Spherical classes and the Dickson algebra

Nguyễn H. V. Hu’ng  Franklin P. Peterson

November 4, 1996

Abstract

We attack the conjecture that the only spherical classes in the homology of Q_{0}S^{0} are Hopf invariant one and Kervaire invariant one elements. We do this by computing products in the E^{2}-term of the unstable Adams spectral sequence converging to π_{∗}(Q_{0}S^{0}) using results about the Dickson algebra and by studying the Lannes–Zarati homomorphism.

1 Introduction

In this paper, we investigate the spherical classes in Q_{0}S^{0}, i.e. the elements belonging to the image of the Hurewicz homomorphism

H : π_{∗}^{s}(S^{0}) ≅ π_{∗}(Q_{0}S^{0}) → H_{∗}(Q_{0}S^{0}; F_{2}) .

We are interested in the following classical conjecture.

Conjecture 1.1 (conjecture on spherical classes). There are no spherical classes in Q_{0}S^{0}, except the Hopf invariant one and the Kervaire invariant one elements.

(See Curtis [7] and Wellington [21] for a discussion.)

Let P_{k} = F_{2}[x_{1}, …, x_{k}] be the polynomial algebra on k generators x_{1}, …, x_{k}, each of dimension 1. The general linear group GL_{k} = GL(k, F_{2}) acts on P_{k} in the usual way. Let D_{k} be the Dickson algebra of k variables, i.e. the algebra of invariants

D_{k} := F_{2}[x_{1}, …, x_{k}]^{GL_{k}} .

As the action of the (mod 2) Steenrod algebra, A, and that of GL_{k} on P_{k} commute with each other, D_{k} is an algebra over A.

11991 Mathematics Subject Classification. Primary 55P47, 55Q45, 55S10, 55T15.

2Key words and phrases. Spherical classes, Loop spaces, Adams spectral sequences, Steenrod algebra, Invariant theory, Dickson algebra.

3The first-named author was supported in part by the DGU through the CRM (Barcelona).

4The second-named author was supported in part by the NSF and the Alexander von Humboldt Foundation.

One way to attack Conjecture 1.1 is to study the Lannes–Zarati homomorphism

φ_{k} : Ext_{A}^{k,k+i}(F_{2}, F_{2}) → (F_{2} ⊗_{A} D_{k})_{i}^{∗} ,

which is compatible with the Hurewicz homomorphism (see [10], [11, p. 46]). The domain of φ_{k} is the E_{2}-term of the stable Adams spectral sequence converging to π_{∗}^{s}(S^{0}). Furthermore, by a theorem of Madsen [13], the range of φ_{k} is E_{k}^{2}, the submodule of H_{∗}(Q_{0}S^{0}; F_{2}) consisting of all A-annihilated primitives which correspond to Dyer–Lashof operations of length k. The direct sum of E_{k}^{2} running over all k makes the filtration 0 line of the E^{2}- term of the unstable Adams spectral sequence converging to π_{∗}(Q_{0}S^{0}). By compatibility of φ_{k} and the Hurewicz homomorphism we mean φ_{k} is a “lifting” of the latter from the “E_{∞}-level” to the “E_{2}-level”.

Let h_{r} denote the Adams element in Ext_{A}^{1,2^{r}}(F_{2}, F_{2}). Lannes and Zarati proved in [11] that φ_{1} is an isomorphism with {φ_{1}(h_{r}) | r ≥ 0} forming a basis of the dual of F_{2} ⊗_{A} D_{1} and

φ_{2} is surjective with {φ_{2}(h_{r}^{2})| r ≥ 0} forming a basis of the dual of F_{2} ⊗_{A} D_{2}. Recall that, from Adams [1], the only Hopf invariant one elements are represented by h_{1}, h_{2}, h_{3} of the stems i = 2^{r} − 1 = 1, 3, 7, respectively. Moreover, by Browder [5], the only dimensions where a Kervaire invariant one element would occur are 2(2^{r} − 1), for r > 0, and it really occurs at this dimension if and only if h_{r}^{2} is a permanent cycle in the Adams spectral sequence for the spheres.

Therefore, Conjecture 1.1 is a consequence of the following conjecture.

Conjecture 1.2 φ_{k} = 0 in any positive stem i for k > 2.

One of our main results is that φ_{3} = 0 in positive stems, φ_{4} = 0 in positive stems < 89 and that φ_{k} vanishes on decomposable elements for 2 < k < 10. The ingredients to prove this are:

(1) show that φ = ⊕φ_{k} is a homomorphism of algebras (this is done in Section 3),

(2) show that E_{k}^{2} has a trivial algebra structure except for E_{1}^{2} ⊗ E_{1}^{2} → E_{2}^{2} (this is done in Theorem 4.1), and

(3) show that φ_{4} vanishes on known indecomposable elements (this is done in Section 5).

Our other main result is the definition and properties of some elements ω_{k}(n) ∈ D_{k}, of dimension n, which are defined if 2^{k−α(n)} | n. These elements seem to be important in the structure of D_{k} as an A-module (see Arnon [3]). These elements are defined in Section 2 and used to prove the triviality of the algebra structure in ⊕E_{k}^{2} in Section 4. Finally, in Section 6 we give another example where D_{k} for k ≥ 2 differs very much from D_{1}.

Acknowledgment. The authors would like to thank Manuel Castellet and all their colleagues at the CRM, Barcelona, for the hospitality and for an excellent working condition.

