ESSENTIAL COHOMOLOGY FOR ELEMENTARY ABELIAN p-GROUPS

FATMA ALTUNBULAK AKSU AND DAVID J. GREEN

Abstract. For an odd prime p the cohomology ring of an elementary abelian p-group is polynomial tensor exterior. We show that the ideal of essential classes is the Steenrod closure of the class generating the top exterior power. As a module over the polynomial algebra, the essential ideal is free on the set of Mùi invariants.

1. Introduction

Let G be a finite group and k a field whose characteristic p divides the order of G. A cohomology class x ∈ H^{n}(G, k) is called essential if its restriction Res_{H}(x) is zero for every proper subgroup H of G. The essential classes form an ideal, called the essential ideal and denoted by Ess(G). It is standard that restriction to a Sylow p-subgroup of G is a split injection (see for example Theorem XII,10.1 of [4]), and so the essential ideal can only be nonzero if G is a p-group. Many p-groups have nonzero essential ideal, for instance the quaternion group of order eight. The essential ideal plays an important role and has therefore been the subject of many studies: two such being Carlson’s work on the depth of a cohomology ring [2], and the cohomological characterization due to Adem and Karagueuzian of those p-groups whose order p elements are all central [1].

The nature of the essential ideal depends crucially on whether or not the pgroup G is elementary abelian. If G is not elementary abelian, then a celebrated result of Quillen (Theorem 7.1 of [10]) implies that Ess(G) is a nilpotent ideal. By contrast, the essential ideal of an elementary abelian p-group contains nonnilpotent classes. Work to date on the essential ideal has concentrated on the non-elementary abelian case. In this paper we give a complete treatment of the outstanding elementary abelian case. As we shall recall in the next section, the case p = 2 is straightforward and well known. So we shall concentrate on the case of an odd prime p.

So let p be an odd prime and V a rank n elementary abelian p-group. We may equally well view V as an n-dimensional 𝔽_{p}-vector space. Recall that the cohomology ring has the form

Date: 25 September 2008.

2000 Mathematics Subject Classification. Primary 20J06; Secondary 13A50, 55S10.

The first author was supported by a Ph.D. research scholarship from the Scientific and Technical Research Council of Turkey (TÜBİTAK-BAYG).

(1)  H^{∗}(V, 𝔽_{p}) ≅ S(V^{∗}) ⊗_{𝔽_{p}} Λ(V^{∗}) ,

where the exterior copy of the dual space V^{∗} is H^{1}(V, 𝔽_{p}), and the polynomial copy lies in H^{2}(V, 𝔽_{p}): specifically, the polynomial copy is the image of the exterior copy under the Bockstein boundary map β. Our first result is as follows:

Theorem 1.1. Let p be an odd prime and V a rank n elementary abelian pgroup. Then the essential ideal Ess(V) is the Steenrod closure of Λ^{n}(V^{∗}). That is, Ess(V) is the smallest ideal in H^{∗}(V, 𝔽_{p}) which contains the one-dimensional space Λ^{n}(V^{∗}) ⊆ H^{n}(V, 𝔽_{p}) and is closed under the action of the Steenrod algebra.

Our second result concerns the structure of Ess(V) as a module over the polynomial subalgebra S(V^{∗}) of H^{∗}(V, 𝔽_{p}). It was conjectured by Carlson (Question 5.4 in [3]) – and earlier in a less precise form by Mùi [8] – that the essential ideal of an arbitrary p-group is free and finitely generated as a module over a certain polynomial subalgebra of the cohomology ring. In [6], the second author demonstrated finite generation, and for most p-groups of a given order was able to prove freeness as well: specifically the method works provided the group is not a direct product in which one factor is elementary abelian of rank at least two. Our second result states that Carlson’s conjecture holds for elementary abelian p-groups too, and gives explicit free generators.

Theorem 1.2. Let p be an odd prime and V a rank n elementary abelian pgroup. Then as a module over the polynomial part S(V^{∗}) of the cohomology ring H^{∗}(V, 𝔽_{p}), the essential ideal Ess(V) is free on the set of Mùi invariants, as defined in Definition 3.3.

