RELATIVE TRACE IDEALS AND COHEN - MACAULAY QUOTIENTS OF MODULAR INVARIANT RINGS

PETER FLEISCHMANN

Abstract. Let G be a finite group, 𝔽 a field whose characteristic p divides the order of G and A^{G} the invariant ring of a finite-dimensional 𝔽G - module V. In analogy to modular representation theory we define for any subgroup H ≤ G the (relative) trace - ideal A_{H}^{G} ◁ A^{G} to be the image of the relative trace map t_{H}^{G} : A^{H} → A^{G}, f ↦ ∑_{g∈[G:H]} g(f). Moreover, for any family X of subgroups of G, we define the relative trace ideals A_{X}^{G} := ∑_{X∈X} A_{X}^{G} ◁ A^{G} and study their behaviour. If X consists of proper subgroups of a Sylow p - group P ≤ G, then A_{X}^{G} is always a proper ideal of A^{G}; in fact, we show that its height is bounded above by the codimension of the fixed point space V^{P}. But we also prove that if V is relatively X - projective, then A_{X}^{G} still contains all invariants of degree not divisible by p. If V is projective then this result applies in particular to the (absolute) trace ideal A_{{e}}^{G}. We also give a ‘geometric analysis’ of trace ideals, in particular of the ideal A_{<P}^{G} := ∑_{Q<P} A_{Q}^{P}, and show that I^{G,P} := \sqrt{A_{<P}^{G}} is a prime ideal which has the geometric interpretation as ‘vanishing ideal’ of G - orbits with length coprime to p. It is shown that A^{G}/I^{G,P} is always a Cohen - Macaulay algebra, if the action of P is defined over the prime field. This generalizes a well known result of Hochster and Eagon for the case P = 1 (see [13]). Moreover we prove that if V is a direct summand of a permutation module (i.e. a ‘trivial source module’), then the A_{X}^{G} are radical ideals and A_{<P}^{G} = I^{G,P}. Hence in this case the ideal and the corresponding Cohen - Macaulay quotient can be constructed using relative trace maps.

1. Introduction

Let R be an arbitrary commutative ring, V a free R- module of finite rank, and G a finite group, acting linearly on V. Then G acts on V^{∗} := Hom_{R}(V, R) by the rule gf := f ∘ g^{−1}. This action can be extended to the symmetric algebra A = R[V] := S(V^{∗}), on which G now acts as graded R - algebra automorphisms. We define A^{G} = ⊕_{i≥0}A_{i}^{G} to be the corresponding graded ring of invariants.

Up to now, the relation between module theoretic properties of the RG - modules V or V^{∗} and the ring and ideal structure of A^{G} does not seem to be well understood. In this paper we investigate these connections by utilizing techniques of modular representation theory to construct certain ideals of A^{G}. For any subgroup H ≤ G we define the (relative) trace - ideal A_{H}^{G} ◁A^{G} to be the image of the relative trace map t_{H}^{G} : A^{H} → A^{G}, f ↦ ∑_{g∈[G:H]} g(f) (see sec.3). In classical invariant theory, where R is a field of characteristic zero, these trace maps, in particular the absolute trace map t_{1}^{G}, have always been a major tool in constructing invariants. This of course is due to the fact that if the index |G : H| is invertible in R, then t_{H}^{G} is surjective. If R is a field of characteristic p | |G : H|, this is no longer true, as A_{H}^{G} cannot contain the constants. Notice that the condition |G : H| ∈ R^{∗} implies that all V and V^{∗} are relatively H - projective in the sense of modular representation theory. So a natural

1991 Mathematics Subject Classification. Primary 20C20, 13A50, 13C14.

question is: How far away from being surjective are relative trace maps for subgroups H whose index in G is not a unit in R? We will investigate this question in a broader context, by defining trace ideals A_{X}^{G} := ∑_{H∈X} A_{H}^{G}, where X is an arbitrary family of subgroups of G. It turns out that, if R = 𝔽 is a field of characteristic p > 0 and the elements of X intersect all Sylow p - subgroups of G properly, then the height of A_{X}^{G} is bounded above by the codimension of the fixed point space codim_{𝔽} V^{P} of a Sylow p - group P. Nevertheless we show, that if V (or V^{∗}) is relatively H - projective (or X - projective), then the ideal A_{H}^{G} (or A_{X}^{G} respectively) is still large enough, as to contain iA_{i}^{G} for all i ∈ ℕ_{0}. If again R is a field of characteristic p, this implies that t_{H}^{G} is surjective in all degrees coprime to p; in particular this applies to projective and regular representations of G. (In the case of regular representations, this result has been obtained independently in [17].)

In the fourth section we investigate in more detail the situation of permutation modules. Generalizing a result of Shank and Wehlau [22] on A_{{e}}^{G}, we prove that in this case the A_{X}^{G} are always radical ideals. In particular if G is a p - group, the ideal A_{<}^{G} := ∑_{U<P} A_{U}^{G} is prime and its quotient is a polynomial ring in the ‘orbit products’ of G on V^{∗} ( for the case of cyclic p - groups see also [18] and [20]). For the regular representation of the cyclic group of order p over a field of characteristic p we show that A^{G} can be generated by elements of a degree less than that given by Goebel’s bound for general invariant rings of permutation modules.

In the last section we assume R = 𝔽 to be a field of characteristic p > 0 and utilize the geometric interpretation of A^{G} as a ring of regular functions on the orbit variety V/G. For a fixed Sylow p - subgroup P of G we will define the relative trace ideal A_{<P}^{G} and the prime

ideal I^{G,P} = \sqrt{A_{<P}^{G}} in A^{G}. The latter one is the ‘vanishing ideal’ of G - orbits with length coprime to p. Using a result of Wilkerson [23], it will be shown that the quotient A^{G}/I^{G,P} is a Cohen - Macaulay ring of Krull dimension dim_{𝔽} V^{P}, whenever the action of P on V is defined over the prime field of 𝔽. Here V^{P} denotes the subspace of P - fixed points in V. If char 𝔽 does not divide |G|, i.e. if P = 1, then I^{G,P} = (0) and we recover the well known fact, proved by Eagon and Hochster ([13]), that in this case A^{G} is Cohen - Macaulay.

All these constructions are viewed as efforts to link the module structure of V to the ring and ideal structure of A^{G}. In particular we show that if V is a ‘trivial source module’ (i.e. a direct summand of a permutation module), then all A_{X}^{G} are radical ideals and A_{<P}^{G} = I^{G,P} is prime (with CM - quotient A^{G}/A_{<P}^{G} of dimension dim_{𝔽} V^{P}).

For all facts about modular representations the reader might refer to [1], [6] or [16].

Througout the paper the groups considered will assumed to be finite.

2. Basic definitions

Definition 2.1. Let R be a commutative ring, A an R - algebra and B an A - bimodule. We let D_{R}(A, B) denote the R - module of R - linear derivations from A to B, i.e.

D_{R}(A, B) := {∂ ∈ Hom_{R}(A, B)|∂(aa′) = a∂(a′) + ∂(a)a′ ∀a, a′ ∈ A}.

With _{RG}mod, mod_{RG} and _{RG}mod_{RG} we denote the categories of RG - (bi) - modules V such that V is also a free R - module of finite rank.

If Z(A) denotes the center of A, then D_{R}(A, B) is a Z(A) - module with (a∂)(f) := a(∂f) for a ∈ Z(A), f ∈ A and ∂ ∈ D_{R}(A, B). If B = _{A}A_{A}, the regular bimodule, then D_{R}(A) := D_{R}(A, A) is a Z(A) - Lie - algebra. An ideal I ◁ A will be called derivation - stable if ∂(i) ∈ I for all i ∈ I and ∂ ∈ D_{R}(A). For a group G, RG denotes the corresponding group ring.

Let V ∈ _{RG}mod. We define

T (V) := T_{R}(V) := ⊕_{i=0}^{∞}V^{⊗i}

to be the graded tensor algebra over V. The R - linear action of G on V together with the universal property of tensor products induces a representation of G as automorphisms of the (non - commutative) graded ring T (V). For any v_{1} ⊗ ... ⊗ v_{n} and g ∈ G we have: g(v_{1} ⊗ ... ⊗ v_{n}) = gv_{1} ⊗ ... ⊗ gv_{n}

If A = T (V), d := rank_{R}V and B is an A − A bimodule then the map

c : Hom_{R}(V, B) ≅ B^{d} → D_{R}(A, B)

α ↦ dα : (v_{i_{1}} ⊗ ... ⊗ v_{i_{k}} ↦ ∑_{j} v_{i_{1}} ⊗ ...α(v_{i_{j}})... ⊗ v_{i_{k}})

is an isomorphism of R - modules. In fact the functors Hom_{R}(V, ), ( )^{d} and D_{R}(A, ) from A - bimodules to R - modules are all isomorphic and exact. If B = T (V), then each α ∈ Hom_{R}(V, V) ⊆ Hom_{R}(V, B) has the unique extension d(α) ∈ D_{R}(A), thus we have the canonical embedding d : E := End_{R}(V) ↪ D_{R}(A); in particular d(Id_{V})(a) = deg(a) · a for homogeneous a ∈ T (V). Let I be a derivation - stable homogeneous ideal of A and c_{I} : A → A/I the canonical epimorphism. Then the exactness of D_{R}(A, ) implies that for any ∂ ∈ D_{R}(A, A/I) there is ˜∂ ∈ D_{R}(A) such that ∂ = c_{I} ∘ ˜∂. Hence ∂(I) = 0 and D_{R}(A, A/I) = D_{R}(A/I, A/I). Again we get isomorphisms

c : Hom_{R}(V, A/I) → D_{R}(A/I, A/I), α ↦ dα.