2 Some new elements in the Dickson algebra

Let S_{k} = P_{k}^{Σ_{k}} = H^{∗}(BO(k); F_{2}) be the subalgebra of P_{k} of elements invariant under the symmetric group. It is well-known that S_{k} = F_{2}[σ_{1}, …, σ_{k}], where σ_{i} denotes the i-th symmetric function on x_{1}, …, x_{k}.

Dickson [8] proved that D_{k} = F_{2}[Q_{k−1}, …, Q_{0}], where |Q_{s}| = 2^{k} − 2^{s}. Note that Q_{s} depends on k and when necessary will be denoted Q_{k,s}.

The following elements seem to be important in studying the Dickson algebra.

Definition 2.1 Let ω(n) = ω_{k}(n) = ∑ x_{1}^{i_{1}} … x_{k}^{i_{k}} ∈ S_{k}, where the sum is over all sequences i_{1}, …, i_{k} with i_{j} either 0 or a power of 2 and ∑ i_{j} = n. This sum is invariant under Σ_{k} and hence lies in S_{k}.

Let α(n) be the number of 1’s occuring in the dyadic expansion of n.

We note that ω_{k}(n) = 0 if k < α(n). The following theorem shows when ω_{k}(n) ∈ D_{k} if k ≥ α(n).

Theorem 2.2 ω_{k}(n) ∈ D_{k} if and only if 2^{k−α(n)} | n.

We need the following easy lemma from elementary linear algebra.

Lemma 2.3 If σ ∈ P_{k} is invariant under the action of Σ_{k} and that of T, where T (x_{1}) = x_{1} + x_{2} and T (x_{i}) = x_{i}, i ≥ 2, then σ is invariant under GL_{k}, in other words, σ ∈ D_{k}.

Proof of Theorem 2.2.

We want to find conditions on ω(n) so that (T − id)(ω(n)) = 0. Here are four easy properties.

(a) (T − id)(x_{2}^{i_{2}} ⋯ x_{k}^{i_{k}}) = 0

(b) If i_{1} ≠ i_{2}, both > 0, then (T − id)(x_{1}^{i_{1}}x_{2}^{i_{2}} ⋯ x_{k}^{i_{k}} + x_{1}^{i_{2}}x_{2}^{i_{1}} ⋯ x_{k}^{i_{k}}) = 2x_{2}^{i_{1}+i_{2}}x_{3}^{i_{3}} ⋯ = 0.

(c) If i_{1} > 0 and i_{2} = 0, then (T − id)(x_{1}^{i_{1}}x_{3}^{i_{3}} ⋯) = x_{2}^{i_{1}}x_{3}^{i_{3}} · · ·.

(d) If i_{2} > 0, i_{2} ≡ 0(mod 2), Then (T − id)(x_{1}^{i_{2}/2}x_{2}^{i_{2}/2}x_{3}^{i_{3}} ⋯) = x_{2}^{i_{2}}x_{3}^{i_{3}} · · ·.

Let α(n) = k, then no i_{j} = 0 and all are different powers of 2. Hence i_{1} ≠ i_{2} and (b) shows ω(n) ∈ D_{k}. Write the dyadic expansion of n = 2^{s−1} + 2^{t_{1}} + 2^{t_{2}} + ⋯ + 2^{t_{α(n)−1}}. Note that T − id annihilates most monomials or pairs by (a) and (b), and by (c) and (d), the only partitions of n which do not cancel are those of the form (1, 0, 2^{r_{1}}, 2^{r_{2}}, …, 2^{r_{q}}) where 1 + 2^{r_{1}} + ⋯ + 2^{r_{q}} = 2^{s−1} + 2^{t_{1}} + ⋯ + 2^{t_{α(n)−1}}. However, q + 2 ≤ k, and to add up to n we must have q ≥ s − 1 + α(n) − 1, or s ≤ k − α(n). This happens only when 2^{k−α(n)} |̸ n. If 2^{k−α(n)} |̸ n, then use the partition (1, 0, 1, 2, …, 2^{s−2}, 2^{t_{1}}, …, 2^{t_{α(n)−1}}).

The theorem is proved.

Corollary 2.4 If 2^{k−α(n)+1} | n, then ω_{k}(n) = (ω_{k}(n/2))^{2} ∈ D_{k}.

Corollary 2.5 If α(n) = k, then Q_{0} | ω(n) in D_{k}. If α(n) = k and 2^{s} | n, then (Q_{0})^{2^{s}} | ω(n) in D_{k}.

Proof. If α(n) = k, then the above proof shows that no i_{j} = 0 and hence x_{1} | ω(n). This together with the fact that ω(n) is invariant under GL_{k} implies that x_{ℓ} | ω(n) and then (x_{ℓ} + λ_{ℓ+1}x_{ℓ+1} + ⋯ + λ_{k}x_{k}) | ω(n) for every 1 ≤ ℓ ≤ k, λ_{i} ∈ F_{2}. So, ω(n) is divisible by the product of all the linear forms. This product is nothing but Q_{0} (see [8]).

The second part of the corollary is similarly proved.

The elements ω_{k}(n) have been used by Arnon [3] to study D_{k}. One of his results is the following theorem.

Theorem 2.6 (Arnon [3]). If ω(n) ∈ D_{k}, then Sq^{i}(ω(n)) = \binom{n+i}{2i}ω(n + i).

We note that this formula is not true in general in S_{k}.

Corollary 2.7 {ω_{k}(n)} ⊂ D_{k} form a cyclic A-module generated by ω_{k}(2^{k−1}) = Q_{k−1}.

Arnon also gives a formula expressing ω_{k}(n) in terms of the Q’s. This is given in the following theorem.

