼_{2} be the extraspecial p-group of order p^{5} and of exponent p. Consider the group Γ = C_{p^{2}} ∗ 𝔼_{2}, the central product of the cyclic group C_{p^{2}} of order p^{2} and 𝔼_{2}, we prove

Theorem. \overline{Ch}(Γ)/\sqrt{0} is strictly contained in H^{∗}(Γ)/\sqrt{0}.

Proof of the theorem

The group Γ given above can be expressed as an extension of 𝔽_{p} by a vector space over 𝔽_{p} as follows. Let V be a vector space over 𝔽_{p} of dimension 5, with basis e, a_{1}, …, a_{4}. Let x, x_{1}, …, x_{4} be a basis of H^{1}(V), dual to that of V and let y = βx, y_{i} = βx_{i} with β the Bockstein homomorphism, so

H = H^{∗}(V) = E[x, x_{1}, …, x_{4}] ⊗ P [y, y_{1}, …, y_{4}],

with E[u, v, …] (resp. P [u, v, …]) the exterior (resp. polynomial) algebra over 𝔽_{p} with generators u, v, … of order 1 (resp. 2). We have the central extension

(Γ)  0 → 𝔽_{p} →^{i} Γ → V → 0

corresponding to the cohomology class z = y + x_{1}x_{2} + x_{3}x_{4} ∈ H^{2}(V).

We shall use the following notation. A will denote the set of maximal elementary abelian p-subgroups of Γ. If W is a vector subspace of V of dimension k and u_{1}, …, u_{k} a system of generators of βH^{1}(W), then Q_{k,m}(W) = Q_{k,m}(u_{1}, …, u_{k}), 0 ≤ m ≤ k, are Dickson invariants with variables u_{1}, …, u_{k} in H^{∗}(W) defined inductively as follows (we shall omit the variables, if no confusion can arise):

1991 Mathematics Subject Classification. 20J06.

Key words and phrases. Cohomology of finite groups.

Q_{k,k} = 1,
Q_{k,0} = ∏_{λ_{j}∈𝔽_{p} λ_{j} not all equal to 0}(λ_{1}u_{1} + … + λ_{k}u_{k})^{p−1},

Q_{k,m} = Q_{k−1,m}(Q_{k,0}/Q_{k−1,0}) + Q_{k−1,m−1}^{p}.

With some abuse of notation, we also denote by u_{i} the element Inf_{Γ}^{W} (u_{i}). Hence the Q_{k,r}(u_{1}, …, u_{k})’s can be considered as elements of H^{∗}(Γ). If ϕ, ψ, … are elements of a ring R, we denote by (ϕ, ψ, …)_{R}, or simply by (ϕ, ψ, …), the ideal in R generated by ϕ, ψ, . . ..

Set Z = i(𝔽_{p}). For every subgroup K of Γ containing Z, by setting K′ = K/Z, we have the induced central extension

(K)  0 → 𝔽_{p} → K → K′ → 1

with factor set z|_{K′} = Res_{K′}^{V}z. Denote by {E_{r}(K), d_{r}} the corresponding Hochschild-Serre spectral sequence. So

E_{2}(K) = H^{∗}(K′) ⊗ H^{∗}(Z),
and  E_{∞}^{i,j}(K) = F^{i}H^{i+j}(K)/F^{i+1}H^{i+j}(K),

with {F^{k}H^{∗}(K)} the corresponding Hochschild-Serre filtration of H^{∗}(K).

It is clear that the restriction on cochain levels from C^{∗}(Γ) to C^{∗}(K) induces a restriction map between spectral sequences {_{r}Res_{K}^{Γ} : E_{r}(Γ) → E_{r}(K)}, and the following diagram is commutative

(∗)  F^{i}H^{i+j}(Γ] ⟶^{pr} E_{∞}^{i,j}(Γ) ^{Res}_{K}^{Γ} ↓  ↓_{∞}^{Res}_{K}^{Γ} F^{i}H^{i+j}(K) ⟶_{pr} E_{∞}^{i,j}(K) ,

where the pr’s are projection maps.

For every X ∈ H^{k}(K), set d_{X} = max{d|0 ≤ d ≤ k, X ∈ F^{d}H^{k}(K)} Consider the projection map

\begin{matrix}θ_{K} : H^{∗}(K) & → E_{∞}(K) \\ X & ↦ X + F^{d_{X}+1}H^{∗}(K) \end{matrix}

(note that θ_{K} is not a homomorphism). Let u and v be elements of H^{∗}(Z) so that H^{∗}(Z) = E[u] ⊗ P [v]. We have the following

Lemma 1. If K is maximal in Γ and 0 ≠ w is a cohomology class of K of even degree, then d_{w} ≠ 1. Furthermore, if d_{w} = 0, then |w| is a multiple of p.