If moreover I ∩V = 0 then V ⊆ A/I and again each α ∈ End_{R}(V) can be uniquely extended to an element dα ∈ D_{R}(A/I) (i.e. satisfying dα(v) = α(v) for v ∈ V ⊆ (A/I)_{1}) such that the element ϑ := d(Id_{V}) satisfies ϑ(f) = deg(f)f for each homogeneous f ∈ A/I.

Now let I be the homogeneous ideal of T (V), generated by the tensors v ⊗ w − w ⊗ v for v, w ∈ V and S_{R}(V) = S(V) the symmetric algebra S(V) := T (V)/I. Then

S(V) := ⊕_{i=1}^{∞}S_{i}(V)

is also a graded R - algebra with S_{i}(V) := < v_{1} ⊗ ... ⊗ v_{i} + I >_{R} and v_{i} ∈ V. We also define Λ_{R}(V) = Λ(V) to be the exterior algebra Λ(V) := T (V)/J where J is the homogeneous ideal generated by the tensors v ⊗ v for v ∈ V and denote v_{1} ∧ ... ∧ v_{i} := v_{1} ⊗ ... ⊗ v_{i} + J. Notice that v_{1} ∧ v_{2} = −v_{2} ∧ v_{1}.

Again

Λ(V) := ⊕_{i=1}^{∞}Λ_{i}(V)

is a graded R - algebra with Λ_{i}(V) := < v_{1} ∧ ... ∧ v_{i} >_{R} and v_{i} ∈ V. In particular the ideals I and J are G - stable and hence we also have a natural action of G on S(V) and Λ(V) as automorphisms of graded R - algebras. Choosing an R - basis B := {v_{1}, ..., v_{d}} of V we can identify S(V) with the polynomial ring R[v_{1}, ..., v_{d}] in the indeterminates v_{1}, ..., v_{d}.

For w_{i} := ∑_{j_{i}} w_{ij_{i}}v_{j_{i}} and i = 1, ..., k we have

w_{1} ∧ ... ∧ w_{k} = ∑_{j_{1},...,j_{k}} w_{1j_{1}} · ... · w_{kj_{k}}v_{j_{1}} ∧ ... ∧ v_{j_{k}}.

If k > d then in each sum the v_{j_{r}} can not be pairwise different, so w_{1} ∧ ... ∧ w_{k} = 0. We see that each Λ_{i}(V) has an R basis

{v_{j_{1}} ∧ ... ∧ v_{j_{i}} | j_{1} < j_{2} < ... < j_{i}, 1 ≤ j_{k} ≤ d}

of cardinality ( \begin{matrix}d \\ i \end{matrix} ). In particular Λ_{i}(V) = 0 for i > d.

Let v, w ∈ V then for any δ ∈ D_{R}(A) we have δ(v ⊗ w − w ⊗ v) = δv ⊗ w + v ⊗ δw −δw⊗v −w⊗δv ∈ I and δ(v ⊗v) = δv ⊗v +v ⊗δv = (δv +v)⊗(δv +v)−δv ⊗δv −v ⊗v ∈ J. Hence I and J are derivation - stable.

If A = ⊕_{i=1}^{∞}S_{i}(V) then Hom_{R}(V, A) ≅ A^{d} with d = rank(V), hence D_{R}(A) is a free A - module of rank d. Now a straightforward induction, using the defining property of derivations (product rule), shows that c_{I} assigns to each vector α := (α_{1}, ..., α_{n}) ∈ A^{n} a unique derivation dα ∈ D_{R}(A) given by the formula

dα(f) = ∑_{i} ∂_{v_{i}}(f)α_{i}  (1)

(Here ∂_{v_{i}} is a shorthand for the formal partial differential operator \frac{∂}{∂_{v_{i}}}.)

3. The trace map

We collect some further notation which will be used in the sequel: Let A be an arbitrary graded R - G - algebra i.e. a graded R - algebra on which G acts as graded algebra isomorphisms. For W ∈ _{RG}mod and any subgroup H ≤ G we define W^{H} := {w ∈ W |hw = w ∀h ∈ H} to be the R - submodule of H - fixed points in W. In particular A^{H} is the algebra of H - invariants in A. We have the following homomorphism of R - modules:

t_{H}^{G} : W^{H} → W^{G}, w ↦ ∑_{g∈[G:H]} gw,

where [G : H] denotes an arbitrary cross section of (left) H - cosets in G such that G = ⊎_{g∈[G:H]}gH. This is called the (relative) trace homomorphism, its image will be denoted with W_{H}^{G}. If the index |G : H| is invertible in R, then t_{H}^{G} is an epimorphism; if W = A = R[V] = S(V^{∗}) then the trace homomorphism preserves degrees. For any a, a′ ∈ A^{G} and b ∈ A^{H} we have t_{H}^{G}(aba′) = at_{H}^{G}(b)a′, showing that the image A_{H}^{G} := t_{H}^{G}(A^{H}) is an ideal in A^{G}. If A = R[V] then A_{U}^{H} is a homogeneous ideal. If X is a set of subgroups of H, then we define W_{X}^{H} := ∑_{U∈X} W_{U}^{H} and W_{<}^{H} := ∑_{U<H} W_{U}^{H}. Notice that again A_{X}^{H} and A_{<}^{H} are ideals in A^{H} and homogenous if A = R[V].

Let U, H ≤ G be subgroups and suppose we are given a cross section [U : G : H] of U, H double - cosets with G = ⊎_{g∈[U:G:H]}UgH. For each g ∈ [U : G : H] the subset UgH is a disjoint union of H - cosets of the form ugH. Although UgH is not a subgroup in general, we can define [UgH : H] to be a cross section in U of these H - cosets, i.e. UgH = ⊎_{t∈[UgH:H]}tgH. Then [UgH : H] is also a cross section [U : U ∩ ^{g}H] of U ∩ ^{g}H cosets in U, i.e. U = ⊎_{u∈[UgH:H]}u(U ∩ ^{g}H). Here, as in the following ^{g}H denotes gHg^{−1}. For each a ∈ A^{U} and b ∈ A^{H} we have the following ‘Mackey - formulae’:

t_{H}^{G}(b) = ∑_{g∈[U:G:H]} t_{U∩ ^{g}H}^{U}(gb) ∈ A_{X}^{U}  (1)

t_{U}^{G}(a) · t_{H}^{G}(b) = ∑_{g∈[U:G:H]} t_{U∩ ^{g}H}^{G}(agb) ∈ A_{X}^{G}  (2)

where X := {U ∩ ^{x}H | x ∈ G} (e.g. see [3], [16]). The group G acts on E(V) := End_{R}(V) by the rule α^{g}(v) = g^{−1}αg(v) for α ∈ E(V), g ∈ G and v ∈ V. Notice that E(V)^{G} = End_{RG}(V).

Now fix a set X of subgroups of G. In representation theory the module V ∈ RG − mod is called relatively X - projective if and only if E(V)^{G} = E(V)_{X}^{G} or in other words, if the identity Id_{V} is contained in E(V)_{X}^{G}. Notice that V is relatively X - projective if and only if so is V^{∗}. If X = {H} then the X - projective module V is called relatively H - projective (e.g. if the index [G : H] is invertible in R). It is easy to see that t_{H}^{G}(a) = t_{^{g}H}^{G}(^{g}a) for any g ∈ G and a ∈ A^{H}; hence A_{H}^{G} = A_{^{g}H}^{G} and a module V is relatively H projective iff it is so for any G - conjugate of H. Notice that V is projective in _{RG}mod if and only if it is relatively 1 - projective, i.e. with respect to the trivial subgroup ([2] 3.6.4).

Here we propose to extend the above machinery to graded R - G - algebras, where the endomorphism ring E(V) is replaced by the R - Lie - algebra of derivations and the identity element by the element ϑ = d(Id). So let A be an R - G - algebra and U ≤ H ≤ G be subgroups. It is clear that G acts on D_{R}(A) by conjugation. Again we can define D_{R}(A)^{H} to be the Lie - subalgebra of H - fixed points and D_{R}(A)_{U}^{H} to be the image of the trace homomorphism

t_{U}^{H} : D_{R}(A)^{U} → D_{R}(A)^{H}, δ ↦ ∑_{h∈[H:U]} hδh^{−1}.

As above we define: D_{R}(A)_{<}^{H} := ∑_{Y <H} D_{R}(A)_{Y}^{H}. Now D_{R}(A)_{U}^{H} and D_{R}(A)_{<}^{H} are Lie - algebra ideals in D_{R}(A)^{H}. For any two subsets D ⊆ D_{R}(A) and X ⊆ A we define D · X := {∂(b) | b ∈ X and ∂ in D}. If δ = ∑_{g∈[G:H]} g∂g^{−1} with ∂ ∈ D_{R}(A)^{H}, then we have for any a ∈ A^{H}, h∂(a) = ∂(ha) = ∂(a) ∈ A^{H}. If moreover a ∈ A^{G} ≤ A^{H}, then

δ(a) = ∑_{g∈[G:H]} g∂g^{−1}(a) = ∑_{g∈[G:H]} g∂(a) ∈ A_{H}^{G}

hence D_{R}(A)_{H}^{G} · A^{G} ⊆ A_{H}^{G}. From this we immediately get D_{R}(A)_{X}^{G} · A^{G} ⊆ A_{X}^{G}. We are now ready to prove:

Proposition 3.1. Let G be a group, X a set of subgroups and V ∈ _{RG}mod. For a homogeneous derivation - stable and G - stable ideal I of T (V) with V ∩I = 0 we set A := T (V)/I. Then the following are equivalent:

(i): V is relatively X - projective;

(ii): ϑ = d(Id_{V}) ∈ D_{R}(A)_{X}^{G};

(iii): Each homogeneous component A_{i} with i invertible in R is relatively X - projective.

