ON THE DIMENSION THEORY OF DOMINANT SUMMANDS

John Martino and Stewart Priddy

Section 0. Introduction.

Let G be a finite group and let p be a prime. The p-local stable decompositon

BG = X_{1} ∨ X_{2} ∨ ⋯ ∨ X_{N}

of the classifying space of G is known for a variety of specific groups [MiP2, HK, Mt] and a general decomposition procedure is available for all finite groups [MP, BF]. Information about the summands such as their cohomology or connectivity [MiP, FS, CW] can sometimes be determined but it is often quite difficult to obtain. The purpose of this paper is to establish some dimension theoretic results for the Poincaré series

P S(X, t) = ∑_{k≥0} dim_{F_{p}}(H^{k}X)t^{k}

of the summands X of BG. Here and throughout the paper cohomology will be taken with coefficients in the prime field F_{p}.

We say that a power series f(t) ∈ Z[[t]] has dimension n, d(f(t)) = n, if it has a pole of order n at t = 1. For a space or spectrum X, the dimension of X is defined by

d(X) = d(P S(X, t)).

We have two main results concerning dimension. If P is a finite p-group we recall that the principal dominant summand X_{0} of BP is the dominant summand associated with the trivial representation (see Section 1). If G has P as a Sylow

2 JOHN MARTINO AND STEWART PRIDDY

p-subgroup then X_{0} is also a summand of BG. Let n = rank of G, that is, the dimension of the largest elementary abelian p-subgroup of G. Theorem A. X_{0} has dimension n.

Our approach to this theorem and the next is via the algebra D(P) of Dickson invariants for the regular representation of P. This is a polynomial algebra of Krull dimension n over which H^{∗}BP is a finitely generated module. Theorem B. Suppose for some k > 0, D(P) acts freely on H^{k}(BP). Then every dominant summand of BP has dimension n.

The hypothesis of Theorem B means D(P)x ⊂ H^{∗}(BP) is a free D(P) module for all nontrivial x ∈ H^{k}(BP). Corollary C. If H^{∗}(BP) is free over D(P), then every dominant summand of BP has dimension n.

Let H(P) denote the subalgebra of H^{∗}BP consisting of even dimensional elements or just H^{∗}(BP) if p = 2. Since H(P) is a commutative algebra, it is more convenient for some purposes. Also H^{∗}(BP) is finitely generated module over H(P) thus there is no loss of generality for dimension theory.

We now recall that a commutative algebra is called Cohen-Macaulay if its depth equals its Krull dimension, that is, it has a regular sequence [Mu] of length n. A recent result of D. Rusin [R] states that H(P) is Cohen-Macaulay if as a D(P)- module the generators of D(P) form a regular sequence. In particular, if H(P) is Cohen-Macaulay then it is free over D(P). Thus we have Corollary D. If H(P) is Cohen-Macaulay then every dominant summand has dimension n. Question. Theorems A and B lead us to ask if every dominant summand of BG has dimension n. At present we know of no counter example and have been unable to settle this question either way.

ON THE DIMENSION THEORY OF DOMINANT SUMMANDS 3

The paper is organized as follows: Section 1 contains some preliminary notions about dimension theory and the Dickson invariants. The proofs of Theorems A and B are given in Sections 2 and 3 respectively. Section 4 is devoted to some related results.

Theorem A answers a question raised by Haynes Miller as to the dimension of the principal dominant summand.

4 JOHN MARTINO AND STEWART PRIDDY

Section 1. Preliminaries.

In this section we establish the preliminary notions which will be needed in the later sections. We begin with a brief sketch of Nishida’s theory of dominant summands [N]. For BP the classifying space of a p-group P, Lewis, May and McClure [LMM], using Carlsson’s solution of the Segal Conjecture [C], have shown that the ring of stable homotopy classes of self-maps {BP, BP } is generated as a free module over the p-adic integers Ẑ_{p} by maps of the form BP ⟶^{tr} BQ ⟶^{Bh} BP, where Q ≤ P, tr : BP → BQ is the reduced transfer (if Q = P then tr is understood to be the identity) and h : Q → P is an outer homomorphism. If