Theorem 2.8 (Arnon [3]). ω_{k}(n) = ∑ ∏_{i=1}^{ℓ} (ω_{k}(2^{k} − 2^{(r_{i}−s_{i})}))^{2^{s_{i}}} where the sum is over all ℓ-tuples (r_{1}, …, r_{ℓ}) with k ≥ r_{i} − s_{i} ≥ 0 for every i and (s_{1}, …, s_{ℓ}) is a fixed ℓ-tuple with 2^{s_{1}} + ⋯ + 2^{s_{ℓ}} = 2^{ℓ−1} for ℓ large.

In Section 4, we will need the following corollary of this result. (A direct proof can be given of this corollary, but it is long.)

Corollary 2.9 Denote Q^{I} = Q_{k−1}^{i_{k−1}} ⋯ Q_{0}^{i_{0}} for I = (i_{k−1}, ..., i_{0}). Write ω_{k}(n) = ∑ Q^{I}. Then each I has at most two odd numbers.

Proof. This comes from the fact that if 2^{ℓ−1} is written as a sum of ℓ powers of 2, at most two of them are one.

3 The Lannes-Zarati map is an algebra homomorphism

We would like to recall how to define the Lannes–Zarati map

φ_{k} : Ext_{A}^{k}(Σ^{−k}F_{2}, F_{2}) → (F_{2} ⊗_{A} D_{k})^{∗} ,

which is compatible with the Hurewicz map (see [10], [11]). To this end, we follow the explanation by the first-named author of the present paper [15].

First we sketch Lannes–Zarati’s work [11] on the derived functors of the destabilization. Let D be the destabilization functor, which sends an A-module M to the unstable Amodule D(M) = M/B(M), where B(M) is the submodule of M generated by all Sq^{i}u with u ∈ M, i > |u|.

D is a right exact functor. Let D_{k} be its k-th derived functor for k ≥ 0.

Suppose M_{1}, M_{2} are A-modules. Lannes and Zarati defined in [11, §2] a homomorphism

\begin{aligned}∩ : & Ext_{A}^{r}(M_{1}, M_{2}) ⊗ D_{s}(M_{1}) & → & D_{s−r}(M_{2}) \\ & (f, z) & ↦ & f ∩ z \end{aligned}

as follows. Let F_{∗}(M_{i}) be a free resolution of M_{i}, i = 1, 2. A class f ∈ Ext_{A}^{r}(M_{1}, M_{2}) can be represented by a chain map F : F_{∗}(M_{1}) → F_{∗−r}(M_{2}) of homological degree −r. We write f = [F]. If z = [Z] is represented by Z ∈ F_{∗}(M_{1}), then by definition f ∩z = [F (Z)].

Let M be an A-module. We set r = s = k, M_{1} = Σ^{−k}M, M_{2} = P_{k} ⊗ M, where as before P_{k} = F_{2}[x_{1}, …, x_{k}], and get the homomorphism

∩ : Ext_{A}^{k}(Σ^{−k}M, P_{k} ⊗ M) ⊗ D_{k}(Σ^{−k}M) → P_{k} ⊗ M .

Now we need to define the Singer element e_{k}(M) ∈ Ext_{A}^{k}(Σ^{−k}M, P_{k} ⊗ M) (see Singer [18, p. 498]). Let P̂_{1} be the submodule of F_{2}[x, x^{−1}] spanned by all powers x^{i} with i ≥ −1, where |x| = 1. The A-module structure on F_{2}[x, x^{−1}] is extending that of P_{1} = F_{2}[x] (see Adams [2], Wilkerson [22]). The inclusion P_{1} ⊂ P̂_{1} gives rise to a short exact sequence of A-modules:

0 → P_{1} → P̂_{1} → Σ^{−1}F_{2} → 0 .

Denote by e_{1} the corresponding element in Ext_{A}^{1}(Σ^{−1}F_{2}, P_{1}).

Definition 3.1 (Singer [18]).

(i) e_{k} = \underbrace{e_{1} ⊗ ⋯ ⊗ e_{1}}_{k times} ∈ Ext_{A}^{k}(Σ^{−k}F_{2}, P_{k}).

(ii) e_{k}(M) = e_{k} ⊗ M ∈ Ext_{A}^{k}(Σ^{−k}M, P_{k} ⊗ M) , for M an A-module. Here we also denote by M the identity map of M.

The cap-product with e_{k}(M) gives rise to the homomorphism

\begin{aligned}e_{k}(M) : & D_{k}(Σ^{−k}M) & → & D_{0}(P_{k} ⊗ M) ≡ P_{k} ⊗ M \\ & e_{k}(M)(z) & = & e_{k}(M) ∩ z . \end{aligned}

As Σ^{n−1}F_{2} is an unstable A-module for n ≥ 1, the following theorem is a special case of the main result in [11].

Theorem 3.2 (Lannes–Zarati [11]). Let D_{k} ⊂ P_{k} be the Dickson algebra of k variables. Then e_{k}(Σ^{n}F_{2}) : D_{k}(Σ^{n−k}F_{2}) → Σ^{n}D_{k} is an isomorphism of internal degree 0.