Structure of the paper. In §2 we briefly cover the well-known case p = 2. We introduce the Mùi invariants in §3. After proving Theorem 1.2 in §4 we consider the action of the Steenrod algebra on the Mùi invariants in order to prove Theorem 1.1 in §5.

2. Elementary abelian p-groups and the case p = 2

The cohomology group H^{1}(G, 𝔽_{p}) may be identified with the set of group homomorphisms Hom(G, 𝔽_{p}). This set is an 𝔽_{p}-vector space, and – assuming that G is a p-group – the maximal subgroups of G are in bijective correspondence with the one-dimensional subspaces: the maximal subgroup corresponding to α: G → 𝔽_{p} being ker(α). Of course, the cohomology class α ∈ H^{1}(G, 𝔽_{p}) has zero restriction to the maximal subgroup ker(α). Note that in order to determine Ess(G) it suffices to consider restrictions to maximal subgroups.

Definition. Denote by L_{n} the polynomial

L_{n}(X_{1}, …, X_{n}) = det |\begin{matrix}X_{1} & X_{2} & ⋯ & X_{n} \\ X_{1}^{p} & X_{2}^{p} & ⋯ & X_{n}^{p} \\ ⋮ & ⋮ & ... & ⋮ \\ X_{1}^{p^{n−1}} & X_{2}^{p^{n−1}} & ⋯ & X_{n}^{p^{n−1}} \end{matrix} | ∈ 𝔽_{p}[X_{1}, …, X_{n}] .

There is a well-known alternative description of L_{n}.

Lemma 2.1. L_{n} is the product of all monic linear forms in X_{1}, …, X_{n}. So for an n-dimensional 𝔽_{p}-vector space V we may define L_{n}(V) ∈ S(V^{∗}) up to a nonzero scalar multiple by

(2)  L_{n}(V) = ∏_{[x]∈ℙV^{∗}} x .

Proof. First part: Here we call a linear form monic if the first nonzero coefficient is one. The right hand side divides the left. Both sides have the same total degree. And the coefficient of X_{1}X_{2}^{p}X_{3}^{p^{2}} ⋯ X_{n}^{p^{n−1}} is +1 in both cases. The second part follows. □

Let V be an elementary abelian 2-group. Then H^{∗}(V, 𝔽_{2}) ≅ S(V^{∗}), where the dual space V^{∗} is identified with H^{1}(V, 𝔽_{2}). Pick x_{1}, …, x_{n} to be a basis for H^{1}(V, 𝔽_{2}). The following is well-known:

Lemma 2.2. For an elementary abelian 2-group V, the essential ideal is the principal ideal in H^{∗}(V, 𝔽_{2}) generated by L_{n}(x_{1}, …, x_{n}).

Moreover, Ess(V) is the free S(V^{∗})-module on L_{n}(V), and the Steenrod closure of this one generator.

Proof. L_{n}(V) is essential, because every nonzero linear form is a factor and every maximal subgroup is the kernel of a nonzero linear form. Now suppose that y is essential, and let x ∈ V^{∗} be a nonzero linear form. Let U ⊆ V^{∗} be a complement of the subspace spanned by x. So y = y′x + y′′ with y′ ∈ S(V^{∗}) and y′′ ∈ S(U). Hence Res_{H}(y′′) = 0 for H = ker(x), as y is essential and Res_{H}(x) = 0. But the map Res_{H}: V^{∗} → H^{∗} satisfies ker(Res_{H}) ∩ U = 0, and so Res_{H} is injective on S(U). Hence y′′ = 0, and x divides y. By unique factorization in S(V^{∗}) it follows that L_{n}(V) divides y. So Ess(V) is the principal ideal generated by L_{n}(V), and the free module on this one generator. Finally, the definition of the essential ideal means that it is closed under the action of the Steenrod algebra. □

