NOETHER NUMBERS FOR SUBREPRESENTATIONS OF CYCLIC GROUPS OF PRIME ORDER

R. JAMES SHANK AND DAVID L. WEHLAU

Abstract. Let W be a finite dimensional Z/p-module over a field, k, of characteristic p. The maximum degree of an indecomposable element of the algebra of invariants, k[W]^{Z/p}, is called the Noether number of the representation and is denoted by β(W). A lower bound for β(W) is derived and it is shown that if U is a Z/p submodule of W then β(U) ≤ β(W). A set of generators, in fact a SAGBI basis, is constructed for k[V_{2} ⊕ V_{3}]^{Z/p} where V_{n} is the indecomposable Z/p-module of dimension n.

1. Introduction

Let V be a finite dimensional vector space over a field k. We choose a basis, {x_{1}, …, x_{n}}, for the dual, V^{∗}, of V. Consider a finite subgroup G of GL(V). The action of G on V induces an action on V^{∗} which extends to an action by algebra automorphisms on the symmetric algebra of V^{∗}, k[V] = k[x_{1}, …, x_{n}]. The ring of invariants of G is the subring of k[V] given by

k[V]^{G} := {f ∈ k[V] | g · f = f for all g ∈ G}.

We say that the representation of G on V is modular if the characteristic of k divides the order of G. We say the representation is non-modular if |G| is invertible in k. For an introduction to the invariant theory of finite groups we recommend [2] or [22].

Suppose that R = ⊕_{i=0}^{∞}R_{i} is a finitely generated graded algebra. Let R_{+} denote the augmentation ideal of R, that is, the ideal generated by the homogeneous elements of positive degree. We call an element of R decomposable if it lies in the ideal (R_{+})^{2}. Otherwise we say the element is indecomposable in R. The Noether number of R, β(R), is the least integer d such that the set ⊕_{i=0}^{d}R_{i} generates R as an algebra. In other words, β(R) is the largest degree of a homogeneous indecomposable element of R. If R = k[V]^{G} we shall often write β(V) in place of β(R) if the group G is clear from the context.

1991 Mathematics Subject Classification. 13A50, 20J06.

The research of the second author is supported by grants from ARP and NSERC.

This work was done during a visit of the second author to the University of Kent at Canterbury.

Emmy Noether [18] proved that if the characteristic of k is zero, or if it exceeds |G|, then β(V) ≤ |G|. Recently Fleischmann [14] and Fogarty [12] proved that the same bound holds for the general non-modular case. Much less is known about β(V) in the modular case. Göbel [15] proved that, for any characteristic, if G acts by permuting a basis of V then β(V) ≤ \binom{dim V}{2}. Recently Hughes and Kemper [16] found bounds for β(V) for any modular representation of Z/p. Here we will show that if W is a representation of Z/p in characteristic p and U is a subrepresentation then β(U) ≤ β(W). We will also give a lower bound for β(V) for any modular representation of Z/p and compute a SAGBI basis for k[V_{2} ⊕ V_{3}]^{Z/p} where k is field of characteristic p and V_{n} is the indecomposable Z/p-module of dimension n. The term SAGBI is an acronym for Subalgebra Analog to Gröbner Bases for Ideals and was introduced by L. Robbiano & M. Sweedler [20]. The concept was introduced independently by D. Kapur & K. Madlener [17]. In the final section of the paper we discuss a conjectured value for the Noether number of the regular modular representation of Z/p.

Let R be a finitely generated graded algebra. We will denote by P(R, λ) the Hilbert series of R: P(R, λ) := ∑_{d=0}^{∞}(dim_{k} R_{d})λ^{d}. A sequence of homogeneous elements h_{1}, h_{2}, …, h_{n} in R_{+} is a homogeneous system of parameters if R is a finitely generated module over the subalgebra generated by h_{1}, h_{2}, …, h_{n}. The Krull dimension of R is the length, n, of a homogeneous system of parameters for R. A sequence of homogeneous elements h_{1}, h_{2}, …, h_{k} in R_{+} is regular if, for each i ≤ k, h_{i} is not a zero-divisor on R/(h_{1}, …, h_{i−1})R. The depth of M is the length of the longest regular sequence on M. The depth of a ring is bounded above by its Krull dimension. A ring is Cohen-Macaulay if the depth equals the dimension. For a detailed discussion of depth and dimension see [9].

2. Preliminaries

In this paper we consider the invariant theory of Z/p, the cyclic group of order p, over a field k of characteristic p. We denote by σ a fixed generator of Z/p. In the group ring, k(Z/p), define Δ := σ − 1 and Tr := ∑_{i=1}^{p} σ^{i}. Note that Tr = Δ^{p−1}. Further note that Tr gives a homomorphism, called the transfer, of k[V]^{Z/p}-modules: Tr^{Z/p} : k[V] → k[V]^{Z/p}.