1 = ∑ e_{i} ∈ {BP, BP }

is a decomposition of one into primitive, orthogonal idempotents then BP ≅ ∨e_{i}BP, where each e_{i}BP = T el(BP ⟶^{e_{i}} BP) is the infinite mapping telescope, is a stable splitting into indecomposable summands. Define J ⊆ {BP, BP } to be the ideal generated by maps of the form BP ⟶^{tr} BQ ⟶^{Bh} BP where Q ≨ P. Then by Nishida [N]

{BP, BP }/J ≅ Ẑ_{p}Out(P)

where Out(P) is the outer automorphism group of P. If 1 = ∑ ẽ_{j} ∈ Ẑ_{p}Out(P) is a decomposition into primitive orthogonal idempotents, then each ẽ_{j} = e_{jo} +⋯+e_{jk} can be further decomposed into primitive, orthogonal idempotents in {BP, BP } where e_{jo} ∉ J and e_{ji} ∈ J for i = 1, …, k. We call e_{jo}BP a dominant summand. Each e_{ji}BP, i = 1, …, k is stably homotopy equivalent to a dominant summand of some BQ, Q ≨ P since e_{ji} ∈ J. The stable homotopy type of a summand is independent of the choice of idempotents. We also call ẽ_{j}BP ≅ e_{jo}BP ∨ (e_{j1}BP ∨ ⋯ ∨ e_{jk}BP) a superdominant summand. Since each ẽ_{j} corresponds to an indecomposable representation of Ẑ_{p}Out(P), we have a correspondence between irreducible representations and dominant summands. We call the domi-

ON THE DIMENSION THEORY OF DOMINANT SUMMANDS 5

nant summand corresponding to the trivial representation the principal dominant summand.

We now develop the main tool we will use to analyze the dimension of a dominant summand. For a finite field k of characteristic p the GL_{n}(k)-invariants of the polynomial algebra k[x_{1}, …, x_{n}] were found by Dickson [Dk] to be a polynomial subalgebra of the same Krull dimension n. In [W], it is shown for k = F_{p} that generators are given by the images of the Chern classes {c_{p^{n}−p^{i}}|i = 0, …, n − 1} ⊆ H^{∗}(BU(p^{n})) under the regular representation T^{n} → U(p^{n}) where T^{n} is a p-torus (or elementary abelian p-group) of F_{p}-dimension n.

We generalize this definition to all finite p-groups. The rank of a p-group P is the maximum F_{p}-dimension of p-tori in P. Let P be a p-group of order p^{r} and rank n. Consider the regular representation reg_{P} : P → ∑(P) ⊆ U(p^{r}) where ∑(P) is the group of set bijections of P.

Definition 1.1. The Dickson invariants D(P) = reg_{P}^{∗} C(n) ⊆ H^{∗}(BP) where C(n) = F_{p}[c_{p^{s}(p^{n}−p^{i})}|i = 0, 1, …, n − 1] ⊆ H^{∗}(BU(p^{r})) and p^{s} = [P : T] for T a maximal p-torus of rank n.

We now develop some basic properties of D(P).

Proposition 1.2. D(P) is a polynomial algebra of Krull dimension n, where n is the rank of P.

Proof. Let T be a maximal p-torus of rank n and [P : T] = p^{s}. P ≅ p^{s}T = T ∐ ⋯ ∐ T (p^{s} copies) as a T -set. Thus

P  ⟶^{reg_{P}} ∑(P)
↖ incl  ↗ p^{s}reg_{T}  commutes.
T

So incl^{∗}reg_{P}^{∗} C(n) = p^{s}reg_{T}^{∗} C(n) ≈ F_{p}[(c_{p^{n}−p^{i}})^{p^{s}}|i = 0, …, n − 1]. □

6 JOHN MARTINO AND STEWART PRIDDY

Proposition 1.3. If x ∈ D(P) and f ∈ Out(P) then f^{∗}(x) = x.

Proof. This is immediate from the commutative diagram

P ⟶^{reg_{P}} ∑(P)
↑^{f} ↑^{c_{f}}
P ⟶^{reg_{P}} ∑(P)

where c_{f} is conjunction by f in ∑(P).