We finish off this section by recalling the action of the Steenrod algebra on the cohomology of an elementary abelian p-group in the case of an odd prime. So let p be an odd prime and V an elementary abelian p-group. Recall that the mod-p-cohomology ring is the free graded commutative algebra

H^{∗}(V, 𝔽_{p}) ≅ 𝔽_{p}[x_{1}, …, x_{n}] ⊗_{𝔽_{p}} Λ(a_{1}, …, a_{n}) ,

where a_{i} ∈ H^{1}(V, 𝔽_{p}), x_{i} ∈ H^{2}(V, 𝔽_{p}), and n is the rank of V. That is, a_{1}, …, a_{n} is a basis of the exterior copy of V^{∗}, and x_{1}, …, x_{n} is a basis of the polynomial copy. The product a_{1}a_{2} ⋯ a_{n} ∈ H^{n}(V, 𝔽_{p}) is a basis of the top exterior power Λ^{n}(V^{∗}). The Steenrod algebra A acts on the cohomology ring, making it an unstable Aalgebra with β(a_{i}) = x_{i} and P^{1}(x_{i}) = x_{i}^{p}. Observe that L_{n}(x_{1}, …, x_{n}) is essential, for the same reason as in the case p = 2.

3. The Mùi invariants

Let k be a finite field and V a finite dimensional k-vector space. Consider the natural action of GL(V) on V^{∗}. The Dickson invariants generate the invariants for the induced action of GL(V) on the polynomial algebra S(V^{∗}). But there is also an induced action on the polynomial tensor exterior algebra S(V^{∗}) ⊗_{k} Λ(V^{∗}), and the Mùi invariants are SL(V)-invariants of this action: see Mùi’s original paper [7] as well as Crabb’s modern treatment [5].

We shall need several properties of the Mùi invariants. For the convenience of the reader, we rederive these from scratch: but see Mùi’s papers [7, 9] and Sum’s work [11].

Notation. Often we shall work with the direct sum decomposition

H^{∗}(V, 𝔽_{p}) = ⨁_{r=0}^{n} N_{r}(V) ,

where n is the rank of V and we set

N_{r}(V) = S(V^{∗}) ⊗_{𝔽_{p}} Λ^{r}(V^{∗}) .

Observe that restriction to each subgroup respects this decomposition. This means that the essential ideal is well-behaved with respect to this decomposition:

(3)  Ess(V) = ⨁_{r=0}^{n} N_{r}(V) ∩ Ess(V) .

Definition. Recall that L_{n}(x_{1}, …, x_{n}) is the determinant of the n × n-matrix

C = (\begin{matrix}x_{1} & x_{2} & ⋯ & x_{n} \\ ⋮ & ⋮ & ... & ⋮ \\ x_{1}^{p^{n−1}} & x_{2}^{p^{n−1}} & ⋯ & x_{n}^{p^{n−1}} \end{matrix} ) ,

where C_{s,i} = x_{i}^{p^{s−1}} for 1 ≤ s ≤ n. For each such s, define E(s) to be the matrix obtained from C by deleting row s and then prefixing (a_{1} a_{2} ⋯ a_{n}) as new first row: so

det E(s) = ∑_{i=1}^{n}(−1)^{i+1}γ_{s,i}a_{i} ,

where γ_{s,i} is the determinant of the minor of C obtained by removing row s and column i.

Now define the Mùi invariant M_{n,s} ∈ H^{∗}(V, 𝔽_{p}) by M_{n,s} = det E(s). Note that our indexing differs from Mùi’s: our M_{n,s} is his M_{n,s−1}.

Example. So M_{4,3} = |\begin{matrix}a_{1} & a_{2} & a_{3} & a_{4} \\ x_{1} & x_{2} & x_{3} & x_{4} \\ x_{1}^{p} & x_{2}^{p} & x_{3}^{p} & x_{4}^{p} \\ x_{1}^{p^{3}} & x_{2}^{p^{3}} & x_{3}^{p^{3}} & x_{4}^{p^{3}} \end{matrix} | and γ_{2,3} = |\begin{matrix}x_{1} & x_{2} & x_{4} \\ x_{1}^{p^{2}} & x_{2}^{p^{2}} & x_{4}^{p^{2}} \\ x_{1}^{p^{3}} & x_{2}^{p^{3}} & x_{4}^{p^{3}} \end{matrix} |.