There are exactly p distinct inequivalent indecomposable representations of Z/p, one of each dimension 1, 2, …, p. We will denote the indecomposable representation of Z/p of dimension n by V_{n}. Note that V_{p} is the unique indecomposable projective Z/p-module and V_{1} is the unique simple Z/p-module. We choose a basis, {e_{1}, …, e_{p}}, for V_{p}, with Δe_{1} = 0 and, for i > 1, Δe_{i} = e_{i−1}. The vector space spanned by {e_{1}, …, e_{n}} is a Z/p-submodule isomorphic to V_{n} and we have Z/p-equivariant inclusions: V_{1} ⊂ V_{2} ⊂ ⋯ ⊂ V_{p}. Note that V_{n}^{Z/p} is isomorphic to V_{1}.

For a Z/p-module M, the cohomology of Z/p with coefficients in M is given by

H^{0}(Z/p, M) = kernel(Δ|_{M}) = M^{Z/p},
H^{1}(Z/p, M) = \frac{kernel(Tr |_{M})}{image( Δ|_{M} )},
H^{2}(Z/p, M) = \frac{kernel( Δ|_{M} )}{image(Tr |_{M})},

and, for i > 0, H^{2i+1}(Z/p, M) = H^{1}(Z/p, M) and H^{2i}(Z/p, M) = H^{2}(Z/p, M). A Z/p-module decomposition of M gives a vector space decomposition of H^{∗}(Z/p, M). Thus it is important to understand H^{∗}(Z/p, V_{n}). Using the fact that Tr = Δ^{p−1}, we see that H^{1}(Z/p, V_{p}) = H^{2}(Z/p, V_{p}) = 0 and, for n < p, the element e_{n} represents a non-zero class in the one dimensional vector space H^{1}(Z/p, V_{n}) while e_{1} represents a non-zero class in the one dimensional vector space H^{2}(Z/p, V_{n}). We are particularly interested in H^{1}(Z/p, k[V]). Since the action of k[V]^{Z/p} on k[V] commutes with both Δ and Tr, H^{1}(Z/p, k[V]) is a k[V]^{Z/p}-module. Furthermore, k[V]^{Z/p} is a Noetherian ring and H^{1}(Z/p, k[V]) is a quotient of a submodule of the finitely generated module k[V]. Therefore H^{1}(Z/p, k[V]) is a finitely generated k[V]^{Z/p}-module. We direct the reader to [11] for a detailed discussion of group cohomology.

Every Z/p-module may be written as a direct sum of copies of indecomposable modules. Note that k[W ⊕ V_{1}]^{Z/p} ≅ k[W]^{Z/p} ⊗ k[V_{1}]. Thus to study the ring of invariants of a module V it suffices to consider modules having no summand isomorphic to V_{1}. We will say that a Z/p-module W is reduced if W contains no summands isomorphic to V_{1}.

Consider the vector space of linear functionals V_{n}^{∗}. Since V_{n}^{∗} is an indecomposable Z/p-module, V_{n}^{∗} and V_{n} are isomorphic. We will call an element, z, of V_{n}^{∗} a distinguished variable for V_{n} if z is a generator of the cyclic Z/p-module V_{n}^{∗}. Equivalently z is a distinguished variable if z restricted to V_{n}^{Z/p} is not identically zero. For any distinguished variable z there is a triangular basis, {z, Δz, Δ^{2}z, …, Δ^{n−1}z}, of V_{n}^{∗}. For any f ∈ k[V_{n}], let deg_{z}(f) denote the degree of f as a polynomial in z with coefficients in k[Δz, Δ^{2}z, …, Δ^{n−1}z]. The special property of the distinguished variable z, and the corresponding triangular basis, which we will exploit, is the fact that deg_{z}(σ(f)) = deg_{z}(f). Dual to the inclusion of V_{n} into V_{n+1} we have a Z/p-equivariant surjection V_{n+1}^{∗} → V_{n}^{∗}. Note that this surjection carries a distinguished variable of V_{n+1} to a distinguished variable of V_{n}.

Consider a Z/p-module W. Decompose W into a direct sum of indecomposable Z/p-summands:

W = ⨁_{i=1}^{t} W_{i}

where W_{i} ≅ V_{dim(W_{i})} for all i. For each i choose a distinguished variable z_{i} ∈ W_{i}^{∗} and use the corresponding triangular basis for W_{i}^{∗}. Let N_{i} denote the norm of z_{i}. Thus N_{i} = z_{i} if W_{i} ≅ V_{1} and N_{i} := ∏_{j=1}^{p} σ^{j}(z_{i}), otherwise.