Proof. Consider the term E_{3}(K). As d_{2}(u) = z|_{K′}, we have

\begin{matrix}E_{3}(K) = & H^{∗}(K′)/(z|_{K′}) ⊗ 𝔽_{p}[v] \\ ⊕ Ann_{H^{∗}(K′)}(z|_{K′}) ⊗ 𝔽_{p}[v]u. \end{matrix}

Direct verifications show that Ann_{H^{∗}(K′)}(z|_{K′})∩H^{1}(K′) = {0}. Hence E_{3}^{1,2k+1}(K) = E_{∞}^{1,2k+1}(K) = {0}. The first part of the lemma follows. As d_{3}(v) = βz|_{K′} ≠ 0 mod(z|_{K′}), it follows that d_{3}(v^{n}) = 0 if and only if n is a multiple of p. So d_{w} = 0 implies that |w| is a multiple of p. The lemma is proved. □

Following [6] (see also [1]), we have Proposition 2 ([1], [6]). If K = Kerx_{3} ∩ Kerx_{4}, then

\begin{aligned}E_{∞}(K) & = H^{∗}(K′)/(y + x_{1}x_{2}, y_{1}x_{2} − y_{2}x_{1}, y_{1}^{p}x_{2} − y_{2}^{p}x_{1}, y_{1}^{p}y_{2} − y_{2}^{p}y_{1}) ⊗ 𝔽_{p}[v^{p}] \\ & ⊕ (𝔽_{p}x_{1}x_{2} ⊕ 𝔽_{p}xx_{1}x_{2}) ⊗ ∑_{i=1}^{p−2} 𝔽_{p}[v^{p}]v^{i}. □ \end{aligned}

The term E_{∞}(Γ) was given in [8] as follows.

Theorem 3 ([8]).

E_{∞}(Γ) = H′/((y_{1}x_{2} − y_{2}x_{1})Q, P^{p}P^{1}βz, βP^{p}P^{1}βz) ⊗ 𝔽_{p}[v^{p^{2}}]

⊕ (𝔽_{p}x_{1}x_{2}x_{3}x_{4} ⊕ 𝔽_{p}xx_{1}x_{2}x_{3}x_{4}) ⊗ 𝔽_{p}[v^{p^{2}}]v^{p(p−1)}

⊕ (y)_{H′} ⊗ ∑_{i=1}^{p−2} 𝔽_{p}[v^{p^{2}}]v^{pi}

⊕ (𝔽_{p}x_{1}x_{2}x_{3}x_{4} ⊕ 𝔽_{p}xx_{1}x_{2}x_{3}x_{4}) ⊗ ∑_{i=1}^{p−3} 𝔽_{p}[v^{p}]v^{i}

⊕ (𝔽_{p}x_{1}x_{2}x_{3}x_{4} ⊕ 𝔽_{p}xx_{1}x_{2}x_{3}x_{4}) ⊗ ∑_{i=1}^{p−1} 𝔽_{p}[v^{p^{2}}]v^{ip−2},

with P^{i} the Steenrod operations, H′ = H/(z, βz, P^{1}βz, βP^{1}βz) and Q = Q_{2,1}(y_{1}, y_{2}) − Q_{2,1}(y_{3}, y_{4}). □

The following is then straightforward from the above theorem.

Corollary 4. If X ∈ H^{2(p^{2}−p)}(Γ) and d_{X} < 2(p^{2} − p), then θ_{Γ}(X) ∈ E_{∞}^{2(p^{2}−(i+1)p),2ip}(Γ), for some 1 ≤ i ≤ p − 2. □

Proposition 5. If 0 ≠ X ∈ H^{2(ip+1)}(Γ), 1 ≤ i ≤ p − 2, and X belongs to the image under transfer from proper subgroups of Γ, then d_{X} > 2.

Proof. Let K be a maximal subgroup of Γ and let Y = t_{Γ}^{K}(w), with t_{Γ}^{K} the transfer map from H^{∗}(K) to H^{∗}(G) and w ∈ H^{2(ip+1)}(K). It follows from Theorem 2 that d_{Y} ≥ 2. If d_{Y} = 2, according to [7, Th. 1.1], we have either d_{w} = 1 or d_{w} = 0; by Lemma 1, d_{w} = 0 and |w| is a multiple of p, which contradicts the fact that |Y | = 2(ip + 1). Hence d_{Y} > 2. The proposition follows, by noting that X is expressed as a sum of the Y ’s. □

We now consider the subring Ch(Γ) of H^{∗}(Γ). By Whitney sum formula, it is clear that Ch(Γ) is generated, as an algebra, by Chern classes of irreducible representations of Γ. Such classes can be obtained by the same argument used in [3] (see also [4], for a construction using the transfer) as follows. Fix a nontrivial linear character χ of Z. We have then an irreducible character χ̂ of Γ given by

χ̂(g) = \{ \begin{matrix}p^{2}χ(g) & , if g ∈ Z, \\ 0 & , otherwise. \end{matrix}

Let ρ be a representation of Γ affording the character χ̂, and let ϱ be its restriction to 𝔼_{2} =Kerx. We have

Proposition 6 (compare [2, Th. 20], [3, Prop. 7], [4, Th. 5.2]). There exist elements κ_{i} ∈ H^{2(p^{2}−p^{i})}(Γ), 0 ≤ i ≤ 1, and ζ ∈ H^{2p^{2}}(Γ) such that