Lemma 3.1. M_{n,s} ∈ N_{1}(V) ∩ Ess(V).

Proof. By construction M_{n,s} ∈ N_{1}(V). Restricting to a maximal subgroup of V involves killing a nonzero linear form on V^{∗}: that is, one imposes a linear dependence on the a_{i} and consequently the same linear dependency on the x_{i}. So one obtains a linear dependency between the columns of E(s), meaning that restriction kills M_{n,s} = det E(s). □

Lemma 3.2. Ess(V)^{2} = L_{n}(V) · Ess(V).

Proof. As L_{n}(V) is essential, the left hand side contains the right. Now let H be a maximal subgroup of V. Then H = ker(a) for some nonzero a ∈ H^{1}(V, 𝔽_{p}). Let x = β(a) ∈ H^{2}. Observe that the kernel of restriction to H is generated by a, x. Suppose that f, g both lie in this kernel: then we may write f = f′a + f′′x, g = g′a + g′′x, and so fg = (f′′g′ ± f′g′′)ax + f′′g′′x^{2}, that is fg = xh for h = (f′′g′ ± f′g′′)a + f′′g′′x ∈ ker Res_{H}.

Since H^{∗}(V, 𝔽_{p}) is a free module over the unique factorization ring S(V^{∗}), this means that fg = L_{n}(V) · y for some y ∈ H^{∗}(V, 𝔽_{p}). So h = \frac{L_{n}(V)}{x} · y. As

Res_{H}(h) = 0 and Res_{H} (\frac{L_{n}(V)}{x}) is a non-zero divisor, we deduce that Res_{H}(y) = 0. So y ∈ Ess(V). □

Definition 3.3. Let S = {s_{1}, …, s_{r}} ⊆ {1, …, n} be a subset with s_{1} < s_{2} < ⋯ < s_{r}. In view of Lemmas 3.1 and 3.2 we may define the Mùi invariant M_{n,S} ∈ N_{r}(V) ∩ Ess(V) by

M_{n,S} = \frac{1}{L_{n}(V)^{r−1}} M_{n,s_{1}}M_{n,s_{2}} ⋯ M_{n,s_{r}} .

Note in particular that M_{n,∅} = L_{n}(V).

Remark. Observe that

(4)  M_{n,S}M_{n,T} = \begin{cases}±L_{n}(V)M_{n,S∪T} & if S ∩ T = ∅; \\ 0 & otherwise. \end{cases}

4. Joint annihilators

In this section we study the joint annihilators of the M_{n,S} with |S| = r as a means to prove Theorem 1.2.

Lemma 4.1. The joint annihilator of M_{n,1}, …, M_{n,n} is N_{n}(V).

Proof. The element a_{1} … a_{n} is a basis for Λ^{n}(V) and is clearly annihilated by each M_{n,s}. Conversely, suppose that y ≠ 0 is annihilated by every M_{n,s}. As M_{n,s}N_{r}(V) ⊆ N_{r+1}(V) we may assume without loss of generality that y ∈ N_{r}(V) for some r. Multiplying once or more by suitably chosen elements a_{i}, we reduce to the case y ∈ N_{n−1}(V).

Denote by K the field of fractions of S(V^{∗}), and let W = K ⊗_{k} Λ^{n−1}(V^{∗}). Each M_{n,s} induces a linear form ϕ_{s}: W → K given by ϕ_{s}(w)a_{1} ⋯ a_{n} = M_{n,s}w. By assumption, y ≠ 0 lies in the kernel of every ϕ_{s}. A basis for W consists of the elements a_{1} ⋯ â_{r} ⋯ a_{n} for 1 ≤ r ≤ n, where the hat denotes omission. Now,

M_{n,s} · a_{1} ⋯ â_{r} ⋯ a_{n} = (−1)^{r+1}γ_{s,r}a_{r} · a_{1} ⋯ â_{r} ⋯ a_{n} ,

and so

ϕ_{s}(a_{1} ⋯ â_{r} ⋯ a_{n}) = γ_{s,r} .