If one of (i),(ii) or (iii) holds, then we have the inclusions:

⊕_{i∈ℕ}i · (A_{i})^{G} ⊆ D_{R}(A)_{X}^{G} · A^{G} ⊆ A_{X}^{G}.

In particular if R = 𝔽 is a field of characteristic p > 0 and V is relatively X - projective, then A_{X}^{G} contains all invariants of degree coprime to p.

Proof: The injection d : E(V) → D_{R}(A), α ↦d(α) is an RG - module homomorphism, so t_{H}^{G}(d(α)) = d(t_{H}^{G}(α)). Moreover Id_{V} ∈ E(V)_{X}^{G} implies that Id_{V} = ∑_{H∈X} t_{H}^{G}(α_{H}) for some α_{H} ∈ E(V)^{H}, hence ϑ = d(Id_{V}) = ∑_{H∈X} t_{H}^{G}(d(α_{H})) ∈ D_{R}(A)_{X}^{G}. This proves ‘(i) ⇒ (ii)’.

(ii) ⇒ (iii): Suppose that i ∈ R^{∗} and ϑ ∈ D_{R}(A)_{X}^{G}. Let π_{i} denote the projection of A onto A_{i}; then π_{i} ∈ Hom_{RG}(A, A_{i}) and for any ∂ ∈ D_{R}(A)^{H} we have π_{i} ∘ ∂_{|A_{i}} ∈ E(A_{i})^{H}. By assumption ϑ = d(Id_{V}) = ∑_{H∈X} t_{H}^{G}(∂_{H}) with ∂_{H} ∈ D_{R}(A)^{H}. So we get

Id_{|A_{i}} = π_{i}(i^{−1}ϑ_{|A_{i}}) = ∑_{H∈X} π_{i}(t_{H}^{G}(i^{−1}∂_{H})) = ∑_{H∈X} t_{H}^{G}(i^{−1}π_{i} ∘ ∂_{H}) ∈ E(A_{i})_{X}^{G}.

Hence A_{i} is relatively X - projective. The implication (iii) ⇒ (i) is obvious, since V ≅ A_{1}. If ϑ ∈ D_{R}(A)_{X}^{G} then for any a ∈ (A_{i})^{G} we get ia = ϑa ∈ D_{R}(A)_{X}^{G} · A^{G} ⊆ A_{X}^{G}.

∘

Corollary 3.2. If R is a field of characteristic p, dividing the group order |G|, V a projective module (e.g. the regular representation of G) and A = R[V] = S(V^{∗}), then the usual (absolute) trace map

t_{<1>}^{G} : A → A^{G}, a ↦ ∑_{g∈G} g(a)

is surjective in all degrees i ≠ 0 mod p.

Remark 3.3. For the case of the regular representation of G the result above also appears in [17].

Corollary 3.4. Let R be a commutative ring and V ∈ _{RG}mod relatively H - projective for the subgroup H ≤ G. Then for any i ∈ ℕ which is invertible in R, the homogeneous components S_{i}(V^{∗}) and Λ_{i}(V^{∗}) of the symmetric or exterior tensors S(V^{∗}) respectively Λ_{i}(V^{∗}) are also relatively H - projective.

Remark 3.5. Suppose that R a field of characteristic p which divides |G| and A = R[V] with V ∈ _{RG}mod.

a) Relative projectivity of V with respect to H will in general not imply relative projectivity of S_{p·i}(V^{∗}) or Λ_{p·i}(V^{∗}). If R is a field of characterisitic p | |G| then each projective RG - module W has dimension divisible by p (because its restriction W_{|P} to a Sylow p - subgroup P is a free RP - module; see [1]). If V has dimension d = p^{k}ℓ with gcd(ℓ, p) = 1, then

S_{p^{k}}(V^{∗}) has dimension ( \begin{matrix}d + p^{k} − 1 \\ p^{k} \end{matrix} ) which can be ≠ 0 mod p; hence S_{p^{k}}(V^{∗}) need not

be projective. Similarly Λ_{p^{k}}(V^{∗}) has dimension ( \begin{matrix}d \\ p^{k} \end{matrix} ) ≠ 0 mod p and Λ(V^{∗})_{p^{k}} is not

projective either.

b) If P is a Sylow p - subgroup of G, then V is relatively P - projective. If moreover V is indecomposable, then there always exists a subgroup Q ≤ P, unique up to conjugacy in G, which is minimal such that V is relatively Q - projective. In modular representation theory, such a subgroup is called a vertex of V (e.g. see [1] pg. 67 ff). In this language, 3.1 says that the relative trace ideal A_{Q}^{G} of a vertex Q of V contains all invariants of degree coprime to p. This also holds for A_{X}^{G}, whenever X contains a vertex of V.

c) Due to a theorem of D.Higman ([2] 3.6.4) V is relatively H - projective if and only if V is a direct summand of an induced module W^{↑G}, where W ∈ _{RH}mod (e.g. W = V_{|H}). In this case A_{H}^{G} contains all invariants of degree coprime to p.

4. Invariant rings of permutation modules

Now let G be a finite group, Ω a finite G - set and R^{Ω} the set of functions from Ω to R. We consider the permutation module V_{Ω} := R^{Ω} ≅ ⊕_{ω∈Ω}R · e_{ω}, where e_{ω}(ω′) = 1 if and only if ω = ω′ and zero otherwise. This module is selfdual, so V_{Ω}^{∗} ≅ V_{Ω} and we can set A = A(R) := S_{R}(V_{Ω}) = ⊕_{f∈ℕ_{0}^{Ω}}Rπ(f), where for each f ∈ ℕ_{0}^{Ω} we define π(f) := ∏_{ω∈Ω} e_{ω}^{f(ω)}. Notice that for α, β ∈ ℕ_{0}^{Ω} we have π(α) · π(β) = π(α + β) where (α + β)(ω) = α(ω) + β(ω), hence we can view S(V_{Ω}) as the semigroup - algebra R[ℕ_{0}^{Ω}].

The group Σ_{Ω} and its subgroup G act on A by permuting the monomials π(f) according to the rule:

σ(π(f)) = ∏_{ω∈Ω} e_{σ(ω)}^{f(ω)} = ∏_{ω∈Ω} e_{ω}^{f∘σ^{−1}(ω)} = π(f ∘ σ^{−1}).

Hence the RG -module S(V_{Ω}) is isomorphic to the permutation module V_{ℕ_{0}^{Ω}} and in each degree i the module S(V_{Ω})_{i} is isomorphic to the permutation module V_{(ℕ_{0}^{Ω})_{i}}, where (ℕ_{0}^{Ω})_{i} := {f ∈ ℕ_{0}^{Ω}| ∑_{ω} f(ω) = i} with the natural action of Σ_{Ω} and G. In particular the R - module A^{G} has a basis consisting of the orbit sums π(f)^{+} := ∑_{g∈[G:G_{f}]} gπ(f) = t_{G_{f}}^{G} (π(f)) for f ∈ ℕ_{0}^{Ω}.

Now the Mackey - formula of section 3 describes the multiplication of orbit sums:

Lemma 4.1 ([9]). For α, β ∈ ℕ_{0}^{Ω} we have in A(R)^{G}:

π(α)^{+} · π(β)^{+} = ∑_{g∈[G_{α}:G:G_{β}]} \frac{|G_{α+g(β)}|}{|G_{α} ∩ ^{g}G_{β}|}π(α + gβ)^{+}.

In particular if R has characteristic p > 0 and G is a p - group we get

π(α)^{+} · π(β)^{+} = ∑_{g∈[G_{α}:G:G_{β}], G_{α+g(β)}=G_{α}∩ ^{g}G_{β}} π(α + gβ)^{+}.

The following is an extended version of a result of [10], where it has been used to obtain degree bounds for the generators of modular invariant rings of trivial source modules:

Theorem 4.2. Let X be a set of subgroups of G, A_{X} := ∑_{H∈X} t_{H}^{G}(A^{H}) ◁ A^{G} and A_{<}^{G} := A_{{H | H<G}}^{G}. Then:

(i): A_{X}^{G} = ⊕_{α∈ℕ_{0}^{Ω}}Rg_{X} (α) · π(α)^{+}, where g_{X} (α) := gcd(\frac{|G_{α}|}{|H_{α}|} | H ∈ X) · 1_{R}.

(ii): For T ⊆ Ω let χ_{T} : Ω → ℕ_{0} be the characteristic function of T (i.e. 1 on T and 0 on Ω\T). Then S := ⟨π(α) | α ∈ ℕ_{0}^{Ω}, G_{α} = G⟩_{R} ≅ R[π(χ_{ω^{G}}) | ω^{G} ∈ Ω/G] with algebraically independent elements π(χ_{ω^{G}}). In other words, S is a polynomial subalgebra of A^{G} in |Ω/G| many variables π(χ_{ω^{G}}).

(iii): A^{G}/A_{<}^{G} ≅ S/(A_{<}^{G} ∩ S) with A_{<}^{G} ∩ S = gcd(\frac{|G|}{|H|} | H < G) · S.