Let f ∈ k[W]^{Z/p}. Since N_{1}, considered as a polynomial in z_{1}, is monic we may divide N_{1} into f to obtain the unique decomposition f = f_{1}N_{1} + r_{1} where the remainder r_{1} has degree at most p−1 in the variable z_{1}. Next we divide r_{1} by N_{2} to obtain a decomposition: f = f_{1}N_{1} + f_{2}N_{2} + r_{2} where deg_{z_{1}}(f_{2}) < p, deg_{z_{1}}(r_{2}) < p and deg_{z_{2}}(r_{2}) < p. Continuing in this manner we obtain a decomposition

f = f_{1}N_{1} + f_{2}N_{2} + … + f_{t}N_{t} + r

where deg_{z_{i}}(f_{j}) < p for all i < j and deg_{z_{i}}(r) < p for all i. Note that r is the normal form of f with respect to the Gröbner basis {N_{1}, N_{2}, …, N_{t}} of the ideal (N_{1}, N_{2}, …, N_{t})k[W]. Furthermore the decomposition f = f_{1}N_{1} + f_{2}N_{2} + … + f_{t}N_{t} + r is a normal decomposition of f with respect to this Gröbner basis. We will call this the norm decomposition of f. Note that the norm decomposition depends upon the choice of the z_{i} but is otherwise unique.

Proposition 2.1. Suppose f ∈ k[W]^{Z/p} and consider its norm decomposition: f = f_{1}N_{1} + f_{2}N_{2} + … + f_{t}N_{t} + r. Then f_{1}, f_{2}, …, f_{t}, r ∈ k[W]^{Z/p}

Proof. Applying σ we have that f = σ(f_{1}) · N_{1} + σ(f_{2}) · N_{2} … + σ(f_{t}) · N_{t} + σ(r). Since deg_{z_{i}}(σ(r)) = deg_{z_{i}}(r) and deg_{z_{i}}(σ(f_{j})) = deg_{z_{i}}(f_{j}) for all i and j, the uniqueness of the norm decomposition shows that σ(r) = r and σ(f_{j}) = f_{j} for all j.

Denote by k[W]^{♯} the ideal of k[W] generated by N_{1}, N_{2}, …, N_{t}. Let k[W]^{♭} := {r ∈ k[W] | deg_{z_{i}}(r) < p for all i = 1, 2, …, t}. Thus k[W]^{♭} is the set of functions f having all coefficients f_{i} = 0 in its norm decomposition. Note that k[W]^{♭} and k[W]^{♯} are both Z/p-stable and we have the decomposition

k[W] = k[W]^{♯} ⊕ k[W]^{♭}.

The ring k[W] has a multi-grading given by the degrees in each W_{i}, that is, induced by k[W] ≅ k[W_{1}] ⊗ k[W_{2}] ⊗ ⋯ ⊗ k[W_{t}]. The action of Z/p preserves this grading and thus k[W]^{Z/p}, k[W]^{♯} and k[W]^{♭} inherit this grading. It is easy to see that

k[W]_{(d_{1},... ,d_{t})}^{♭} ≅ k[W_{1}]_{d_{1}}^{♭} ⊗ k[W_{2}]_{d_{2}}^{♭} ⊗ ⋯ ⊗ k[W_{t}]_{d_{t}}^{♭}.

Furthermore k[V_{n}]_{d}^{♭} is a free Z/p-module for all d ≥ p − n + 1 (see [1] or [16, Lemma 2.10]). Thus k[W]_{(d_{1},... ,d_{t})}^{♭} is free if any d_{i} ≥ p − dim(W_{i}) + 1.

3. Lower bounds

It is well known (see, for example, [5] or [19]) that k[2 V_{2}]^{Z/p} = k[x_{1}, N_{1}, x_{2}, N_{2}, u] where N_{1} = y_{1}^{p} − y_{1}x_{1}^{p−1}, N_{2} = y_{2}^{p} − y_{2}x_{2}^{p−1}, u = x_{2}y_{1} − x_{1}y_{2}, and {x_{1}, y_{1}, x_{2}, y_{2}} is a basis for (2 V_{2})^{∗} with Δy_{1} = x_{1}, Δx_{1} = 0, Δy_{2} = x_{2}, and Δx_{2} = 0. In particular, β(2 V_{2}) = p. The next proposition shows that this is the only reduced decomposable representation of Z/p with such a low value of β.

Proposition 3.1. Suppose that W is a non-zero reduced Z/p-module with W ≅ 2 V_{2}. Then

β(W) ≥ max{p, (p − 1) dim W^{Z/p}}.

Proof. Let t denote dim W^{Z/p}. Note that p ≤ t(p − 1) unless t = 1. Decompose W into a direct sum of indecomposable Z/p-modules:

W = ⨁_{i=1}^{t} W_{i}.

Denote by z_{i} a distinguished variable for W_{i}. Suppose first that t = 1. By hypothesis W_{1} = V_{n} for some n > 1. Thus there is an inclusion of Z/p-modules V_{2} ↪ V_{n} = W_{1} giving rise to a surjection of Z/p-modules, W_{1}^{∗} → V_{2}^{∗}. The surjection takes the distinguished variable z_{1} to a distinguished variable, say y. The induced map k[V_{n}]^{Z/p} → k[V_{2}]^{Z/p} takes N_{1}, the norm of z_{1}, to N = y^{p} − (Δy)^{p−1}y, the norm of y. A decomposition of N_{1} would give a decomposition of N. However, it is well known that k[V_{2}]^{Z/p} = k[Δ(y), N] (see, for example, [5],

[19] or [22, Ch. 5, §6, Example 3]) and thus N is indecomposable. Therefore N_{1} is an indecomposable invariant of degree p and the proposition follows for the case t = 1.