Now consider the matrix Γ ∈ M_{n}(K) given by Γ_{s,r} = γ_{s,r}. If one transposes and then multiplies the ith row by (−1)^{i} and the jth column by (−1)^{j}, then one obtains the adjugate matrix of C. As the determinant of C is L_{n}(V) and in particular nonzero, it follows that det Γ ≠ 0.

So by construction of Γ, the ϕ_{s} form a basis of W^{∗}. So their common kernel is zero, contradicting our assumption on y. □

Corollary 4.2. The joint annihilator of {M_{n,S}: |S| = r} is ⨁_{s≥n−r+1} N_{s}(V).

Proof. By induction on r, Lemma 4.1 being the case r = 1. As M_{n,S} ∈ N_{|S|}(V) and N_{r}(V)N_{s}(V) ⊆ N_{r+s}(V), the annihilator is at least as large as claimed. Now suppose that y ∈ H^{∗}(V, 𝔽_{p}) does not lie in ⨁_{s≥n−r+1} N_{s}(V). We may therefore write

y = ∑_{s=0}^{n} y_{s}

with y_{s} ∈ N_{s}(V), and we know that s_{0} ≤ n − r for s_{0} = min{s | y_{s} ≠ 0}. As y_{s_{0}} ≠ 0 and y_{s_{0}} ∉ N_{n}(V), Lemma 4.1 tells us that y_{s_{0}}M_{n,t} ≠ 0 for some 1 ≤ t ≤ n. As y_{s_{0}}M_{n,t} ∈ N_{s_{0}+1}(V), we conclude that yM_{n,t} lies outside ⨁_{s≥n−r+2} N_{s}(V). So the inductive hypothesis means that there is some T with |T | = r − 1 and yM_{n,t}M_{n,T} ≠ 0. So yM_{n,S} ≠ 0 for S = T ∪ {t} and |S| = r: note that t ∈ T is impossible. □

Corollary 4.3. Every M_{n,S} is nonzero. For S = \underline{n} = \{1, …, n\} we have

M_{n,\underline{n}} is a nonzero scalar multiple of a_{1}a_{2} ⋯ a_{n}.

Proof. Observe that M_{n,\underline{n}} is a scalar multiple of a_{1} ⋯ a_{n} for degree reasons. The case r = n of Corollary 4.2 says that 1 ∈ N_{0}(V) does not annihilate M_{n,\underline{n}} and therefore M_{n,\underline{n}} ≠ 0. But from Eqn (4) we see that every M_{n,S} divides L_{n}(V)M_{n,\underline{n}} ≠ 0. □

Proof of Theorem 1.2. In view of Eqn (3) it suffices to show that for each r the Mùi invariants M_{n,S} with |S| = r are a basis of the S(V^{∗})-module N_{r}(V)∩Ess(V). We observed in Definition 3.3 that these M_{n,S} lie in this module.

So suppose that y ∈ N_{r}(V) ∩ Ess(V). We should like there to be f_{S} ∈ S(V^{∗}) such that

(5)  y = ∑_{|S|=r} f_{S}M_{n,S} .

Note that for T = \underline{n} − S we have M_{n,S}M_{n,T} = ±L_{n}(V)M_{n,\underline{n}} by Eqn (4). Define ε_{S} ∈ \{+1, −1\} by M_{n,S}M_{n,T} = ε_{S}L_{n}(V)M_{n,\underline{n}}. So Eqn. (5) implies that we should define f_{S} by

f_{S}M_{n,\underline{n}} = \frac{1}{L_{n}(V)}ε_{S}yM_{n,T} ,

since T ∩ S′ ≠ ∅ and therefore M_{n,S′}M_{n,T} = 0 for all S′ ≠ S with |S| = r. Note that this definition of f_{S} makes sense, as yM_{n,T} lies in both N_{r}(V)N_{n−r}(V) = N_{n}(V) and L_{n}(V) Ess(V), the latter inclusion coming from Lemma 3.2.