Corollary 1.4. If ẽ ∈ Ẑ_{p}Out(P) then H^{∗}(ẽBP) = e^{∗}H^{∗}(BP) is a finitely generated D(P)-module.

For X a dominant summand of BP, we now want to establish that our definition of dimension d(X) is well-defined. We use

Theorem 1.5 (Hilbert, Serre) [AM]. Let A = ⊕A_{n} be a Noetherian graded ring generated as an A_{0}-algebra by the homogeneous elements x_{1}, …, x_{s} of positive degree k_{1}, …, k_{s} respectively. If M is a finitely generated graded module over A then

P S(M, t) = \frac{f(t)}{∏_{i=1}^{s}(1 − t^{k_{i}})}

where f(t) ∈ Z[t].

Proposition 1.6. If X is a summand of BP then P S(X, t) is a rational function.

Proof. By induction on the order of P we may assume that X is a dominant summand. Let ẽ ∈ Ẑ_{p}Out(P) be a primitive idempotent corresponding to X then by Corollary 1.4 and Theorem 1.5 P S(ẽBP, t) is a rational function. But ẽBP = X ∨ Y_{1} ∨ ⋯ ∨ Y_{k} where the Y_{i}’s are nondominant summands. Each Y_{i} is a dominant summand of BQ for some Q ≨ P. By the inductive hypothesis P S(Y_{i}, t) is a rational function. Since P S(ẽBP, t) = P S(X, t) + ∑_{i=1}^{k}P S(Y_{i}, t) we have that P S(X, t) is a rational function. □

ON THE DIMENSION THEORY OF DOMINANT SUMMANDS 7

Thus d(X) is a well-defined number. There is an immediate upper bound for d(X). Theorem 1.7 (Quillen) [Q2]. d(BP) = Krull dimension of H(P) = rank of P. So d(X) ≤ rank of P for every summand X of BP.

Another theorem of Quillen’s will prove useful in the following sections. Theorem 1.8 (Quillen) [Q1]. H^{∗}(BP) \underrightarrow{∏ incl^{∗}} ∏ H^{∗}(BT) where the product ranges over maximal p-tori, is a monomorphism modulo nilpotent elements.

8 JOHN MARTINO AND STEWART PRIDDY

Section 2. The proof of Theorem A.

We begin by recalling the definition of the principal dominant summand X_{0} of BP. Let e ∈ F_{p}Out(P) be a primitive idempotent with augmentation 1, i.e. ε(e) = 1, where ε : F_{p}Out(P) → F_{p} is the usual augmentation defined by ε(g) = 1 for all g ∈ Out(P). Let ẽ ∈ Ẑ_{p}Out(P) be a primitive idempotent lifting e, then ẽ decomposes in {BP, BP } into a sum of primitive idempotents

ẽ = e_{0} + e_{1} + ⋯ + e_{k}

with X_{0} ≃ e_{0}BP the principal dominant summand and e_{i}BP non-dominant summands for i ≥ 1. The p-local homotopy type of X_{0} is independent of these choices.

Proof of Theorem A. By Proposition 1.3 the ring D(P) of Dickson invariants consists of Out(P) invariants, hence e^{∗}D(P) = D(P) and so

D(P) ⊂ H^{∗}eBP = e^{∗}(H^{∗}BP).

Thus the theorem follows provided

dim e_{i}^{∗}(D(P)) < n

for all i ≥ 1. By the Segal Conjecture [LMM] each e_{i}, i ≥ 1, is a Ẑ_{p} linear combination of maps of the form

f : BP ⟶^{tr} BQ ⟶^{Bh} BP

for various subgroups Q ≤ P and homomorphisms h. Thus the theorem follows from

Lemma 2.1. If Q ≨ P or if h is not injective then f^{∗}(D(P)) has dimension < n.

Proof. dim f^{∗}D(P) ≤ dim H^{∗}BQ ≤ n. Hence, we are reduced to assuming dim H^{∗}BQ = n. By Theorem 1.7, the p-rank of Q is also n.