Let N be an A-module. By definition of the functor D, we have a natural homomorphism: D(N) → F_{2} ⊗_{A} N. Suppose F_{∗}(N) is a free resolution of N. Then the above natural homomorphism induces a commutative diagram

[figure: ⋯; F_{2} ⊗_{A} F_{k}(N); F_{2} ⊗_{A} F_{k−1}(N); · · · .; i_{k}; i_{k−1}; ⋯; DF_{k}(N); DF_{k−1}(N); ⋯]

Here the horizontal arrows are induced from the differential in F_{∗}(N), and

i_{k}[Z] = [1 ⊗_{A} Z]

for Z ∈ F_{k}(N). Passing to homology, we get a homomorphism

\begin{aligned}i_{k} : & F_{2} ⊗_{A} D_{k}(N) & → & Tor_{k}^{A}(F_{2}, N) \\ & 1 ⊗_{A} [Z] & ↦ & [1 ⊗_{A} Z] . \end{aligned}

Taking N = Σ^{1−k}F_{2}, we obtain a homomorphism

i_{k} : F_{2} ⊗_{A} D_{k}(Σ^{1−k}F_{2}) → Tor_{k}^{A}(F_{2}, Σ^{1−k}F_{2}) .

Note that the suspension Σ : F_{2} ⊗_{A} D_{k} → F_{2} ⊗_{A} ΣD_{k} and the desuspension

Σ^{−1} : Tor_{k}^{A}(F_{2}, Σ^{1−k}F_{2}) ⟶^{≅} Tor_{k}^{A}(F_{2}, Σ^{−k}F_{2})

are isomorphisms of internal degree 1 and (−1), respectively. This leads us to

Definition 3.3 (Lannes–Zarati [11]). The homomorphism φ_{k} of internal degree 0 is the dual of

φ_{k}^{∗} = Σ^{−1}i_{k} (1 ⊗_{A} e_{k}^{−1}(ΣF_{2})) Σ : F_{2} ⊗_{A} D_{k} → Tor_{k}^{A}(F_{2}, Σ^{−k}F_{2}) .

Let k = p + q. The usual inclusion GL_{k} ⊃ GL_{p} × GL_{q} and the usual diagonal Δ : P_{k} → P_{p} ⊗ P_{q} induce the diagonal

\overline{Δ} : F_{2} ⊗_{A} D_{k} → (F_{2} ⊗_{A} D_{p}) ⊗ (F_{2} ⊗_{A} D_{q}) .

As consequence, ⊕_{k} (F_{2} ⊗_{A} D_{k})^{∗} is equipped with an algebra structure. On the other hand,

⊕_{k} Ext_{A}^{k}(Σ^{−k}F_{2}, F_{2}) has the usual algebra structure with respect to the cup product. The following proposition was stated to the authors by J. Lannes.

Proposition 3.4 The Lannes-Zarati map

φ = ⨁ φ_{k} : ⨁_{k} Ext_{A}^{k}(Σ^{−k}F_{2}, F_{2}) → ⨁_{k}(F_{2} ⊗_{A} D_{k})^{∗}

is an algebra homomorphism.

Proof. We prove the dual statement that

φ^{∗} = ⨁_{k} φ_{k}^{∗} : ⨁_{k}(F_{2} ⊗_{A} D_{k}) → ⨁_{k} Tor_{k}^{A}(F_{2}, Σ^{−k}F_{2})

is a coalgebra homomorphism. The key point is to show that φ_{k}^{∗} is also defined by means of Σ^{2} instead of Σ.

Step 1. From Theorem 3.2, e_{k}(Σ^{2}F_{2}) : D_{k}(Σ^{2−k}F_{2}) → Σ^{2}D_{k} is an isomorphism. By definition, we have the commutative diagram

[figure: F_{2} ⊗_{A} Σ^{2}D_{k}; 1⊗_{A}e{−1 \atop k}(Σ^{2}F_{2}); \overline{D_{k}(2)}; i_{k}(Σ^{2−k}F_{2}); Tor_{k}^{A}(F_{2}, Σ^{2−k}F_{2}) ,; Σ; Σ; Σ; F_{2} ⊗_{A} ΣD_{k}; 1⊗_{A}e{−1 \atop k}(ΣF_{2}); \overline{D_{k}(1)}; i_{k}(Σ^{1−k}F_{2}); Tor_{k}^{A}(F_{2}, Σ^{1−k}F_{2})]

where \overline{D_{k}(1)} := F_{2} ⊗_{A} D_{k}(Σ^{1−k}F_{2}) and \overline{D_{k}(2)} := F_{2} ⊗_{A} D_{k}(Σ^{2−k}F_{2}).

Therefore

φ_{k}^{∗} = Σ^{−2}i_{k}(Σ^{2−k}F_{2})(1 ⊗_{A} e_{k}^{−1}(Σ^{2}F_{2}))Σ^{2} .

Step 2. We discuss the coproduct on the involving objects. Suppose k = p + q.

(a) The usual diagonal Δ : D_{k} → D_{p} ⊗ D_{q} induces a diagonal

Δ : Σ^{2}D_{k} → ΣD_{p} ⊗ ΣD_{q} .

(b) Let M, N be two left A-modules and F_{∗}(M), F_{∗}(N) their free resolutions. Then F_{∗}(M) ⊗ F_{∗}(N) is a free resolution of M ⊗ N. We can choose F_{∗}(M ⊗ N) = F_{∗}(M) ⊗ F_{∗}(N). Passing to the derived functors D_{∗} we get a diagonal

Δ = Δ_{p,q} : D_{k}(M ⊗ N) → D_{p}(M) ⊗ D_{q}(N) .

(c) Similarly, we have

Δ : Tor_{k}^{A}(F_{2}, M ⊗ N) → Tor_{p}^{A}(F_{2}, M) ⊗ Tor_{q}^{A}(F_{2}, N) .

Note that, for M = Σ^{r}F_{2}, N = Σ^{s}F_{2}, this Δ is the usual coproduct on Tor_{∗}^{A}(F_{2}, F_{2}).