With this definition of f_{S} we have

(y − ∑_{|S|=r} f_{S}M_{n,S}) M_{n,T} = 0

for every |T | = n − r. As y − ∑_{|S|=r} f_{S}M_{n,S} lies in N_{r}(V), this means that y = ∑_{|S|=r} f_{S}M_{n,S} by Corollary 4.2.

Finally we show linear independence. Suppose that g_{S} ∈ S(V^{∗}) are such that ∑_{|S|=r} g_{S}M_{n,S} = 0. Pick one S and set T = \underline{n} − S. Multiplying by M_{n,T}, we deduce that g_{S} = 0. □

5. The action of the Steenrod algebra

To prepare for the proof of Theorem 1.1 we shall study the operation of the Steenrod algebra on the Mùi invariants.

Lemma 5.1.

(6)  β(M_{n,s}) = \begin{cases}L_{n}(V) & s = 1 \\ 0 & otherwise \end{cases}  β(L_{n}(V)) = 0 .

For 0 ≤ s ≤ n − 2 we have:

(7)  P^{p^{s}}(M_{n,r}) = \begin{cases}M_{n,r−1} & r = s + 2 \\ 0 & otherwise \end{cases}  P^{p^{s}}(L_{n}(V)) = 0 .

Proof. One sees Eqn (6) by inspecting the determinants in the definition of M_{n,s} and L_{n}(V). The proof of Eqn (7) is also based on an inspection of these determinants. Recall that P^{m}(a_{i}) = 0 for every m > 0, and that P^{m}(x_{i}^{p^{s}}) is zero too except for P^{p^{s}}(x_{i}^{p^{s}}) = x_{i}^{p^{s+1}}. We may use the Cartan formula

P^{m}(xy) = ∑_{a+b=m} P^{a}(x)P^{b}(y)

to distribute P^{p^{s}} over the rows of the determinant. As p^{s} cannot be expressed as a sum of distinct smaller powers of p, we only have to consider summands where all of P^{p^{s}} is applied to one row and the other rows are unchanged. This will result in two rows being equal unless it is the row consisting of the x_{i}^{p^{s+1}} that is missing. □

Lemma 5.2. Let S = {s_{1}, …, s_{r}} with 1 ≤ s_{1} < s_{2} < ⋯ < s_{r} ≤ n.

(1) Suppose that 1 ∉ S. Then M_{n,S} = β(M_{n,S∪{1}}).

(2) L_{n}(V)^{r−1}P^{m}(M_{n,S}) = P^{m}(M_{n,s_{1}} ⋯ M_{n,s_{r}}) for each m < p^{n−1}.

(3) For 2 ≤ u ≤ n set X = {s ∈ S | s ≤ u} and Y = {s ∈ S | s > u}. Then

L_{n}(V)P^{p^{u−2}}(M_{n,S}) = P^{p^{u−2}}(M_{n,X}) · M_{n,Y} .

(4) For 1 ≤ r ≤ n and 0 < m < p^{n−1} one has P^{m}(M_{n,{1,...,r}}) = 0.

(5) For 2 ≤ u ≤ n one has P^{p^{u−2}}(M_{n,{1,...,u−2,u}}) = M_{n,{1,...,u−1}}.

Proof. Recall that

(8)  L_{n}(V)^{r}M_{n,S} = L_{n}(V)M_{n,s_{1}} ⋯ M_{n,s_{r}} .

The first two parts follow by applying Equations (6) and (7).

Recall that by the Adem relations each P^{m} may be expressed in terms of the P^{p^{s}} with p^{s} ≤ m. So the third part follows from the second, since we deduce from Eqn. (7) that P^{m}(M_{n,s}) = 0 if 0 < m ≤ p^{u−2} and s > u.