Proof: (i): Let H ∈ X and a ∈ A^{H}, then a = ∑_{α} c_{α}t_{H_{α}}^{H}(π(α)). Hence t_{H}^{G}(a) = ∑_{α} c_{α}t_{H_{α}}^{G}(π(α)) = ∑_{α} c_{α}t_{G_{α}}^{G}(t_{H_{α}}^{G_{α}}(π(α))) = ∑_{α} c_{α} \frac{|G_{α}|}{|H_{α}|}t_{G_{α}}^{G}(π(α)). This is

contained in the set on the right hand side of (i). Let g_{X} = ∑_{H∈X} \frac{|G_{α}|}{|H_{α}|}c_{H} with c_{H} ∈ ℤ. Then g_{X} π(α)^{+} =

∑_{H∈X} \frac{|G_{α}|}{|H_{α}|}c_{H}t_{G_{α}}^{G}(π(α)) = ∑_{H∈X} c_{H}t_{G_{α}}^{G}t_{H_{α}}^{G_{α}}(π(α)) = ∑_{H∈X} c_{H}t_{H}^{G}t_{H_{α}}^{H}(π(α)) ∈ A_{X}^{G}.

(ii): If α ∈ ℕ_{0}^{Ω} with G_{α} = G, then α is constant on G - orbits in Ω. In this case α induces a function α̂ ∈ ℕ_{0}^{Ω/G}, ω^{G} ↦ α(ω) and we get π(α) = ∏_{ω^{G}∈Ω/G}(π(χ_{ω^{G}})^{α̂(ω^{G})}. Since different G - orbits are disjoint, the set {π(χ_{ω^{G}}) | ω ∈ Ω/G} is algebraically independent.

(iii): Obviously A^{G} = A_{<}^{G} + S with A_{<}^{G} ∩ S = ⊕_{α∈ℕ_{0}^{Ω}, G_{α}=G}Rg_{<}(α) · π(α)^{+} = gcd( \frac{|G|}{|H|} | H < G) · S. ∘

The next result shows, that in case of permutation modules over fields of positive characteristic, the A_{X}^{G} are always radical ideals. This generalizes Theorem 6.1 of [22]:

Proposition 4.3. Let 𝔽 be a field of characteristic p > 0, G ≤ Σ_{Ω} a permutation group, X a set of subgroups of G and S_{X} := ⊕_{α∈ℕ_{0}^{Ω}, p | |G_{α}:H_{α}|, ∀H∈X} 𝔽π(α)^{+}. Then A^{G} = A_{X}^{G} ⊕ S_{X} and

\sqrt{A_{X}^{G}} = A_{X}^{G} = ⊕_{α∈ℕ_{0}^{Ω}, p does not divide |G_{α}:H_{α}|, for some H∈X} 𝔽π(α)^{+}.

Proof: Obviously g_{X} (α) ≠ 0 if and only if p does not divide |G_{α} : H_{α}| for some H ∈ X so we get A^{G} = A_{X}^{G} ⊕S_{X} and the second equation for A_{X}^{G}. Thus we only have to prove that A_{X}^{G}

is a radical ideal. If this is false, we can assume without loss, that there is 0 ≠ s ∈ S_{X} ∩\sqrt{A_{X}^{G}} with s = c_{1}π(α_{1})^{+} + ⋯ + c_{k}π(α_{k})^{+} ∈ S_{X}. Let p^{m}α_{i} denote the p^{m} - fold sum of α_{i} in the semigroup ℕ_{0}^{Ω}. Then (π(α_{i})^{+})^{p^{m}} = (t_{G_{α_{i}}}^{G} (π(α_{i})))^{p^{m}} = t_{G_{α_{i}}}^{G} (π(p^{m}α_{i})) = t_{G_{p^{m}α_{i}}}^{G} (π(p^{m}α_{i})) =

π(p^{m}α_{i})^{+}. Hence each s^{p^{m}} involves π(p^{m}α_{i}) with coefficient c_{i}^{p^{m}}. Since G_{p^{m}α_{i}} = G_{α_{i}}, we conclude s^{p^{m}} ∉ A_{X}^{G} for all m ∈ ℕ, a contradiction. ∘

From 4.2 it is easy to see, that if R = 𝔽 is a field of characteristic p > 0 and P is a p - group, then A_{<}^{G} ∩ S = 0 and we get the following result:

Theorem 4.4. If P is a p - group, R = 𝔽 a field of characteristic p > 0 and V = 𝔽^{Ω} is a permutation module, then the relative trace ideal A_{<}^{P} of A^{P} is a prime ideal of height |Ω|−|Ω/P |; moreover A^{P} is generated by A_{<}^{P} together with the subalgebra S of orbit products π(ω^{G}). In particular the quotient A^{P} /A_{<}^{P} ≅ S is a polynomial ring in |Ω/P | variables. This holds for any projective 𝔽P - module V, in particular for the regular representation V ≅ 𝔽P.

Remark 4.5. If V = V_{Ω} is a permutation module of the p - group P, then

V = ⊕_{ω^{P} ∈Ω/P} V_{ω^{P}} ,

where the V_{ω^{P}} ≅ R_{P_{ω}}^{↑P} are induced from the trivial module of the stabilizer P_{ω}, so they are P_{ω} - projective. The group P acts on Ω without fixed points (i.e. orbits of size 1) if and only if V_{Ω} is X - projective where X is the set of all proper subgroups of P. In this case 4.4 illustrates the statement of 3.1 that A_{<}^{P} contains all invariants of degree coprime to p.

4.1. The regular representation of ℤ_{p}. Now we consider more closely the groups of prime order. So let p be a prime and P = ℤ_{p} = < g >. In this situation A_{<}^{P} coincides with the absolute trace ideal A_{{e}}^{G} of A^{G}. Hence 4.4 says that if R is a field of characteristic p, then for any permutation representation of ℤ_{p} on a set Ω, the trace ideal is prime with quotient being a polynomial algebra in as many variables as there are P orbits on Ω. We want to look at the regular representation V = RP in more detail.

We can choose g = (1 2 … p) ∈ Σ_{p} and view V as the natural permutation module R\underline{p} where \underline{p} := \{1, 2, …, p\}; hence A^{G}/A_{<}^{G} ≅ R[N(e_{1})] where N(e_{1}) is the orbit product e_{1}e_{2} … e_{p} = e_{1}g(e_{1})g^{2}(e_{1}) … g^{p−1}(e_{1}). Due to a fundamental result of M. Göbel on invariant rings of permutation modules, the invariant ring A^{P} is generated by orbit sums of monomials

whose degree is less than or equal to ( \begin{matrix}p \\ 2 \end{matrix} ). In our special case we can do better:

Proposition 4.6. Let R be an arbitrary commutative ring and A = S(V) with V = RP and P ≅ ℤ_{p}. Then A^{G} is generated by N(e_{1}) together with elements of the trace ideal A_{{e}}^{P}

of degree less than or equal to max\{3, ( \begin{matrix}p − 1 \\ 2 \end{matrix} ) + 1\}.

Proof: For p ∈ {2, 3} the result can be verified easily. So we can assume that p ≥ 5. As before we will identify the monomial e_{1}^{f(1)} … e_{p}^{f(p)} with the exponent function f ∈ ℕ\underline{p} ; 0. A homogeneous invariant a ∈ A^{P} of degree i will be called indecomposable if it cannot be written as a sum of products of invariants of strictly lower degree; if it is not indecomposable it is called decomposable. Two homogeneous invariants of the same degree will be called equivalent if their difference is decomposable.

Following Göbel [12] we call an orbit sum f^{+} := ∑_{g∈P_{f}} g(f) ‘special’ if f(\underline{p}) is an interval containing 0. So f = (0, 1, 2, 2, 3) is special while h = (0, 1, 3, 4, 4) and h′ = (1, 2, 3, 4, 5) are not. In [12] it is proved that A^{P} is generated by special orbit sums, thus giving the degree bound 0 + 1 + ⋯ + p − 1. So we have to investigate the special orbit sums for indecomposability. Now the following lemma yields our result: ∘

Lemma 4.7. Let p ≥ 5 and f^{+} ∈ A_{<}^{P} be an indecomposable and special orbit sum. Then ( #f^{−1}({0}) ≥ 2) or (#f^{−1}({0}) = 1 and #f^{−1}({1}) ≥ 2) or (#f^{−1}({0}) = #f^{−1}({1}) = 1 and #f^{−1}({2}) = p − 2). In particular f^{+} is of degree less or equal to ≤ 0 + 1 + 1 + 2 + 3 + ⋯ + p − 2.

Proof: As f is special, #f^{−1}({0}) and #f^{−1}({1}) ≥ 1 and we can assume that #f^{−1}({0}) = #f^{−1}(\{1\}) = 1. Let f(i_{1}) = 1 and f(i_{0}) = 0 with i_{0} ≡ i_{1} + c and p > c ∈ \underline{p}. (By cycling we can assume that 1 ≤ i_{1} < i_{0} ≤ p). Then we get

f^{+} = (f_{1} − 1, f_{2} − 1, …, 1, …, 0, …, f_{p} − 1)^{+} · (1, 1, …, 1, 0, 1, …, 1, 0, 1, …, 1)^{+}−

∑_{g^{i}≠1}[(f_{1} − 1, f_{2} − 1, …, 1, …, 0, …, f_{p} − 1) · g^{i}((1, 1, …, 0, 1, …, 1, 0, 1, …, 1))]^{+}.