For the remainder of the proof we suppose that t ≥ 2. Define

f := Tr^{Z/p} (z_{1}^{p−1} · z_{2}^{p−1} ⋯ z_{t}^{p−1}) .

We will show that f is an indecomposable invariant and therefore β(W) ≥ deg(f) = t(p − 1).

There is a Z/p-module injection of t V_{2} into W. This induces a surjection ρ : k[W] → k[t V_{2}]. If t ≥ 3 then ρ(f) is an indecomposable invariant in k[t V_{2}]^{Z/p} (see [19, Proposition 0.6, page 31] or [5, Corollary, page 4]). Since ρ induces an algebra map of the rings of invariants, f is indecomposable in k[W]^{Z/p}.

The only remaining case is t = 2. Since W ≅ 2̸ V_{2}, there is a Z/p-module injection of V_{2} ⊕ V_{3} into W. This induces a surjection τ : k[W] → k[V_{2} ⊕ V_{3}]. By Corollary 5.2 below, τ(f) is indecomposable in k[V_{2} ⊕ V_{3}]^{Z/p} and thus f is an indecomposable invariant.

4. The Main Theorem

Suppose that U is a Z/p-submodule of W. Decomposing U and W into indecomposable Z/p-modules gives

W = ⨁_{i=1}^{t} W_{i}

and

U = ⨁_{i=1}^{s} U_{i}

where we label the summands so that dim(W_{i}) ≥ dim(W_{i+1}) and dim(U_{i}) ≥ dim(U_{i+1}).

Lemma 4.1. There exists a Z/p-module monomorphism, ϕ : U → W, satisfying ϕ(U_{i}) ⊆ W_{i} for i = 1, 2, …, s.

Proof. Decomposing U and W gives

W ≅ ⨁_{j=1}^{p} n_{j}V_{j}

and

U ≅ ⨁_{j=1}^{p} m_{j}V_{j}

where n_{j} and m_{j} are non-negative integers. Clearly if ∑_{j=ℓ}^{p} n_{j} ≥ ∑_{j=ℓ}^{p} m_{j} for all ℓ, then ϕ can be constructed by starting with the largest indecomposable summands of U and working down.

Let k be the largest integer such that n_{k} ≠ 0. We prove ∑_{j=ℓ}^{p} n_{j} ≥ ∑_{j=ℓ}^{p} m_{j} for all ℓ by induction on k. If k = 1 then W = W^{Z/p} and the result follows. Suppose k > 1. Observe that U^{Z/p} = U ∩ W^{Z/p} and therefore inclusion induces a monomorphism from U/U^{Z/p} to W/W^{Z/p}. Taking a quotient by the fixed points maps V_{j} to V_{j−1} for j > 1 and V_{1} to the zero space. Thus

W/W^{Z/p} ≅ ⨁_{j=2}^{p} n_{j}V_{j−1}

and

U/U^{Z/p} ≅ ⨁_{j=2}^{p} m_{j}V_{j−1}.

Therefore the induction hypothesis gives ∑_{j=ℓ}^{p} n_{j} ≥ ∑_{j=ℓ}^{p} m_{j} for ℓ > 1. For ℓ = 1, ∑_{j=1}^{p} n_{j} = dim(W^{Z/p}) ≥ dim(U^{Z/p}) = ∑_{j=ℓ}^{p} m_{j}.

Theorem 4.2. Suppose that U is a Z/p-submodule of W. Then β(W) ≥ β(U).

Proof. Decompose W into a direct sum of indecomposable Z/p-modules,

W = ⨁_{i=1}^{t} W_{i}.

Since β is preserved by a Z/p-module isomorphism, we may replace U with the image of U under the monomorphism ϕ given by Lemma 4.1. Having made this replacement we have

U = ⨁_{i=1}^{s} U_{i}

where U_{i} = U ∩ W_{i}.

Note that if U is a summand of W, the result is clear and as a consequence we may assume that U is not contained in any proper summand of W, that is, we may assume s = t. Clearly we may assume that both U and W are reduced Z/pmodules. If U ≅ 2 V_{2} then β(U) = p (see the discussion preceding Proposition 3.1) and thus by Proposition 3.1, β(U) ≤ β(W). Hence we also suppose that U ≅ 2̸ V_{2}.

The inclusion of U into W induces a surjection of algebras ρ : k[W] → k[U]. Let J denote the kernel of ρ. The short exact sequence of Z/p-modules

0 → J →^{γ} k[W] →^{ρ} k[U] → 0

gives rise to a long exact sequence in group cohomology

0 → J^{Z/p} → k[W]^{Z/p} →^{ρ} k[U]^{Z/p} → H^{1}(Z/p, J) →^{γ^{1}} H^{1}(Z/p, k[W]) → · · · .