Taking M = Σ^{1−i}F_{2}, N = Σ^{1−j}F_{2} and consider

Δ_{p,q} : D_{k}(Σ^{2−(i+j)}F_{2}) → D_{p}(Σ^{1−i}F_{2}) ⊗ D_{q}(Σ^{1−j}F_{2}) ,
Δ_{p−i,q−j} : D_{k−(i+j)}(Σ^{2}P_{i+j}) → D_{p−i}(ΣP_{i}) ⊗ D_{q−j}(ΣP_{j}) .

We claim that

3.5

(f ⊗ g) ∩ Δ_{p,q}(x) = Δ_{p−i,q−j}[(f ⊗ g) ∩ x]

in D_{p−i}(ΣP_{i}) ⊗ D_{q−j}(ΣP_{j}) for f ∈ Ext_{A}^{i}(Σ^{1−i}F_{2}, ΣP_{i}), g ∈ Ext_{A}^{j}(Σ^{1−j}F_{2}, ΣP_{j}) and x ∈ D_{k}(Σ^{2−(i+j)}F_{2}). (Compare with Singer [18, p. 501].) To prove it, represent x as a cycle in D(F_{∗}(Σ^{1−i}F_{2}) ⊗ F_{∗}(Σ^{1−j}F_{2})).

Now we apply this for i = p, j = q, f = e_{p}(ΣF_{2}), g = e_{q}(ΣF_{2}). By definition, e_{k}(Σ^{2}F_{2}) = e_{p}(ΣF_{2}) ⊗ e_{q}(ΣF_{2}). So, formula 3.5 leads us to the commutative diagram

[figure: (F_{2} ⊗_{A} ΣD_{p}) ⊗ (F_{2} ⊗_{A} ΣD_{q}); 1⊗_{A}e_{p}^{−1}(ΣF_{2})⊗1⊗_{A}e_{q}^{−1}(ΣF_{2}); \overline{D_{p}(1)} ⊗ \overline{D_{q}(1)} .; Δ; Δ_{p,q}; (F_{2} ⊗ Σ^{2}D_{k}); 1⊗_{A}e_{k}^{−1}(Σ^{2}F_{2}); \overline{D_{k}(2)}]

Furthermore, by definition of i_{k}(−), we easily check that the following diagram is commutative.

[figure: \overline{D_{p}(1)} ⊗ \overline{D_{q}(1)}; i_{p}(Σ^{1−p}F_{2})⊗i_{q}(Σ^{1−q}F_{2}); Tor_{p}^{A}(F_{2}, Σ^{1−p}F_{2}) ⊗ Tor_{q}^{A}(F_{2}, Σ^{1−q}F_{2}) .; Δ_{p,q}; Δ; \overline{D_{k}(2)}; i_{k}(Σ^{2−k}F_{2}); Tor_{k}^{A}(F_{2}, Σ^{2−k}F_{2})]

Combining the two commutative diagrams with the formula at the end of Step 1 we have proved that φ^{∗} is a coalgebra homomorphism. The proposition is proved.

4 The diagonal map

In this section we study the diagonal map Δ = Δ_{p,q} : D_{k} → D_{p} ⊗ D_{q}, for k = p + q, and the induced homomorphism \overline{Δ} : \overline{D}_{k} → \overline{D}_{p} ⊗ \overline{D}_{q}, where \overline{D}_{k} = F_{2} ⊗_{A} D_{k}. Our main result is the following theorem.

Theorem 4.1 Let k > 2. Then \overline{Δ} = 0 in positive dimensions unless either p = 0 or q = 0.

Remark 4.2 When p = q = 1, then it is easy to see that \overline{Δ}(Q_{2,1}^{2^{s}−1}) = Q_{1,0}^{2^{s}−1} ⊗ Q_{1,0}^{2^{s}−1}. Note that Q_{1,0}^{2^{s}−1} and Q_{2,1}^{2^{s}−1} are non-zero in \overline{D}_{1} and \overline{D}_{2} respectively, so the cases k = 2 and k > 2 are different.

One of the motivations for defining ω_{k}(n) is the following easy theorem.

Theorem 4.3 If ω_{k}(n) ∈ D_{k}, then Δ(ω_{k}(n)) = ∑_{n_{1}+n_{2}=n} ω_{p}(n_{1}) ⊗ ω_{q}(n_{2}).

Note: Since 2^{k−α(n)} | n, it is easy to check that both ω_{p}(n_{1}) ∈ D_{p} and ω_{q}(n_{2}) ∈ D_{q} or one of them is 0 using the fact that α(n_{1} + n_{2}) ≤ α(n_{1}) + α(n_{2}).

We now study Δ when p > 0 and q > 0.

Proposition 4.4 Δ(ω_{k}(2^{k−1})) ∈ (D_{p})^{2} ⊗ D_{q} + D_{p} ⊗ (Q_{q,0}) + 1 ⊗ (D_{q})^{2}.

Proof. The term 1⊗ω_{q}(2^{k−1}) ∈ 1⊗(D_{q})^{2} because q < k. Let s = p−α(n_{1}) and assume 2^{s} | n_{1} but 2^{s+1} |̸ n_{1}. Hence ω_{p}(n_{1}) ∉ (D_{p})^{2}, 2^{k−1} = n_{1} +n_{2} = 2^{k−2} +2^{k−3} +⋯+2^{s+1} +2^{s} +2^{s}, α(n_{1}) + α(n_{2}) = k − s = k − p + α(n_{1}). Thus α(n_{2}) = k − p = q and by Corollary 2.5, Q_{q,0}^{2^{s}} | ω_{q}(n_{2}).

Since (D_{p})^{2} and (Q_{q,0}) are closed under the Steenrod algebra, we get the following corollary using Corollary 2.7.