All terms that have to be subtracted are decomposable, except for i = c, when g^{c}((1, 1, …, 0, 1, …, 1, 0, 1, …, 1)) has its zero entries at i_{1}+c = i_{0} and i_{0}+c ≠ i_{1} ( mod p). If f(i_{0} + c) > 2, then

[(f_{1} − 1, f_{2} − 1, …, 1, …, 0, …, f_{p} − 1) · g^{c}((1, 1, …, 0, 1, …, 1, 0, 1, …, 1))]^{+}

has no entry 1, so is decomposable again; we conclude f(i_{0} + c) = 2 and −f^{+} is equivalent to

f′^{+} := (f_{1}, f_{2}, …, 2, …, 0, …, 1, …, f_{p})^{+}

with f′(i_{1}) = 2, f′(i_{0}) = 0, f′(i_{0} + c) = 1 and f′(j) = f(j) for all the other j’s. Now consider f′^{+} =

(f_{1} − 1, f_{2} − 1, …, 1, …, 0, …, 0, …, f_{p} − 1)^{+} · (1, 1, …, 1, …, 0, …, 1, …, 1)^{+}−

∑_{g^{i}≠1}[(f_{1} − 1, f_{2} − 1, …, 1, …, 0, …, 0, …, f_{p} − 1) · g^{i}((1, 1, …, 1, …, 0, …, 1, …, 1))]^{+}.

A similar argument as before shows that −f′^{+} is equivalent to

f′′^{+} := (f_{1}, f_{2}, …, 2, …, 1, …, 0, …, f_{p})^{+}

with f′′(i_{1}) = 2, f′′(i_{0}) = 1, f′′(i_{0} + c) = 0 and f′′(j) = f(j) for all the other j’s. Now iterating this process shows that f(i_{1} + ℓ · c) = 2 for all ℓ ∈ \underline{p}\\{1, p\} (modulo p). Hence f(i) = 2 for all i ≠ i_{1}, i_{0}. ∘ Remark 4.8. If p = 5 the bound in 4.6 gives 7, which in this case can be shown to be sharp by computer calculations (this has been verified by Gregor Kemper using a MAGMA implementation of his algorithms, see also [14] ).

5. A canonical trace ideal and Cohen-Macaulay quotients

In this section we will discuss a special feature of modular invariant theory over a field 𝔽 of positive characteristic 0 < p | |G|. As before let V be a finitely generated 𝔽G - module for the finite group G, A := A(V) := 𝔽[V] its symmetric algebra and A^{G} the invariant ring. Let \overline{𝔽} denote the algebraic closure of 𝔽 and Â := A ⊗_{𝔽} \overline{𝔽}. Then clearly V̂ := V ⊗_{𝔽} \overline{𝔽} can be viewed as an algebraic 𝔽 - variety and the maximal ideal spectrum max − spec Â is in bijection to the points in V̂.

Now let U, H ≤ X ≤ G be subgroups; notice that A is integral and finite over A^{H} and A as well as A^{H} are integrally closed in their quotient fields Q(A) and Q(A)^{H} respectively. Moreover Q(A) : Q(A)^{H} is a Galois extension with Galois group H. The dual i^{∗} := i_{H}^{∗} of the inclusion map i_{H} : A^{H} ↪ A is the map spec A → spec A^{H}, m ↦ m ∩A^{H}. In this situation the classical ‘lying over’, ‘going up’, ‘going down’ and transitivity theorems of commutative algebra apply (see [3] Theorem 1.4.4. on pg. 7). Notice that A^{G} ⊗_{𝔽} \overline{𝔽} = (A⊗_{𝔽} \overline{𝔽})^{G} = Â^{G} and the above statements also hold for Â instead of A. Hence max − spec Â^{H} can be identified with the set V̂ /H of H - orbits on V̂ and î_{H}^{∗}, the dual of î_{H} := i_{H} ⊗_{𝔽} \overline{𝔽}, can be identified with the orbit map v ↦v^{H}. Thus a strict geometric quotient V̂ /H exists and the invariant ring Â^{H} can be interpreted as the ring of regular functions O(V̂ /H) of the quotient variety (e.g. see [11] Theorem 5.52 pg. 187 or [3] pg.8 ). Notice that all H - orbits are Zariski - closed, since H is finite. The orbit map î_{H}^{∗}, being the comorphism of a finite embedding, is surjective and closed, i.e. sends closed irreducible subvarieties of V̂ to closed irreducible subvarieties of V̂ /H.

The U - fixed points in V̂ form a linear subspace, hence an irreducible closed 𝔽 - subvariety V̂^{U} of V̂. The image î_{H}^{∗}(V̂^{U}) is therefore a closed and irreducible subvariety of V̂ /H; it coincides with the set of orbits v^{H} such that U ≤_{H} G_{v}. Here and in the following, U ≤_{H} X means that an H - conjugate ^{h}U = hUh^{−1} of U is contained in the subgroup X ≤ G.

In particular if H = P is a Sylow p - subgroup of G, then

î_{G}^{∗}(V̂^{P}) := {v^{G} | p |̸ |v^{G}|} = {v^{G} | P ≤_{G} G_{v}} ⊆ V̂ /G.

For any v^{H} ∈ î_{H}^{∗}(V̂^{U}), the fibre (î_{H}^{∗})_{|V̂^{U}}^{−1} (v^{H}) consists of all v′ ∈ V̂^{U} that are H - conjugate to v, hence it is a finite set and we conclude that the dimension of the variety î_{H}^{∗}(V̂^{U}) ⊆ V̂ /H is equal to the dimension of the \overline{𝔽} - vector space V̂^{U} or of the 𝔽 - vector space V^{U}. Notice that A and A^{H} still induce regular functions on V and V/H respectively, but the maps A → O(V) and A^{H} → O(V/H) are no longer injective and neither O(V) nor O(V/H) need to be integral domains.

Now let θ^{∗} : Ŷ ↪ V̂ be the inclusion of a closed G - stable 𝔽 - subvariety with O(Ŷ) = B̂ = B ⊗_{𝔽} \overline{𝔽} = θ(Â) ≅ Â/Î and Î := ker θ. Again there is a strict quotient Ŷ /G and î^{∗} ∘ θ^{∗} : Ŷ → V̂ /G factors through the quotient map j^{∗} : Ŷ → Ŷ /G, which is the dual of the inclusion O(Ŷ)^{G} ↪ O(Ŷ). Then Ẑ := î^{∗} ∘ θ^{∗}(Ŷ) is a closed 𝔽 - subvariety of V̂ /G whose corresponding ring of regular functions O(Ẑ) =: Ĉ is a certain quotient of Â^{G}. We summarize the situation in the following commutative diagrams:

û^{∗} Ŷ /G ⟶ V̂ /G ↘  ↗ ĵ^{∗} ↑  Ẑ  ↑ î^{∗} ↗ Ŷ  ↪  V̂ θ^{∗}  û B̂^{G}  ⟵ Â^{G} ↖  ↙ ĵ ↓  Ĉ  ↓ î ↙ B̂  ←  Â θ

Since the morphism ω^{∗} = i^{∗} ∘ θ^{∗} : Ŷ → Ẑ is surjective, its comorphism ω : Ĉ → B̂ is an injection and the kernel of ϕ : Â^{G} → Ĉ coincides with the kernel of û, so Ĉ ≅ Â^{G}/(ker θî_{G}) = Â^{G}/(ker θ ∩ Â^{G}). Although the morphism γ^{∗} : Ŷ /G → Ẑ is bijective by construction, it is in general not an isomorphism. Indeed, the comorphism γ : Ĉ ↪ B̂^{G} makes B̂^{G} a purely inseparable, integral extension. This result can be found (without proof) in [21] pg. 51. For convenience we include a short proof here:

Lemma 5.1. The injection γ : A^{G}/(ker θ ∩ A^{G}) ≅ C ↪ B^{G} is a purely inseparable, integral ring extension.

Proof: Let b ∈ B^{G} and a ∈ A with θ(a) = b. Fix a Sylow p - group P of G and define m := ∏_{h∈P} h(a) ∈ A^{P}. Then θ(m) = ∏_{h∈P} (hθ(a)) = b^{|P |}. Hence \frac{1}{|G:P |}θ(t_{P}^{G}(m)) = \frac{1}{|G:P |}t_{P}^{G}(θ(m)) = b^{|P |} ∈ θi_{G}(A^{G}) = γ(C). ∘

Remark 5.2. Notice that we have not yet used the linear structure of V̂, so all the statements above remain valid, if we start with an arbitrary affine 𝔽 - variety X̂ instead of the linear variety V̂.

We identify A and A^{G} with A ⊗ 1 ⊆ Â and A^{G} ⊗ 1 ⊆ Â^{G} respectively and define

Î^{H,U} := I(î_{H}^{∗}(V̂^{U})) = {f ∈ Â^{H} | f(v^{H}) = 0, ∀ v^{H} ∈ î_{H}^{∗}(V̂^{U})}

i.e. the (prime) ideal in Â^{H} corresponding to the irreducible and closed subset î_{H}^{∗}(V̂^{U}) ⊆ V̂ /H; hence Â^{H}/Î^{H,U} ≅ O(î_{H}^{∗}(V̂^{U})). We also define I^{H,U} to be the prime ideal Î^{H,U} ∩A◁A. In particular we have I^{H,1} = 0.

Lemma 5.3. For subgroups U, H ≤ X ≤ G we have I^{X,U} = I^{H,U} ∩ A^{X} and 𝔽[V^{H}] is a polynomial ring which is a finite integral and purely inseperable extension of A^{H}/I^{H,H}.