The maps in the above sequence are all k[W]^{Z/p}-module homomorphisms.

The ring of invariants k[U]^{Z/p} is generated as an algebra by ρ(k[W]^{Z/p}) together with the preimage of kernel(γ^{1}). The assumptions we made above, together with Proposition 3.1, imply β(U) ≥ t(p−1). Therefore, to prove the theorem it suffices to show that the kernel of γ^{1} is generated, as a k[W]^{Z/p}-module, in degrees less than t(p − 1).

For each summand W_{i} choose a distinguished variable, z_{i}, and let N_{i} denote its norm. Define z̃_{i} := ρ(z_{i}) and Ñ_{i} := ρ(N_{i}). Note that z̃_{i} is a distinguished variable for U_{i} and the norm of z̃_{i} is Ñ_{i}. Therefore the map ρ respects the decompositions k[U] = k[U]^{♯} ⊕ k[U]^{♭} and k[W] = k[W]^{♯} ⊕ k[W]^{♭}.

Define J^{♯} := J · k[W]^{♯}. Clearly J^{♯} ⊆ J ∩ k[W]^{♯}. We will now prove that these ideals are in fact equal.

Choose f ∈ J ∩ k[W]^{♯} and let f = f_{1}N_{1} + f_{2}N_{2} + … + f_{t}N_{t} be its norm decomposition. Then 0 = ρ(f) = ρ(f_{1})(Ñ_{1}) + ρ(f_{2})(Ñ_{2}) + … + ρ(f_{t})(Ñ_{t}). For i < j we have deg_{z_{i}}(f_{j}) < p and thus also degz̃_{i}(ρ(f_{j})) < p. By the uniqueness of the norm decomposition of 0 in k[U], this implies that ρ(f_{1}) = ρ(f_{2}) = … = ρ(f_{t}) = 0, that is, f_{1}, f_{2}, …, f_{t} ∈ J. Therefore f ∈ J · k[W]^{♯} and this shows that J ∩ k[W]^{♯} = J · k[W]^{♯}.

We define J^{♭} = J ∩ k[W]^{♭} and obtain J = J^{♯} ⊕ J^{♭}.

Thus our short exact sequence separates into two short exact sequences:

0 → J^{♯} →^{γ^{♯}} k[W]^{♯} →^{ρ^{♯}} k[U]^{♯} → 0

and

0 → J^{♭} →^{γ^{♭}} k[W]^{♭} →^{ρ^{♭}} k[U]^{♭} → 0.

Furthermore γ^{1} separates into the two graded vector space homomorphisms:

(γ^{1})^{♯} : H^{1}(Z/p, J^{♯}) → H^{1}(Z/p, k[W]^{♯})

and

(γ^{1})^{♭} : H^{1}(Z/p, J^{♭}) → H^{1}(Z/p, k[W]^{♭}).

Consider the latter homomorphism first. We know that k[W]_{(d_{1},d_{2},... ,d_{t})}^{♭} is free if some d_{i} ≥ p−dim W_{i}+1. Similarly k[U]_{(d_{1},d_{2},... ,d_{t})}^{♭} is free if some d_{i} ≥ p−dim U_{i}+1. Therefore both k[W]_{(d_{1},d_{2},... ,d_{t})}^{♭} and k[U]_{(d_{1},d_{2},... ,d_{t})}^{♭} are free if some d_{i} ≥ p − 1. In particular, both k[W]_{d}^{♭} and k[U]_{d}^{♭} are free if d ≥ t(p − 1). Therefore, for d ≥ t(p − 1), ρ_{d}^{♭} is a surjection between free modules. Thus the map splits and the kernel, J_{d}^{♭}, is also free. Therefore H^{1}(Z/p, J^{♭})_{d} = 0 for all d ≥ t(p − 1). In particular, kernel(γ^{1}) ∩ H^{1}(Z/p, J^{♭})_{d} = {0} for all d ≥ t(p − 1).

Thus we have reduced to considering the kernel of (γ^{1})^{♯}. Consider a cohomology class [f] ∈ kernel((γ^{1})^{♯}). We will show that [f] is not a k[W]^{Z/p}-module generator of kernel(γ^{1}). We have f ∈ J ∩ k[W]^{♯}. Let f = f_{1}N_{1} + f_{2}N_{2} + … + f_{t}N_{t} be the norm decomposition of f. By the proof that J ∩ k[W]^{♯} = J · k[W]^{♯}, we have that f_{1}, f_{2}, …, f_{t} ∈ J. Since f represents a cohomology class, Tr(f) = 0. Thus Tr(f_{1})N_{1} + Tr(f_{2})N_{2} + … + Tr(f_{t})N_{t} = 0. Since deg_{z_{i}}(Tr(f_{j})) ≤ deg_{z_{i}}(f_{j}) < p for all i < j, this must be the norm decomposition of 0, that is, Tr(f_{1}) = Tr(f_{2}) = ⋯ = Tr(f_{t}) = 0. Therefore, each of f_{1}, f_{2}, …, f_{t} itself represents a cohomology class in H^{1}(Z/p, J).