Proposition 4.5 Δ(ω_{k}(n)) ∈ (D_{p})^{2} ⊗ D_{q} + D_{p} ⊗ (Q_{q,0}) + 1 ⊗ (D_{q})^{2}.

Lemma 4.6 If Δ(Q_{k,0}) = ∑ Q^{K} ⊗ Q^{L}, then either k_{0} = 1 and L is even, or ℓ_{0} = 1 and K is even.

Proof. By Theorem 2.2,

Q_{k,0} = ∑_{i_{1}+···+i_{k}=2^{k}−1 i_{s}=0 or a 2-power} x_{1}^{i_{1}} ⋯ x_{k}^{i_{k}}

The unique (up to an order) decomposition of 2^{k} − 1 into a sum of k 2-powers is 2^{k} − 1 = 1 + 2 + ⋯ + 2^{k−1}. Then each term in the expansion is divisible by (x_{1} ⋯ x_{k}) and not divisible by (x_{1} ⋯ x_{k})^{2}. So, for each term Q^{K} ⊗ Q^{L} in ΔQ_{k,0}, we have (x_{1} ⋯ x_{p}) | Q^{K} and (x_{1} ⋯ x_{q}) | Q^{L}. Hence k_{0} > 0, ℓ_{0} > 0. Furthermore, we get min(k_{0}, ℓ_{0}) = 1, otherwise (x_{1} ⋯ x_{k})^{2} | Q^{K} ⊗ Q^{L}, a contradiction.

Suppose ℓ_{0} = 1. Each term in Q^{K} is a part of some term in the above expansion of Q_{k,0}. So α(m) = p, where m = dimQ^{K}. Also, by the expansion and the fact ℓ_{0} = 1, (x_{1} ⋯ x_{p})^{2} | Q^{K}. Hence 2^{p−α(m)+1} = 2^{1} | m. Then, by Corollary 2.4, Q^{K} = ω_{p}(m/2)^{2}. As ω_{p}(m/2) satisfies 2^{p−α(m/2)} | m/2, it is a GL_{p}-invariant. Thus K is even.

The case of k_{0} = 1 is symmetrically considered. The lemma is proved.

Let \overline{A} be the submodule of all elements of positive degree in A. Hence \overline{D}_{p} = D_{p}/\overline{A}D_{p}.

Proposition 4.7 Let Q^{J} ∈ D_{p}, J = (j_{p−1}, ..., j_{0}) with j_{0} > 0. If j_{m} ≡ 0(mod 2) for some m > 0 or h(J) := ∑ j_{m} ≡ 0(mod 2), then Q^{J} ∈ \overline{A}D_{p}.

Proof. We prove this by induction on the smallest m > 0 with j_{m} ≡ 0(mod 2). If m = 1, then Sq^{1}Q^{(j_{p−1},...,j_{1}+1,j_{0}−1)} = Q^{J}. For the induction step,

Sq^{2^{m−1}}Q^{(j_{p−1},...,j_{m}+1,j_{m−1}−1,...,j_{0})} = Q^{J} + ∑ Q^{L},

where ℓ_{m−1} ≡ 0(mod 2) and ℓ_{0} > 0. The last step of the induction is slightly different. If all j_{m} ≡ 1(mod 2), m > 0, but h(J) ≡ 0(mod 2), then

Sq^{2^{p−1}}Q^{(j_{p−1}−1,j_{k−2},...,j_{0})} = Q^{J} + ∑ Q^{L},

where ℓ_{p−1} ≡ 0(mod 2) and ℓ_{0} > 0. The proposition is proved.

Theorem 4.8 If p ≥ q, then

ImΔ ⊂ \overline{A}D_{p} ⊗ D_{q} + D_{p} ⊗ \overline{A}D_{q} + D_{p} ⊗ (Q_{q,0}^{q−1}) .

Proof. Let Q^{I} ∈ D_{k}. Q^{I} = ∏_{i} ω_{k}(n_{i}). Let Δ(Q^{I}) = ∑ u⊗v. We call v the “D_{q}-coefficient” of u in Δ(Q^{I}). We study the terms where u ∉ \overline{A}D_{p}. Then we want to show either v ∈ \overline{A}D_{q} or Q_{q,0}^{q−1} | v.

Let u = Q^{J} be such a term in the sum. We first show that either v ∈ \overline{A}D_{q} or j_{0} > 0. By Proposition 4.5, Δ(ω_{k}(n)) ∈ (Q_{p,0}) ⊗ D_{q} + D_{p} ⊗ (D_{q})^{2} + (D_{p})^{2} ⊗ 1. In u, if there is no term from (Q_{p,0}) ⊗ D_{q}, then v is a square and hence v ∈ \overline{A}D_{q}. Thus, we have left to consider the case j_{0} > 0. By Proposition 4.7, we may assume j_{ℓ} ≡ 1(mod 2) for ℓ > 0.

¿From Corollary 2.9, there are at most two odd exponents for Q_{p,ℓ} and Q_{p,m} (ℓ ≠ m) appearing in a term of ΔQ_{k,s} for any s. In case if there are exactly such two odd exponents with ℓ > 0 and m > 0, then we have s > 0, by Lemma 4.6. Therefore, by dimensional information and Proposition 4.5, the “D_{q}-coefficient” of the term containing the two odd exponents should be divisible by Q_{q,0}^{2}.

Otherwise, in case if there is only one odd exponent for Q_{p,ℓ} (ℓ > 0) appearing in a term of some ΔQ_{k,r}, then the “D_{q}-coefficient” of this term is divisible by Q_{q,0}, according to Proposition 4.5.