Case 1. Q ≨ P and h is injective: Let [P : Q] = p^{s}, s > 0. Then P ≈ p^{s}Q = Q ∐ ⋯ ∐ Q (p^{s} copies) as a Q set. Thus

(2.2)

P  ⟶^{reg_{P}} ∑(P)
i_{Q,P} ↖  ↗ p^{s}reg_{Q}  commutes.
Q

Similarly for Q′ = h(Q). Let h′ = p^{s}h : p^{s}Q → p^{s}Q′. Then h′ is a permutation of P and we claim that the diagram

(2.3)  \begin{matrix}Q & ⟶^{p^{s}reg_{Q}} & ∑(P) \\ ^{h}↓ & & ↓^{c_{h′}} \\ Q′ & ⟶^{p^{s}reg_{Q′}} & ∑(P) \end{matrix}

commutes where c_{h′} is conjugation by h′. For x ∈ Q, y ∈ p^{s}Q′, (c_{h′}∘p^{s}reg_{Q}(x))(y) = h(p^{s}reg_{Q}(x)(h^{−1}(y))) = h(x.h^{−1}(y)) = h(x).y = p^{s}reg_{Q′}h(x)(y).

Now we claim that

(2.4)  h^{∗}i_{Q′,P}^{∗} D(P) = i_{Q,P}^{∗} D(P).

To see this recall from (1.1) that D(P) = reg_{P}^{∗} C(n), thus i_{Q′,P}^{∗} D(P) = i_{Q′,P}^{∗} reg_{P}^{∗} C(n) = p^{s}reg_{Q′}^{∗}C(n) by (2.2). Hence by (2.3), h^{∗}i_{Q′,P}^{∗} D(P) = h^{∗}p^{s}reg_{Q′}^{∗}C(n) = (p^{s}reg_{Q}^{∗})c_{h′}^{∗}C(n) = (p^{s}req_{Q})^{∗}C(n) = i_{Q,P}^{∗} reg_{P}^{∗} C(n) = i_{Q,P}^{∗} D(P). Hence f^{∗}D(P) = tr^{∗}h^{∗}i_{Q′,P}^{∗} D(P) = tr^{∗}i_{Q,P}^{∗} D(P) = [P : Q]D(P) = 0 and so certainly dim f^{∗}D(P) < n. Case 2. h is not injective: Let T ⟶^{j} Q be a p-torus of rank n. Then T is a maximal p-torus and it follows easily that hj(T) has rank < n. Hence j^{∗}h^{∗}D(P) has dim < n. However by Theorem 1.8,

∏ j^{∗} : H^{∗}BQ → ∏ H^{∗}BT

is injective modulo nilpotent elements, where the union is taken over all maximal p-tori T < Q. Hence dim h^{∗}D(P) < n and so dim f^{∗}D(P) < n.

10 JOHN MARTINO AND STEWART PRIDDY

Section 3. The proof of Theorem B.

Let ẽ ∈ Ẑ_{p}Out(P) be a primitive idempotent and let X be the corresponding dominant summand. We can write ẽ = e_{0} + e_{1} where e_{0}, e_{1} are orthogonal idempotents in {BP, BP }, X = e_{0}BP, and e_{1} ∈ J. To prove Theorem B, it suffices to show that there is a copy of D(P) in H^{∗}(X) as a vector space.

Now e_{1}, as is any element of {BP, BP }, is a linear combination of maps of

the form BP ⟶^{tr} BQ ⟶^{Bh} BP. We write e_{1} = ∑_{α}Bh_{α}tr + ∑_{β}Bk_{β}tr where the

h_{α}’s are monomorphisms and the k_{β}’s are not. Since e_{1} ∈ J for h_{α} : Q → P a monomorphism we must have Q ≨ P.