Fourth part: By induction on r. Follows for r = 1 from the Adem relations and Eqn (7). Inductive step: Enough to consider P^{p^{s}} for 0 ≤ s ≤ n − 2. By the inductive hypothesis and a similar argument to the third part, deduce that

L_{n}(V)P^{p^{s}}(M_{n,{1,...,r}}) = M_{n,{1,...,r−1}}P^{p^{s}}(M_{n,r}) .

But this is zero by Eqn (7), since M_{n,{1,...,r−1}}M_{n,r−1} = 0.

Fifth part: Using the fourth part and an argument similar to the third part, deduce that

L_{n}(V)P^{p^{u−2}}(M_{n,{1,...,u−2,u}}) = M_{n,{1,...,u−2}}P^{p^{u−2}}(M_{n,u}) = M_{n,{1,...,u−2}}M_{n,u−1} :

but this is L_{n}(V)M_{n,{1,...,u−1}}. □

Proof of Theorem 1.1. We shall show that for every M_{n,S} there is an element θ of the Steenrod algebra with M_{n,S} = θ(M_{n,\underline{n}}). We do this by decreasing induction on r = |S|. It is trivially true for r = n, so assume now that r < n. Amongst the S with |S| = r we shall proceed by induction over u, the smallest element of \underline{n} − S. So

S = {1, …, u − 1} ∪ Y with s > u for every s ∈ Y .

Part 1 of Lemma 5.2 covers the case u = 1, so assume that u ≥ 2. Set T = {1, …, u − 2, u}. We complete the induction by showing that M_{n,S} = P^{p^{u−2}}(M_{n,T ∪Y}). Part 3 of Lemma 5.2 gives us

L_{n}(V)P^{p^{u−2}}(M_{n,T ∪Y}) = P^{p^{u−2}}(M_{n,T})M_{n,Y} .

But P^{p^{u−2}}(M_{n,T}) = M_{n,{1,...,u−1}}, by Part 5 of that lemma. So P^{p^{u−2}}(M_{n,T ∪Y}) = M_{n,S}, as claimed. □

Remark. Theorem 1.2 shows that the S(V^{∗})-module generated by the Mùi invariants M_{n,S} is the essential ideal and therefore closed under the action of the Steenrod algebra. One may however see more directly that this S(V^{∗})-module is Steenrod closed. This is observed for example in [11]. In view of Lemma 5.2 and Equations (6) and (7) it only remains to show that P^{p^{n−1}}(M_{n,s}) lies in our S(V^{∗})-module. Now P^{p^{n−1}}(M_{n,n}) = 0 by the unstable condition, so suppose s < n. Recall that M_{n,s} is a determinant, the last row of the matrix having entries x_{i}^{p^{n−1}}. So applying P^{p^{n−1}} replaces these entries by x_{i}^{p^{n}}. But it is well known that x_{i}^{p^{n}} is an S(V^{∗})-linear combination of the x_{i}^{p^{r}} for r ≤ n − 1, and that the coefficients are independent of i: this is the “fundamental equation” in the sense of [12], and the coefficients are the Dickson invariants c_{n,r} in S(V^{∗}). Applying S(V^{∗})-linearity of the determinant in the bottom row of the matrix, one deduces that P^{p^{n−1}}(M_{n,s}) is an S(V^{∗})-linear combination of the M_{n,r}.

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[12] C. Wilkerson. A primer on the Dickson invariants. In H. R. Miller and S. B. Priddy, editors, Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982), Contemporary Math., vol. 19, pages 421–434, Providence, RI, 1983. Amer. Math.

  Soc. Revised version: http://hopf.math.purdue.edu//Wilkerson-80s/dickson.pdf.

Dept of Mathematics, Bilkent University, Bilkent, 06800, Ankara, Turkey

E-mail address: fatma@fen.bilkent.edu.tr

Dept of Mathematics, Friedrich-Schiller-Universität Jena, 07737 Jena, Germany

E-mail address: David.Green@uni-jena.de