Consequently, since Q^{J} has (p − 1) odd exponents j_{ℓ} with ℓ > 0, the “D_{q}-coefficient” v of Q^{J} in ΔQ^{I} is divisible by Q_{q,0}^{p−1}. Hence, Q_{q,0}^{q−1} divides v, as p ≥ q. The theorem is proved.

Recently, Giambalvo and Peterson [9] have proved the following theorem, which was conjectured in Remark 4.8 of the earlier version [17] of the present paper.

Theorem 4.9 Let q > 1 and assume that Q_{q,0}^{q−1} | Q^{J} in D_{q}. Then Q^{J} ∈ \overline{A}D_{q}.

Proof of Theorem 4.1. Theorems 4.8 and 4.9 prove 4.1 in case both p and q are ≥ 2. We have left to consider the case p > q = 1. Note that ω_{1}(n) ≠ 0 implies n is a power of 2. Hence, in Proposition 4.4 we get Δ(ω_{k}(2^{k−1})) ∈ (D_{k−1})^{2} ⊗ D_{1} + ω_{k−1}(2^{k−1} − 1) ⊗ (Q_{1,0}) + 1 ⊗ (D_{1})^{2}, and ω_{k−1}(2^{k−1} − 1) ∈ (Q_{k−1,0}). Thus, in Proposition 4.5 we get Δ(ω_{k}(n)) ∈ (D_{k−1})^{2} ⊗ D_{1} + (Q_{k−1,0}) ⊗ (Q_{1,0}) + 1 ⊗ (D_{1})^{2}. Now use the same argument as that in the proof of Theorem 4.8 to see that Q_{k−1,0}^{k−2} | u. But k ≥ 3, so k − 1 ≥ 2 and u ∈ \overline{A}D_{k−1} by Theorem 4.9. The theorem is proved.

Remark 4.10 Recently Campbell, Hughes, Shank and Wehlau have shown that the image of the trace, tr : P_{k} → D_{k} is the ideal (Q_{k,0}^{k−1}). Presumably there is some relation between this and Theorem 4.9.

5 On the vanishing of φ_{k}

Theorem 5.1 φ_{3} : Ext_{A}^{3,3+i}(F_{2}, F_{2}) → (F_{2} ⊗_{A} D_{3})_{i}^{∗} is zero in positive stems.

Proof. Let V_{k} be an elementary abelian 2-group of rank k.

¿From Boardman [4], Ext_{A}^{3,3+i}(F_{2}, F_{2}) ≅ F_{2} ⊗_{GL_{3}}P H_{i}(BV_{3}; F_{2}) via the algebraic transfer

defined by Singer [18], where P H_{∗}(BV_{k}; F_{2}) denotes the submodule of all A-annihilated elements in H_{∗}(BV_{k}; F_{2}). Therefore, the following proof is a version of that of Hu’ng [15, Th. 3.2].

Let h_{r} ∈ Ext_{A}^{1,2^{r}}(F_{2}, F_{2}) be the Adams element. According to Wang[20], Ext_{A}^{3,∗}(F_{2}, F_{2}) has a basis consisting of some products of the form h_{r}h_{s}h_{t}, where r, s, t are non-negative integers (but not all such appear), and some elements c_{i} (i ≥ 0) with stem(c_{i}) = 2^{i+3} + 2^{i+1} + 2^{i} − 3.

By Theorem 4.1, any decomposable element in (F_{2} ⊗_{A} D_{3})_{i}^{∗} is zero. Then, since φ = ⊕φ_{k} is an algebra homomorphism, φ_{3} sends any element of the form h_{r}h_{s}h_{t} to zero.

On the other hand, we showed in [16] that F_{2} ⊗_{A} D_{3} is concentrated in dimensions

2^{s+2} − 4 (s ≥ 0) and 2^{r+2} + 2^{s+1} − 3 (r > s > 0). A routine computation shows that these dimensions are different from stem(c_{i}) for any i. Then φ_{3} also sends c_{i} to zero. The theorem is proved.

So far as we know, Ext_{A}^{4,∗}(F_{2}, F_{2}) is unknown. In the literature, its value is available only for stems < 89. (See Tangora [19], Bruner [6].)

Theorem 5.2 φ_{4} : Ext_{A}^{4,4+i}(F_{2}, F_{2}) → (F_{2} ⊗_{A} D_{4})_{i}^{∗} is zero in positive stem i < 89.

Proof. Using Theorem 4.1 and the same argument as that given in the proof of the last theorem, we know that φ_{4} vanishes on the decomposable part.

According to Tangora [19], Ext_{A}^{4,∗}(F_{2}, F_{2}) has the following indecomposable generators up to stem ∗ − 4 < 71:

d_{0}(14), e_{0}(17), f_{0}(18), g_{1}(20), d_{1}(32), p_{0}(33), e_{1}(38),
f_{1}(40), g_{2}(44), D_{3}(61), d_{2}(68), p′(69), p_{1}(70) .

Here the number in parenthesis shows the stem of the corresponding element, for example stem(d_{0}) = 14. (Accidentally, the element in stem 61 is also denoted by D_{3}, but it is not the Dickson algebra of 3 variables.) Furthermore, the chart in Bruner [6] shows that Ext_{A}^{4,∗}(F_{2}, F_{2}), for 71 ≤ ∗ − 4 < 89, has at most indecomposable generators in stems 71, 80, 82, 84. Personally, he informed us that the only indecomposable generators in these stems are:

e_{2}(80), f_{2}(84) .