Now (γ^{1})^{♯}([f]) = 0 means that there exists some h ∈ k[W]^{♯} such that Δ(h) = f. Let h = h_{1}N_{1} + h_{2}N_{2} + … + h_{t}N_{t} be the norm decomposition of h. Then Δ(h) = Δ(h_{1})N_{1} + Δ(h_{2})N_{2} + … + Δ(h_{t})N_{t} where deg_{z_{i}}(Δ(h_{j})) ≤ deg_{z_{i}}(h_{j}) < p for all i < j. Therefore, by the uniqueness of the norm decomposition of f we have Δ(h_{j}) = f_{j} and γ^{1}([f_{j}]) = 0. Thus [f] = [f_{1}]N_{1} + [f_{2}]N_{2} + … + [f_{t}]N_{t} with [f_{j}] ∈ kernel(γ^{1}) for all j = 1, …, t and therefore [f] is not a generator of the k[W]^{Z/p}-module, kernel(γ^{1}).

5. V_{2} ⊕ V_{3}

Choose distinguished variables y_{1} ∈ V_{2}^{∗} and z_{2} ∈ V_{3}^{∗}. Define x_{1} := Δ(y_{1}), y_{2} := Δ(z_{2}) and x_{2} := Δ(y_{2}). Then {x_{1}, y_{1}, x_{2}, y_{2}, z_{2}} is a basis for (V_{2} ⊕ V_{3})^{∗} and elements of k[V_{2} ⊕ V_{3}] are polynomials in these five variables. In k[V_{2} ⊕ V_{3}]^{Z/p} there are five so-called “rational” invariants : x_{1}, x_{2}, u := y_{1}x_{2} − x_{1}y_{2}, d := y_{2}^{2} − 2x_{2}z_{2} − x_{2}y_{2} and w := y_{1}^{2}x_{2} + x_{1}y_{1}x_{2} − 2x_{1}y_{1}y_{2} + 2x_{1}^{2}z_{2}. (These invariants are called rational invariants because they admit prime independent descriptions which represent invariants under a lifting of the Z/p-action to a Z-action on Z[x_{1}, y_{1}, x_{2}, y_{2}, z_{2}].) We also have the norms of the distinguished variables: N_{1} := ∏_{i=1}^{p} σ^{i}(y_{1}) = y_{1}^{p} − x_{1}^{p−1}y_{1} and N_{2} := ∏_{i=1}^{p} σ^{i}(z_{2}).

This section is devoted to proving the following theorem and corollary.

Theorem 5.1. The ring of invariants k[V_{2} ⊕V_{3}]^{Z/p} is generated by x_{1}, x_{2}, d, N_{1}, N_{2}, u, w and the following two families of transfers:

(i) {Tr^{Z/p}(y_{1}^{i}y_{2}z_{2}^{p−1}) | 0 ≤ i ≤ p − 1} and (ii) {Tr^{Z/p}(y_{1}^{i}z_{2}^{ℓ}) | 1 ≤ i ≤ p − 1, p − 1 ≥ ℓ ≥ p − ⌈i/2⌉}.

Furthermore, this generating set is a SAGBI basis for k[V_{2} ⊕ V_{3}]^{Z/p} using the graded reverse lexicographic monomial order with x_{1} < y_{1} < x_{2} < y_{2} < z_{2}.

Corollary 5.2. The invariants Tr^{Z/p}(y_{1}^{p−1}y_{2}z_{2}^{p−1}) and Tr^{Z/p}(y_{1}^{p−1}z_{2}^{p−1}) are indecomposable elements of k[V_{2} ⊕ V_{3}]^{Z/p}. In particular β(V_{2} ⊕ V_{3}) ≥ 2p − 1.

To prove Theorem 5.1 we will use the method which the first author used in [21] to compute k[V_{4}]^{Z/p} and k[V_{5}]^{Z/p}. The method is to construct a subalgebra Q of k[V_{2} ⊕ V_{3}]^{Z/p} and then to show that Q = k[V_{2} ⊕ V_{3}]^{Z/p} by showing that the two algebras have the same Hilbert series.

Our Hilbert series computations make use of the theory of SAGBI bases, a generalization to subalgebras of the theory of Gröbner bases. We direct the reader to [8, Chapter 2] for a detailed discussion of monomial orders. We use the convention that a monomial is a product of variables and that a term is a monomial with a non-zero coefficient. For f ∈ k[V] we use LT(f) to denote the lead term of f and LM(f) to denote the lead monomial of f.

Suppose that Q is a subalgebra of k[V]. Let LT(Q) denote the vector space spanned by the lead terms of elements of Q. Note that LT(Q) is a subalgebra of k[V]. If C is a subset of k[V] then let LM(C) denote the set of lead monomials of elements of C. If C is a subset of Q such that LM(C) generates the algebra LT(Q) then C generates Q and C is called a SAGBI basis for Q. For more on SAGBI bases see [17], [20] or [23, Chapter 11].