Proof: Notice that I^{X,U} = A^{X} ∩ I^{1,U} = A^{X} ∩ A^{H} ∩ I^{1,U} = A^{X} ∩ I^{H,U}. Now let θ^{∗} : Ŷ := V̂^{H} ↪ V̂ be the canonical injection. Clearly Ŷ = Ŷ /H and we have B̂ = B̂^{H} ≅ Â/Î^{1,H} in our notation above. Since Ŷ is a linear 𝔽 - variety, both algebras B = 𝔽[V^{H}] and B̂ ≅ B⊗_{𝔽} \overline{𝔽} are polynomial subalgebras of A and Â respectively. We also have C ≅ A^{H}/(A^{H} ∩ I^{1,H}) = A^{H}/I^{H,H}. From 5.1 we know that B = B^{H} is a finite, integral and purely inseparable extension of C. ∘

The following example shows that A^{H}/I^{H,H} will in general be a proper subring of 𝔽[V^{H}].

Example 5.4. We use the notation of section 3. Let V = 𝔽^{Ω} ≅ V^{∗} be a permutation module and P a p - group. Then V^{P} = ⟨v^{+} | v ∈ V ⟩_{𝔽} is the span of the orbit sums in V. If δ_{ω} ∈ V^{∗} denotes the element dual to e_{ω}, then I^{1,P} = (δ_{ω} − δ_{g(ω)} | g ∈ P). Hence 𝔽[V^{P}] = 𝔽[V]/I^{1,P} ≅ 𝔽[δ_{[ω]} | [ω] ∈ Ω/P }. On the other hand I^{P,P} = 𝔽[V]_{<}^{P} (see 5.10), hence

𝔽[V]^{P} /I^{P,P} ≅ 𝔽[π(ω^{P}) | ω^{P} ∈ Ω/P]

and its image in 𝔽[V^{P}] coincides with the isomorphic but proper subalgebra 𝔽[δ_{[ω]}^{|ω^{P} |} | [ω] ∈ Ω/P }.

In the example above, 𝔽[V]^{P} /I^{P,P} and 𝔽[V^{P}] are still abstractly isomorphic polynomial rings. As the next result shows, this is always true for finite groups whose action on V is defined over the prime field. This follows from results of Wilkerson on inseparable extensions of algebras over the Steenrod algebra [23]. At this point I like to thank David Wehlau and Jim Shank for pointing out a gap in an earlier version of 5.6 and also for putting my attention to the paper [23].

Let 𝔽_{p} denote the prime field of characteristic p, W be a finite - dimensional 𝔽_{p} - space and A_{p} the mod(p) - Steenrod algebra. The following is part of Theorem II in [23]:

Theorem 5.5. (Wilkerson) Let R^{∗} be a sub - A_{p} - algebra of 𝔽_{p}[W] such that the graded field of quotients F 𝔽_{p}[W] is algebraic over F R^{∗}. Define a filtration U_{i}(R^{∗}) of W by the rule that v ∈ U_{i}(R^{∗}) if and only if v^{p^{i}} is separable over F R^{∗}. Let D^{∗}(R^{∗}) be the ‘diagonal’ subalgebra of F 𝔽_{p}[W] generated by U_{i}(R^{∗})^{p^{i}}.

Then D^{∗}(R^{∗}) is separable over R^{∗} and D^{∗}(R^{∗}) is isomorphic to

⊗_{i}(𝔽_{p}[U_{i}(R^{∗})/U_{i−1}(R^{∗})])^{p^{i}}

as A_{p} - algebras.

Proposition 5.6. Let G be a finite group, and V ∈ 𝔽G−mod such that the G - action is defined over the prime field. Then 𝔽[V]^{G}/I^{G,G} and 𝔽[V^{G}] are abstracly isomorphic polynomial rings.

Proof: By flat base change we can assume that 𝔽 = 𝔽_{p} is the prime field. It is well known that the polynomial ring 𝔽[V], as well as its subring 𝔽[V]^{G} are unstable A_{p} - modules (e.g. see [19]). The ideal I^{1,G} is generated by the elements in (V^{G})^{⊥}, so it is A_{p} - stable as well as I^{G,G} = I^{1,G} ∩ A^{G}. Hence 𝔽[V]^{G}/I^{G,G} is a sub - A_{p} - algebra of 𝔽[V^{G}] and, by 5.3, the extension is purely inseparable. Using the notation of 5.5, and the extension of this result to A_{p}, we see that 𝔽[V]^{G} = D^{∗}(𝔽[V]^{G}) is a polynomial ring. Since its transcendence degree is the same as that of 𝔽[V^{G}], both rings are isomorphic. ∘

Remark 5.7. In [23] only the case 𝔽 = 𝔽_{p} is considered. Replacing the mod(p) - Steenrod algebra by the Steenrod algebra A_{q} with q = p^{s} as in [19] chapter 11, we expect 5.5 and 5.6 to hold as well for 𝔽_{q} instead of 𝔽_{p} but leave the corresponding details to a subsequent paper.

For any ideal J ◁ A^{G} let V(J) := {v^{G} ∈ V/G | f(v) = 0, ∀f ∈ J }. In [22] it has been shown that V(A_{\{e\}}^{G})) = \{v^{G} | p divides |G_{v}|\} and that \sqrt{A_{\{e\}}^{G}} = P_{G} ∩ A^{G}, where P_{G} is the ideal in A, generated by (g − 1)V^{∗} for all elements g ∈ G of order p. (Part of this result is also contained in [8]). We are, in some sense, looking at the other extreme, namely the subvariety of orbits whose stabilisers contain a Sylow p - group of G. Obviously we have

i^{∗}(V^{P}) = {v^{G} | |P | ≤ |G_{v}|} = {v^{G} ∈ V/G | p does not divide |v^{G}|},

where P denotes a fixed Sylow p - group of G. In order to make the corresponding ideal I^{G,P} computable, we like to describe it in terms of relative traces. To do this, we first go the other way round and give a geometrical description of relative trace ideals. Notice that, by Hilbert’s Nullstellensatz, we have for any ideals I, J ◁ A = A^{\{e\}}, \sqrt{I} = \sqrt{J} if and only if V(I) = V(J). Using the orbit map i_{G}^{∗} one can easily see that the same is true for ideals in A^{G}. We need the following

Lemma 5.8. (i): For U ≤ G, f ∈ A^{U} and v ∈ V^{U} we have

t_{U}^{G}(f)(v) = f(t_{U}^{G}(v)).

(ii): For every v ∈ V and every Sylow p - group Q of the stabilizer G_{v}, there is an element f ∈ A^{Q} with t_{Q}^{G}(f)(v) ≠ 0.

Proof: (i): We have t_{U}^{G}(f)(v) = ∑_{g∈[G:U]}(gf)(v), where [G : U] denotes an arbitrary system of representatives for the right cosets of U in G (since f ∈ A^{U} the value t_{U}^{G}(f) is independent of the choice of coset representatives). Since in this situation we have to consider the G - action on V and on V^{∗} simultaneously we take some care in choosing a system which is appropriate for our needs: it is well known that for any subgroup U ≤ G there always exists a system R of coset representatives for U in G which represents at the same time left - and right U - cosets, i.e. G = ⊎_{x∈R}xU = ⊎_{x∈R}Ux = ⊎_{x∈R}x^{−1}U. This is a combinatorical consequence of Ph. Hall’s ‘marriage theorem’.

Now we get t_{U}^{G}(f)(v) = ∑_{x∈R}(xf)(v) = ∑_{x∈R}(x^{−1}f)(v) (because of the special property of R ) = ∑_{x∈R} f(xv) = f(t_{U}^{G}(v)).

(ii): This can be seen in a similar way as in the proof of Theorem 2.1. of [22]: let A = 𝔽[X_{1} …, X_{n}], and for V ∋ w ≠ v define i(w) := min {i | X_{i}(w − v) ≠ 0} and f̃ := ∏_{w∈v^{G}, w≠v}(X_{i(w)} − X_{i(w)}(w)). Then f̃(w) = 0 for all w ∈ v^{G}\{v} and f̃(v) ≠ 0. Now let f := ∏_{q∈Q} q(f̃) ∈ A^{Q}; then f(v) = ∏_{q∈Q} f̃(q^{−1}v) = (f̃(v))^{|Q|} ≠ 0 and for every w = gv ≠ v: f(w) = f̃(w) ∏_{q∈Q\{1}} f̃(q^{−1}w) = 0. Hence t_{Q}^{G}(f)(v) = f(t_{Q}^{G}(v)) = f(t_{G_{v}}^{G}(t_{Q}^{G_{v}}(v))) = |G_{v} : Q| ∑_{g∈[G:G_{v}]} f(gv) = |G_{v} : Q| · f(v) ≠ 0. ∘

Theorem 5.9. Let 𝔽 be a field of characteristic p > 0 and A := 𝔽[V] for V ∈ 𝔽G − mod. Let X be a set of subgroups of G, P ∈ Syl_{p}(G), the set of Sylow p - subgroups of G and X_{P} := {U ≤ P ∩ ^{g}X | X ∈ X , g ∈ G}. Then we have for A_{X}^{G} := ∑_{X∈X} t_{X}^{G}(A^{X}):

(i):

A_{X}^{G} = A_{X_{P}}^{G} .

(ii):

V(A_{X}^{G}) = {v^{G} | P_{v} ∈ Syl_{p}(G_{v}), P_{v} ∩ ^{g}X < P_{v}, ∀X ∈ X , g ∈ G} =
{v^{G} | p | |G_{v} : X_{v}|, ∀X ∈ X_{P} } = i_{G}^{∗}(∪_{Q≤P, minimal with Q∉X_{P}} V^{Q}).

