ON THE DIMENSION THEORY OF DOMINANT SUMMANDS

John Martino and Stewart Priddy

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 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) is 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.

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 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) is 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.

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 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) is 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.

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 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) is 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.