Before proving Theorem 5.1 we prove a lemma and its corollary which we need in the proof of Theorem 5.1.

Lemma 5.3. Let H := k[x, y] and let I ⊂ H be an ideal generated by monomials. Let γ_{1}, γ_{2}, …, γ_{s} be monomials which minimally generate I where γ_{1} < γ_{2} < … < γ_{s} with respect to lexicographic order with x < y. Then the Hilbert series of the module I is

P(I, λ) = \frac{∑_{i=1}^{s} λ^{deg(γ_{i})} − ∑_{i=1}^{s−1} λ^{deg(LCM(γ_{i},γ_{i+1}))}}{(1 − λ)^{2}}

where LCM(γ_{i}, γ_{i+1}) denotes the least common multiple of {γ_{i}, γ_{i+1}}.

Proof. Let Γ_{1}, Γ_{2}, …, Γ_{s} and R_{1,2}, R_{2,3}, …, R_{s−1,s} be formal symbols and define r_{i,j} := \frac{LCM(γ_{i},γ_{j})}{γ_{i}}Γ_{i} − \frac{LCM(γ_{i},γ_{j})}{γ_{j}}Γ_{j}.

We claim that the following is a resolution of I by free H-modules:

0 ⟶ ⨁_{i=1}^{s−1} HR_{i,i+1} ⟶^{Ψ_{1}} ⨁_{i=1}^{s} HΓ_{i} ⟶^{Ψ_{0}} I ⟶ 0

with Ψ_{0}(Γ_{i}) = γ_{i} and Ψ_{1}(R_{i,i+1}) = r_{i,i+1}.

The kernel of Ψ_{0} is generated as an H-module by {r_{i,j} | 1 ≤ i < j ≤ s}. For a proof of this fact see [3, Proposition 6.1] or [8, Ch. 2, §9, Proposition 8]. If we write γ_{i} = x^{e_{i}}y^{f_{i}} then the ordering on the γ_{i} implies that f_{i} ≤ f_{i+1} for all i. But then the hypothesis that the γ_{i} minimally generate I implies that f_{1} < f_{2} < ⋯ < f_{s} and e_{1} > e_{2} > ⋯ > e_{s}. Therefore if i < j then LCM(γ_{i}, γ_{j}) = x^{e_{i}}y^{f_{j}}.

Now consider r_{i,j} where j ≥ i + 2. Then

r_{i,j} = \frac{LCM(γ_{i}, γ_{j})}{LCM(γ_{i}, γ_{j−1})}r_{i,j−1} + \frac{LCM(γ_{i}, γ_{j})}{LCM(γ_{j−1}, γ_{j})}r_{j−1,j}

where \frac{LCM(γ_{i},γ_{j})}{LCM(γ_{i},γ_{j−1})}, \frac{LCM(γ_{i},γ_{j})}{LCM(γ_{j−1},γ_{j})} ∈ H. Thus \{r_{1,2}, r_{2,3}, …, r_{s−1,s}\} generates the kernel of Ψ_{0}.

Next consider Ψ_{1}. We have

Ψ_{1} (∑_{i=1}^{s−1} f_{i}R_{i,i+1}) =  (f_{1} \frac{LCM(γ_{1}, γ_{2})}{γ_{1}}) Γ_{1}

+ (f_{2} \frac{LCM(γ_{2}, γ_{3})}{γ_{2}} − f_{1} \frac{LCM(γ_{1}, γ_{2})}{γ_{2}}) Γ_{2}
⋮
+ (f_{s−1} \frac{LCM(γ_{s−1}, γ_{s})}{γ_{s−1}} − f_{s−2} \frac{LCM(γ_{s−2}, γ_{s−1})}{γ_{s−1}}) Γ_{s−1}
− (f_{s−1} \frac{LCM(γ_{s−1}, γ_{s})}{γ_{s}}) Γ_{s}.

Thus if Ψ_{1}(∑_{i=1}^{s−1} f_{i}R_{i,i+1}) = 0 then we must have f_{1} = f_{2} = … = f_{s−1} = 0 and thus Ψ_{1} is injective.

Therefore

P(I, λ) = P(⊕_{i=1}^{s}HΓ_{i}, λ) − P(⊕_{i=1}^{s−1}HR_{i,i+1}, λ)
= P(H, λ) ∑_{i=1}^{s} λ^{deg(γ_{i})} − P(H, λ) ∑_{i=1}^{s−1} λ^{deg(LCM(γ_{i},γ_{i+1}))}

= (∑_{i=1}^{s} λ^{deg(γ_{i})} − ∑_{i=1}^{s−1} λ^{deg(LCM(γ_{i},γ_{i+1}))}) /(1 − λ)^{2}.

The following corollary is a consequence of the lemma and the fact that submodules of rank 1 free modules correspond to ideals.