(iii):

\sqrt{A_{X}^{G}} = ∩_{Q≤P, minimal with Q∉X_{P}} I^{G,Q} =
(∩_{Q≤P, minimal with Q∉X_{P}} ((g − 1)V^{∗} | g ∈ Q)A) ∩ A^{G}.

Proof: (i): Let X ∈ X and a ∈ A^{X}, then the first Mackey formula of section 3 yields: t_{X}^{G}(a) = ∑_{g∈[P:G:X]} t_{P ∩ ^{g}X}^{P}(ga) ∈ A^{P} . Hence

|G : P |t_{X}^{G}(a) = t_{P}^{G}(t_{X}^{G}(a)) = ∑_{g∈[P:G:X]} t_{P ∩ ^{g}X}^{G}(gb) ∈ A_{X_{P}}^{G} .

This shows ‘⊆’. The other inclusion follows immediately from transitivity and conjugacy relations of trace maps.

(ii): The second equality is clear. So we only have to show the first and last ones. Let RHS denote the right hand side of the first equation. To show ‘⊆’ we assume v ∈ V such that v^{G} ∉ RHS. By Sylow’s theorem, all Sylow p - groups of G are conjugate, so any G - orbit v^{G} can be represented by a suitable v ∈ V, such that Q := P_{v} is a Sylow p - group of G_{v}. Then there is X ∈ X and g ∈ G with Q = Q ∩ ^{g}X. From 5.8 (ii) there is f ∈ A^{Q} with 0 ≠ t_{Q}^{G}(f)(v) = t_{^{g}X}^{G}(t_{Q}^{^{g}X}(f))(v). But for ϕ := t_{Q}^{^{g}X}(f) ∈ A^{^{g}X} we have g^{−1}ϕ = t_{^{g^{−1}} Q}^{X}(^{g^{−1}}f) ∈ A^{X} and t_{X}^{G}(^{g^{−1}}ϕ) = t_{^{g}X}^{G}(ϕ) = t_{Q}^{G}(f) ∈ A_{X}^{G} and v ∉ V(A_{X}^{G}).

To prove ‘⊇’ let v^{G} ∈ RHS, X ∈ X and f ∈ A_{X}^{G}. Again we can assume that Q := P_{v} ∈ Syl_{p}(G_{v}) and that f = t_{X}^{G}(b) with b ∈ A^{X}. Then from the first Mackey formula we get f = ∑_{g∈[Q:G:X]} t_{Q∩ ^{g}X}^{Q}(gb) and t_{Q∩ ^{g}X}^{Q}(gb)(v) = (gb)(t_{Q∩ ^{g}X}^{Q}(v)) = \frac{|Q|}{|Q∩ ^{g}X|}(gb)(v) = 0. Hence v ∈ V(A_{X}^{G}).

To prove the third equality notice that X_{P} is closed under taking subgroups, so V(A_{X}^{G}) = ∪_{Q≤P, Q∉X_{P}} {v^{G} | Q ∈ Syl_{p}(G_{v})} = ∪_{Q≤P, Q∉X_{P}} i_{G}^{∗}(V^{Q}) = ∪_{Q≤P, minimal with Q∉X_{P}} i_{G}^{∗}(V^{Q}).

(iii): The first equality follows from (ii) and Hilbert’s Nullstellensatz. To prove the second statement, notice that V^{Q} is the socle soc(V_{|Q}) of V restricted to Q and it is well known that (soc(V_{|Q}))^{⊥} = Rad(𝔽[Q]) · V_{|Q}^{∗}, where Rad denotes the Jacobson radical. In case of p - groups in characteristic p > 0, the Jacobson radical coincides with the augmentation ideal 𝔽[Q](h − 1 | h ∈ Q), hence (soc(V_{|Q}))^{⊥} = ⟨(h − 1)V^{∗} | h ∈ Q ⟩_{𝔽}. So we conclude that Î(V^{Q}) = Î^{1,Q} = Â((h − 1)V^{∗} | h ∈ Q). Now the rest follows by intersection with A and A^{G}. ∘

The following consequences include a characterization of the ideal I^{G,P} in terms of relative traces:

Proposition 5.10. Let P be a fixed Sylow p - group of G. Then the following hold:

(i): If X = {H} with a normal subgroup H ◁ G, then X_{P} = {U ≤ H ∩ P } and V(A_{H}^{G}) = i_{G}^{∗}(∪_{x∈P \(H∩P), x^{p}∈H∩P} V^{x}) and

\sqrt{A_{H}^{G}} = (∩_{x∈P \(H∩P), x^{p}∈H∩P} (x − 1)V^{∗}A) ∩ A^{G}.

In particular we obtain Feshbach’s theorem (see [8]):

\sqrt{A_{1}^{G}} = (∩_{x∈P \\{e\}, x^{p}=1} (x − 1)V^{∗}A) ∩ A^{G}.

(ii): For X := {Q | Q < P } and A_{X}^{G} = A_{<P}^{G} we have:

V(A_{<P}^{G}) = V(I^{G,P}) = {v^{G} ∈ V/G | p does not divide |v^{G}|} = i_{G}^{∗}(V^{P}).
\sqrt{A_{<P}^{G}} = I^{G,P} = A^{G} ∩ (hf − f | f ∈ A, h ∈ P)A.

In particular height A_{<P}^{G} = codim i^{∗}(V̂^{P}) = codim_{𝔽} V^{P}.

(iii): For every family X with X_{P} consisting of proper subgroups of P we have

height A_{X}^{G} ≤ codim_{𝔽} V^{P} .

Proof: Most of these statements are immediate consequences of 5.9.

In (i) let Q = ⟨q_{1}, …, q_{s}⟩ be a subgroup of P minimal subject to the condition not lying in X_{P}, i.e. not being a subgroup of H ∩ P; suppose moreover that the q_{i} form a minimal generating set. Then some q_{i} ∉ H ∩ P, hence Q = ⟨q_{i}⟩ and q_{i}^{p} ∈ H ∩ P.

(ii) is clear by 5.9 and the fact that height A_{<P}^{G} = height \sqrt{A_{<P}^{G}} = height I^{G,P} = codim V(I^{G,P}) = codim i_{G}^{∗}(V^{P}) = codim_{𝔽}(V^{P}).

(iii) follows from the fact A_{X}^{G} = A_{X_{P}}^{G} ≤ A_{<P}^{G}. ∘

If V is projective relative to a proper subgroup of a Sylow p - group of G, and hence with respect to some Q < P, we know from 3.1, that A_{<P}^{G} is ‘large’ in the sense that it contains all invariants of degree not divisible by p.

If G has a cyclic Sylow p - group P, then all trace ideals can be described in an easy way: notice that in this case, the lattice of subgroups of P is linearly ordered, so, for any family X there is a unique subgroup U ≤ P such that X_{P} = {X ≤ U}. If U < P then there is also a unique subgroup Q ≤ P containing U as a maximal subgroup. So we get:

Corollary 5.11. Let P ∈ Syl_{p}(G) be cyclic, then every trace ideal A_{X}^{G} is either A^{G} (if P ∈ X_{P}), or of the form A_{<Q}^{G} for a subgroup Q ≤ P. Moreover we have

V(A_{<Q}^{G}) = i_{G}^{∗}(V^{Q})

and \sqrt{A_{<Q}^{G}} is the prime ideal I^{G,Q} = ((V^{Q})^{⊥}A) ∩ A^{G} of height codim_{𝔽} V^{Q}.

In the following we are going to construct certain Cohen - Macaulay quotients of modular invariant rings. Recall that a (positively) graded 𝔽 - algebra R is Cohen - Macaulay (CM) if and only if it is free as a module over the subalgebra generated by a homogeneous system of parameters (h.s.o.p.) of R. In that case R is free over any such h.s.o.p..

Definition 5.12. (see [4] pg.270) Let S ≤ R be a subring; a Reynolds operator for the pair (R, S) is a split epimorphism of S modules π : _{S}R → S, i.e. with a right inverse ρ : S ↪ R. For example t_{P}^{G} is a Reynolds operator for the pair (A^{P} , A^{G}).

Due to a theorem of Eagon and Hochster ([13]) it is well known that A^{G} is always a Cohen - Macaulay ring if char 𝔽 is coprime to |G|. Using the Reynolds operator t_{P}^{G} this has been generalized by Campbell, Hughes and Pollack, who proved that if char 𝔽 = p > 0, the invariant ring A^{G} is still Cohen - Macaulay provided A^{P} is ([5]). The next proposition will be a further generalization thereof.

Crucial for all these results is the following theorem of Hochster-Eagon (see [4] 6.4.5):

Theorem 5.13. Suppose S is a subring of R such that R is integral over S and π is a Reynolds operator for (R, S). Then, if R is CM, so is S.

First notice that if I ◁ A^{H} is an ideal then, as A^{G} ≤ A^{H} we have t_{H}^{G}(I)A^{G} = t_{H}^{G}(IA^{G}) ⊆ t_{H}^{G}(I); hence t_{H}^{G}(I) is an ideal of A^{G}.

Proposition 5.14. Let 𝔽 be a field of characteristic p and P a Sylow p - subgroup of G. Suppose that I ◁ A^{P} is an ideal satisfying

t_{P ∩ ^{g}P}^{P} (gI) ⊆ I (∗)

for all g ∈ G. Then t_{P}^{G}(I) = I ∩ A^{G} ◁ A^{G} and t_{P}^{G} induces a Reynolds operator for the pair (A^{P} /I, A^{G}/(I ∩ A^{G})). In particular if A^{P} /I is Cohen - Macaulay so is A^{G}/(I ∩ A^{G}).