On the other hand, we proved in [16] that F_{2} ⊗_{A} D_{4} is concentrated in dimensions

\begin{matrix}2^{s+3} − 8 & (s ≥ 0) \\ 2^{r+3} + 2^{s+2} − 6 & (r > s > 0) \\ 2^{t+3} + 2^{r+2} + 2^{s+1} − 4 & (t > r > s > 1) \\ 2^{r+3} + 2^{s+1} − 4 & (r > s + 1 > 2) . \end{matrix}

One can check that these dimensions are completly different from the stems of the indecomposable generators listed above. For instance, we will check it here for stems 68, 84 and leave the remaining cases to the reader. (Our claim is true even for stems 71 and 82, in which all generators are reported to be decomposable.)

For stem 68:

2^{s+3} = 8 + 68 = 76, no solution.
2^{r+3} + 2^{s+2} = 6 + 68 = 64 + 8 + 2, no solution.
2^{t+3} + 2^{r+2} + 2^{s+1} = 4 + 68 = 64 + 8, no solution.
2^{r+3} + 2^{s+1} = 4 + 68 = 64 + 8, r = 3, s = 2,

it does not satisfy r > s + 1.

For stem 84:

2^{s+3} = 8 + 84 = 92, no solution.
2^{r+3} + 2^{s+2} = 6 + 84 = 64 + 16 + 8 + 2, no solution.
2^{t+3} + 2^{r+2} + 2^{s+1} = 4 + 84 = 64 + 16 + 8, t = 3, r = 2, s = 2,

it does not satisfy r > s.

2^{r+3} + 2^{s+1} = 4 + 84 = 64 + 16 + 8, no solution.

So, φ_{4} vanishes on the above indecomposable generators. The theorem is proved.

Remark 5.3 R. Bruner informed us an unpublished result by himself which claims that Ext_{A}^{4,∗}(F_{2}, F_{2}), for 89 ≤ ∗ − 4 < 120, has exactly one indecomposable generator in stem 92. It is easy to check by the same argument as given above that D_{4} has no A-generator in dimension 92. Therefore, if Bruner’s result is correct, Theorem 5.2 is extendingly valid up to positive stems < 120.

The following theorem is an immediate consequence of Theorem 4.1.

Theorem 5.4 φ_{k} : Ext_{A}^{k,k+i}(F_{2}, F_{2}) → (F_{2} ⊗_{A} D_{k})_{i}^{∗} vanishes on decomposable elements for k > 2.

6 A-generators for a localization of the Dickson algebra

In this section we prove a curious result which gives another instance where D_{k} is quite different when k = 1 from k > 1 (4.1 and 4.8 already show such.)

Let D_{k}^{∧} = D_{k}[Q_{0}^{−1}], that is the Dickson algebra localized by inverting Q_{0}. ¿From Hu’ng [14], D_{k}^{∧} is an A-module by defining

Sq(Q_{0}^{−1}) = (SqQ_{0})^{−1} = [Q_{0}(1 + Q_{k−1} + Q_{k−2} + ⋯ + Q_{0})]^{−1}
= Q_{0}^{−1}(1 + Q_{k−1} + ⋯ + Q_{0})^{−1}
= Q_{0}^{−1} + Q_{0}^{−1}(Q_{k−1} + ⋯ + Q_{0}) + Q_{0}^{−1}(Q_{k−1} + ⋯ + Q_{0})^{2} + · · · .

W. H. Lin [12] proved that F_{2} ⊗_{A} D_{1}^{∧} has one generator, Q_{0}^{−1}.

Theorem 6.1 If k > 1, then F_{2} ⊗_{A} D_{k}^{∧} = 0.

Proof. We follow the notation of [16]. Let Q^{I}, I = (i_{k−1}, …, i_{0}), i_{j} ≥ 0 if j ≥ 1 and i_{0} ∈ Z, be a monomial in D_{k}^{∧}. Set f_{j} = s(i_{j}) + j − 1 for j ≥ 1 and f(I) = min{f_{j}}, j ≥ 1. Given j ≥ 2 with f_{j} = f(I), then

Sq^{2^{f(I)}}Q^{(i_{k−1},...,i_{j}+2^{s(i_{j})},i_{j−1}−2^{s(i_{j})},...,i_{0})} = Q^{I} + ΣQ^{L}.

Now s(i_{j−1})+j −2 = f_{j−1} ≥ f_{j} = s(i_{j})+j −1, so s(i_{j−1}) ≥ s(i_{j})+1. Hence s(ℓ_{j−1}) ≤ s(i_{j}) and thus f(ℓ_{j−1}) = s(ℓ_{j−1}) + j − 2 < s(i_{j}) + j − 1 = f(i_{j}) and we have f(L) < f(I). To finish the induction step, we must consider the case f_{1} = f(I) and f_{j} > f(I) if 1 < j < k. In this case

Sq^{2^{f_{1}}}Q^{(i_{k−1},...,i_{1}+2^{s(i_{1})},i_{0}−2^{s(i_{1})})} = Q^{I} + ΣQ^{L}.

If no f(ℓ_{j}), 1 ≤ j < k, is < f_{1}, then Sq^{2^{f_{1}}} acts only to increase the power of Q_{0}. However, 2^{k} − 1 |̸ 2^{f_{1}} if k > 1.

To start the induction, assume f(I) = 0. Then i_{1} ≡ 0(mod 2), and

Sq^{1}Q^{(i_{k−1},...,i_{1}+1,i_{0}−1)} = Q^{I}.

The theorem is proved.

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Department of Mathematics, National University of Hanoi, 90 Nguyên Trai Street, Hanoi, Vietnam. E-mail address: nhvhung@it-hu.ac.vn

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 01239. E-mail address: fpp@math.mit.edu