Since {X, X} is a local ring [Mg] and X is a dominant summand the map

X ⟶^{i} BP \underrightarrow{∑ Bh_{α}tr} BP ⟶^{π} X

is nilpotent (i, π are inclusion and projection maps respectively). Thus, for any y ∈ H^{k}(X), (∑ Bh_{α}tr)^{∗}(y) = z ≠ y. Let w ∈ D(P) be the top-most Dickson invariant, that is the element which restricts to the image of (c_{p^{n}−1})^{p^{s}} on a maximal p-torus of rank n. Since the image of c_{p^{n}−1} vanishes on any p-torus of rank less than n [W], for any k_{β} : Q → P which is not a monomorphism Bk_{β}^{∗}(w) ∈ H^{∗}(BQ) is not detected on the cohomology of p-tori and by Quillen’s Theorem 1.8, Bk_{β}^{∗}(w) is nilpotent. Therefore, for some high power of w, Bk_{β}^{∗}(w^{t}) = 0 for every k_{β}. Let w̄ = w^{t}.

Since ẽ ∈ Ẑ_{p}Out(P) we know e^{∗}(w̄y) = w̄e^{∗}(y) = w̄y by Corollary 1.4. Moreover

e_{1}^{∗}(w̄y) = (∑ Bh_{α}tr)^{∗}(w̄y) + (∑ Bk_{β}tr)^{∗}(w̄y) = w̄(∑ Bh_{α}tr)^{∗}(y) + 0

by Frobenius reciprocity since Bh_{α}^{∗}(w̄) = incl_{Q}^{∗}(w̄) by (2.4). Thus e_{1}^{∗}(w̄y) = w̄z implies

e_{0}^{∗}(w̄y) = e^{∗}(w̄y) − e_{1}^{∗}(w̄y) = w̄(y − z) ≠ 0

since y − z ≠ 0 and D(P) acts freely on y − z by hypothesis.

Let u = w̄(y − z) and v ∈ D(P). e^{∗}(vu) = ve^{∗}(u) = ve_{0}^{∗}(u) = vu, but

e^{∗}(vu) = e_{0}^{∗}(vu) + e_{1}^{∗}(vu) = e_{0}^{∗}(vu) + v(∑ Bh_{α}tr)^{∗}(u) by Frobenius reciprocity

since (∑ Bk_{β}tr)^{∗}(u) = 0. Thus e^{∗}(vu) = e_{0}^{∗}(vu) + ve_{1}^{∗}(u) = e_{0}^{∗}(vu) since e_{1}^{∗}(u) = 0.

Hence e_{0}^{∗}(vu) = ve_{0}^{∗}(u) for all v ∈ D(P) ⇒ D(P)u ⊆ H^{∗}(X). □

12 JOHN MARTINO AND STEWART PRIDDY

Section 4. Addendum.

Finally we include a couple of results on superdominant summands along the lines of Theorems A and B.

Proposition 4.1. Let Z be a superdominant summand of BP, P a finite p-group. Then d(Z) ≥ rank of center of P.

Proof. The depth of H(P) is the maximum length of a regular sequence. By Duflot [Df] d=depth of H(P) ≥ rank of center of P. Let u_{1}, …, u_{d} be the bottom-most Dickson invariants, i.e. the elements which restrict to the images of (c_{p^{r}−p^{i}})^{p^{s}}, i = n − d, …, n − 1 on a p-torus. By Rusin [R], u_{1}, …, u_{d} form a regular sequence in H(P). From which it follows that for Z = ẽBP, ẽ ∈ Ẑ_{p}Out(P), H^{∗}(ẽBP) is a free module over F_{p}[u_{1}, …, u_{d}] which implies d(Z) ≥ d. □

An element z ∈ H^{n}(Z) is called nilpotent if Sq^{2^{k}n} … Sq^{n}z = 0 for some k (or P^{p^{k}n} … P^{n}β^{ϵ}z = 0 for p > 2, where ε = 0, 1 and z ∈ H^{2n−ϵ}(Z)).

Proposition 4.2. Let Z be a superdominant summand of BP, P a finite p-group. If z ∈ H^{∗}(Z) is not nilpotent then d(Z) ≥ minimum rank of maximal p-tori of P.

Proof. Since z is non-nilpotent, for some maximal p-torus T the restriction of z to H^{∗}(T) does not vanish (Theorem 1.8). Since incl_{T}^{∗} (D(P)) has the same dimension as D(T), d(Z) ≥ d(D(P)z) ≥ d(D(T)incl_{T}^{∗} (z)) = rank of T. □

ON THE DIMENSION THEORY OF DOMINANT SUMMANDS 13

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