(i) c(ρ) = 1 + κ_{1} + κ_{0} + ζ mod \sqrt{0}; (ii) Ch(Γ)/\sqrt{0} is generated by y, y_{1}, y_{2}, y_{3}, y_{4}, κ_{0}, κ_{1}, ζ; (iii) by setting κ′_{i} = Res_{𝔼_{2}}^{Γ}κ_{i} and ζ′ = Res_{𝔼_{2}}^{Γ}ζ, we have c(ϱ) = 1 + κ′_{1} + κ′_{0} + ζ′ mod \sqrt{0}. Furthermore,

as elements of H^{∗}(𝔼_{2})/\sqrt{0}, we have P^{1}κ′_{1} = κ′_{0}; (iv) for every 0 ≤ i ≤ 1 and for every A ∈ A, we have Res_{A}^{Γ}(κ_{i}) = Q_{2,i}(A′), with A′ = A/Z;

(v) κ_{i} ∉ ImInf_{Γ}^{V}, as elements of Ch(Γ)/\sqrt{0}.

Proof. Parts (i), (ii) can be proved by arguments similar to the ones used in proving [3, Prop.7]. (iii) follows from [3, Lemma 3 and Prop. 7] and [4, Th. 5.2]. (iv) follows from [3, Prop. 7] and the fact that Res_{A}^{Γ}(κ_{i}) = Res_{A}^{𝔼_{2}}(κ′_{i}). Finally, it follows from [2, Th. 20] that κ′_{0} ∉ ImInf_{𝔼_{2}}^{𝔼′_{2}}, hence κ_{0} ∉ ImInf_{Γ}^{V};

furthermore, if κ_{1} ∈ ImInf_{Γ}^{V}, this implies that κ′_{1} belongs to ImInf_{𝔼_{2}}^{𝔼′_{2}}, hence, by (iii), so does κ′_{0} = P^{1}κ′_{1}, a contradiction. The proposition follows. □

We have then

Proposition 7. If X ∈ H^{2(ip+1)}(Γ), 1 ≤ i ≤ p − 2, and d_{X} = 2, then X ∉ Ch(Γ).

Proof. Set K =Kerx_{3}∩ Kerx_{4}. By the commutative diagram (∗) and by Proposition 2, we have 0 ≠ μy ⊗ v^{ip} = θ_{K}Res_{K}^{Γ}(X) ∈ E_{∞}^{2,2ip}(K), with μ ∈ 𝔽_{p}. Therefore Res_{K}^{Γ}(X) ≠ 0. Assume that X ∈ Ch(Γ). Note that, if σ is a irreducible representation of Γ and if c_{j}(σ) ≠ 0 with 1 ≤ j < p^{2}, it follows from Theorem 3 that d_{c_{j}(σ)} ≥ 2. As Ch(G) is generated by Chern classes of irreducible representations of Γ, d_{X} = 2 implies that X = λc_{1+pi}(ρ), with 0 ≠ λ ∈ 𝔽_{p}. Furthermore, since Res_{K}^{Γ}(ρ) = pα, with α a suitable representation of K, and Res_{K}^{Γ}(X) ≠ 0, it follows that |X| is a multiple of p, which contradicts the fact that |X| = 2(ip + 1). So X ∉ Ch(Γ). The proposition follows. □

Proof of the main theorem. Let us now fix homogeneous elements X_{1}, …, X_{p−2} of H^{∗}(Γ) satisfying θ_{Γ}(X_{i}) = y ⊗ v^{ip} ∈ E_{∞}^{2,2ip}(Γ). Assume that X_{i} ∈ \overline{Ch}(Γ), for 1 ≤ i ≤ p − 2. It follows from Proposition 5 that there exists Y_{i} ∈ Ch^{2(ip+1)}(Γ) satisfying d_{Γ}(X_{i}) = d_{Γ}(Y_{i}) = 2, which contradicts Proposition 7. It remains only to prove that there exists i, 1 ≤ i ≤ p − 2, such that X_{i} is not nilpotent. By Corollary 4, there exist Y ∈ H^{2(p^{2}−p)}(V) and Y_{i} ∈ H^{2(p^{2}−(i+1)p−1)}(V), 1 ≤ i ≤ p − 2, satisfying

κ_{1} = Y + Y_{1}X_{1} + ⋯ + Y_{p−2}X_{p−2}.

If all the X_{i}’s are nilpotent, there exists an integer m > 0 such that (κ_{1} − Y)^{p^{m}} = 0. Let Ỹ be the sum of all terms of Y belonging to P [y, y_{1}, …, y_{4}]. It follows that (κ_{1} − Ỹ)^{p^{m}} = 0. Hence, for every A ∈ A, by Proposition 6 (iv), Res_{A}^{Γ}(Ỹ^{p^{m}}) = Q_{2,1}(A′)^{p^{m}}, which implies that Res_{A}^{Γ}(Ỹ) = Q_{2,1}(A′). This contradicts Proposition 6 (v). The theorem is then proved.

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