Proof: Let a ∈ I∩A^{G}, then |G : P |a = t_{P}^{G}(a), hence a ∈ t_{P}^{G}(I). Let b ∈ t_{P}^{G}(I); then, by the Mackey - formula, b = t_{P}^{G}(i) = ∑_{g∈[P:G:P]} t_{P ∩ ^{g}P}^{P} (gi) ⊆ I ∩ A^{G}. Now we have a surjection t_{P}^{G} : A^{P} /I → A^{G}/t_{P}^{G}(I) = A^{G}/(I∩A^{G}) and a canonical embedding ρ : A^{G}/I∩A^{G} ↪ A^{P} /I; both are homomorphisms of A^{G}/(I ∩ A^{G}) - modules with t_{P}^{G} ∘ ρ = |G : P | · Id invertible. Hence t_{P}^{G} induces a Reynolds operator for the pair (A^{P} /I, (A^{G}/I ∩ A^{G})). Clearly A^{P} /I is integral over A^{G}/(I ∩ A^{G}) and the claim follows from 5.13. ∘

Remark 5.15. (i) Notice that for I = 0 we recover the above mentioned result of Campbell, Hughes and Pollack.

(ii) The condition (∗) above is satisfied whenever gI = I for all g in the normalizer N_{G}(P) and A_{<}^{P} ≤ I.

(iii) Let Y be a family of subgroups of P ∈ Syl_{p}(G) such that Y ∈ Y ⇒ P ∩ ^{g}Y ∈ Y for every g ∈ G, then condition (∗) of 5.14 is satisfied for I = A_{Y}^{P}. Indeed, for Y ∈ Y, a ∈ A^{Y} we have t_{P ∩ ^{g}P}^{P} (gt_{Y}^{P} (a)) = t_{P ∩ ^{g}P}^{P} (t_{^{g}Y}^{^{g}P} (ga)) = ∑_{h∈[P ∩ ^{g}P:^{g}P:^{g}Y]} t_{P ∩ ^{g}P}^{P} (t_{P ∩ ^{g}P ∩ ^{hg}Y}^{P ∩ ^{g}P} (hga)) =

∑_{h∈[P ∩ ^{g}P:^{g}P:^{g}Y]} t_{P ∩ ^{g}P ∩ ^{hg}Y}^{P} (hga) = ∑_{h∈[P ∩ ^{g}P:^{g}P:^{g}Y]} t_{P ∩ ^{hg}Y}^{P} (t_{P ∩ ^{g}P ∩ ^{hg}Y}^{P ∩ ^{hg}P} (hga)) ∈ A_{Y}^{P} .

Now we can prove the following theorem on Cohen - Macaulay quotients of modular invariant rings, which generalizes the theorem of Hochster and Eagon on the CM - property of nonmodular invariant rings:

Theorem 5.16. Let char 𝔽 = p and P a Sylow p - subgroup of G. Suppose that the action of P on V is defined over the prime field. Then I^{G,P} = t_{P}^{G}(I^{P,P}) = I^{P,P} ∩A^{G} and A^{G}/I^{G,P} is a Cohen - Macaulay ring.

Proof: From 5.10 we see that A_{<}^{P} ≤ I^{P,P}. Clearly for any g ∈ N_{G}(P) we have gI_{P}^{P} = I_{P}^{P}, so I^{P,P} satisifies the assumption of 5.14. By 5.6 A^{P} /I_{P}^{P} is CM, so the result follows from 5.14. Notice that if P = 1 we get I_{P}^{P} = I(V^{∗}) = (0) ◁ A and A^{G} is CM. ∘

We consider the ideals A_{<P}^{G} and I^{G,P} as links between the 𝔽G - module structure of V and the ideal structure of 𝔽[V]^{G} and therefore propose to study conditions on V, implying the equality

A_{<P}^{G} = I^{G,P} (∗),

which, by 5.10, is equivalent to the primality of A_{<P}^{G}. From the point of view of constructive modular invariant theory the analysis of 𝔽[V]^{G} might benefit in this case from the fact that A_{<P}^{G} is constructible using relative trace maps and the quotient can be handled by methods, developed for Cohen - Macaulay rings.

By the results above, to prove equality (∗) it suffices to consider the case where G is a p - group: in fact A_{<}^{P} = I^{P,P} implies that I^{G,P} = t_{P}^{G}(I^{P,P}) = t_{P}^{G}(A_{<}^{P}) = A_{<P}^{G}. But the following example shows, that for p - groups in general, A_{<P}^{G} can be stricly smaller than

\sqrt{A_{<P}^{G}}:

Example 5.17. (Kathrin Kuhnigk [15]) Let p > 2, 𝔽 of characteristic p and G = ⟨g⟩ ≅ ℤ_{p}, acting on 𝔽[X, Y] by g^{i}(X) = X + iY and g^{i}(Y) = Y for every i. Then, since t_{1}^{G}(X^{p}) = t_{1}^{G}(X)^{p}, it is clear that A_{1}^{G} is generated by t_{1}^{G}(X^{i}) for i = 1, …, p − 1. We get t_{1}^{G}(X^{i}) =

∑_{ℓ=0}^{p−1}(X + ℓY)^{i} = ∑_{ℓ=0}^{p−1} ∑_{j=0}^{i} ( \begin{matrix}i \\ j \end{matrix} ) X^{i−j}Y^{j}ℓ^{j} = ∑_{j=0}^{i} ( \begin{matrix}i \\ j \end{matrix} ) X^{i−j}Y^{j} (∑_{ℓ=0}^{p−1} ℓ^{j}). When-

ever 0 < j < p − 1, the subset {ℓ^{j} | 0 < ℓ < p} of 𝔽_{p}^{∗} is a nontrivial multiplicative subgroup S. If S^{+} is the sum of its elements, then (s − 1)S^{+} = S^{+} for 1 ≠ s ∈ S, hence S^{+} = 0 = ∑_{ℓ=0}^{p−1} ℓ^{j}. This last equation also holds for j = 0, while for j = p − 1 we get ∑_{ℓ=0}^{p−1} ℓ^{j} = −1 ∈ 𝔽. Hence t_{1}^{G}(X^{i}) = −Y^{p−1} for i = p − 1 and zero otherwise. We conclude that

A_{<P}^{G} = (Y^{p−1}) < (Y) = \sqrt{A_{<P}^{G}} ,

We have already seen in 4.3 that all relative trace ideals A_{X}^{G} are radical ideals if V is a permutation representation. In the following theorem we will prove that this is still true for direct summands of permutation modules, in particular for projective modules.

Theorem 5.18. If V ∈ _{F G}mod is a direct summand of a permutation module, then all (relative) trace ideals A_{X}^{G} are radical ideals. In particular if P ∈ Syl_{p}(G), then

A_{<P}^{G} = I^{G,P} .

is a prime ideal of height codim V^{P} and the quotient A^{G}/A_{<P}^{G} is a Cohen - Macaulay ring of Krull dimension dim V^{P}.

If moreover P is cyclic, then each trace ideal A_{X}^{G} with P ∉ X_{P} is a prime ideal of the form I^{G,Q} with height codim_{𝔽} V^{Q} and Q ≤ P minimal subject to ^{g}X ∩ P < Q for all X ∈ X and g ∈ G.

Proof: From 5.9 and the remark 5.15 (iii) we see that

A_{X}^{G} = A_{X_{P}}^{G} = t_{P}^{G}(A_{X_{P}}^{P}) = A_{X_{P}}^{P} ∩ A^{G}.

So, together with ‘lying over’ and ‘going up and down’ for prime ideals and integral extensions, we can restrict to the Sylow p - group P, and hence assume right away that G is a p - group. For p - groups in characteristic p any transitive permutation module is indecomposable and so, by the Krull - Schmidt theorem, any direct summand of a permutation module is again a permutation module (see [7] Chapter IX, 3.4.). Hence we can assume that V is a permutation module and in this case we know the results already from 4.3 and 4.4.

The last statement is an immediate consequence of 5.11. ∘

Remark 5.19. (i) Notice that the proof of 5.18 is independent of 5.6 and hence of the result 5.5 in [23].

(ii) The last statement of 5.18 generalizes a result of Shank and Wehlau [22] Corollary 6.2.

(iii) Let V be indecomposable with vertex Q ≤ P, then V is a direct summand of W^{↑G} where W can be chosen to be an indecomposable summand of V_{|Q} and then is unique up to isomorphism and conjugation. In modular representation theory such an 𝔽Q - module W is called a source of V. It is well known (and proved by ‘Mackey decomposition’) that V is a direct summand of a permutation module if and only if the trivial module 𝔽_{Q} is a source of V (see e.g. [16], chapter II, 12.5). Thus we have shown that A_{<P}^{G} = I^{G,P} for ‘trivial source modules’. In particular if Q < P (which is equivalent to the fact that the trivial 𝔽P - module is not a direct summand of V_{P}), then A^{G} is generated by A_{<P}^{G} together with preimages of generators for the CM - algebra A^{G}/A_{<P}^{G}; these latter ones can all be chosen to be of degree divisible by p.

Acknowledgements: I thank Jacques Alev (Reims), David Green (Wuppertal) and Gregor Kemper (Heidelberg) for stimulating and fruitful discussions during the process of writing this paper. I also thank David Wehlau and Jim Shank (Kingston, Ontario) for their very valuable and constructive criticism.

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Institute for Experimental Mathematics, University of Essen, Ellernstr. 29, 45326 Essen, Germany

E-mail address: peter @ exp-math.uni-essen.de