Corollary 5.4. Let R := k[x, y, a_{1}, a_{2}, ..., a_{n}] and suppose that α generates a free R-module and that I is an ideal in k[x, y] minimally generated by monomials γ_{1} < ⋯ < γ_{s}. Let M be the R-module generated by {αγ_{1}, αγ_{2}, …, αγ_{s}}. Then

P(M, λ) = λ^{deg(α)} (∑_{i=1}^{s} λ^{deg(γ_{i})} − ∑_{i=1}^{s−1} λ^{deg(LCM(γ_{i},γ_{i+1}))}) P(R, λ).

Now we proceed with the proof of Theorem 5.1 We will use the graded reverse lexicographic monomial order with x_{1} < y_{1} < x_{2} < y_{2} < z_{2}. Let C denote the collection of invariants given in the statement of Theorem 5.1 and let Q denote the subalgebra of k[V]^{Z/p} generated by C. We will show that C is a SAGBI basis for Q and that Q = k[V]^{Z/p}.

Let A denote the algebra generated by LM(C). Note that {x_{1}, N_{1}, x_{2}, d, N_{2}} is a homogeneous system of parameters for k[V] as is LM({x_{1}, N_{1}, x_{2}, d, N_{2}}) = {x_{1}, y_{1}^{p}, x_{2}, y_{2}^{2}, z_{2}^{p}}. Let R := k[x_{1}, y_{1}^{p}, x_{2}, y_{2}^{2}, z_{2}^{p}]. Then A is a finite R-module and we will exploit the structure of A as an R-module in order to compute its Hilbert series.

Note LT(u) = x_{2}y_{1} and LT(w) = x_{2}y_{1}^{2}. By [21, Theorem 3.3], we have LM(Tr^{Z/p}(y_{2}z_{2}^{p−1})) = y_{2}^{p} and by [21, Theorem 3.2], LM(Tr^{Z/p}(z_{2}^{ℓ})) = y_{2}^{2ℓ−p+1}x_{2}^{p−ℓ−1}. From these two facts, using a simple generalization of [21, Theorem 3.6], we obtain LM(Tr^{Z/p}(y_{1}^{i}y_{2}z_{2}^{p−1})) = y_{1}^{i}y_{2}^{p} and LM(Tr^{Z/p}(y_{1}^{i}z_{2}^{ℓ})) = y_{1}^{i}y_{2}^{2ℓ−p+1}x_{2}^{p−ℓ−1}.

We define a (Z/p × Z/2)-grading on k[V_{2} ⊕ V_{3}] by declaring that a monomial x_{1}^{i_{1}}y_{1}^{i_{2}}x_{1}^{j_{1}}y_{2}^{j_{2}}z_{2}^{j_{3}} has bi-degree (i_{2}, j_{2}) ∈ (Z/p × Z/2). Let A_{(i,j)} denote the elements of A whose bi-degree is (i, j). Note that the action of R preserves bi-degree and thus since A is generated by monomials, A is a (Z/p × Z/2)-graded R-module. Furthermore, this means that the A_{(i,j)} are themselves R-submodules of A and thus A decomposes as an R-module: A = ⊕_{i=0}^{p−1} ⊕_{j=0}^{1} A_{(i,j)}. Therefore we may compute the Hilbert series of the R-module A by summing the Hilbert series of the individual R-modules, A_{(i,j)}. We will consider each bi-degree (i, j) in turn.

Since A_{(0,0)} is the free R-module generated by the unit, 1, its Hilbert series is P(R, λ).

Similarly for each i ∈ Z/p we see that A_{(i,1)} is the free R-module generated by y_{1}^{i}y_{2}^{p} = LM(Tr^{Z/p}(y_{1}^{i}y_{2}z_{2}^{p−1})). Therefore the Hilbert series of A_{(i,1)} is λ^{p+i}P(R, λ).

For i = 2t with 1 ≤ t ≤ (p − 1)/2 the R-module A_{(2t,0)} is generated by the t + 1 elements: y_{1}^{i}y_{2}^{p−1}, y_{1}^{i}x_{2}y_{2}^{p−3}, y_{1}^{i}x_{2}^{2}y_{2}^{p−5}, …, y_{1}^{i}x_{2}^{t−1}y_{2}^{p−2t+1} and y_{1}^{i}x_{2}^{t}. These are the lead monomials of Tr^{Z/p}(y_{1}^{i}z_{2}^{p−1}), Tr^{Z/p}(y_{1}^{i}z_{2}^{p−2}), Tr^{Z/p}(y_{1}^{i}z_{2}^{p−3}), …, Tr^{Z/p}(y_{1}^{i}z_{2}^{p−t}) and w^{t} respectively. Applying Corollary 5.4 with α = y_{1}^{i}, x = x_{2} and y = y_{2}^{2}, we see that the Hilbert series of A_{(2t,0)} is

P(A_{(2t,0)}, λ) = λ^{i}((λ^{p−1} + λ^{p−2} + … + λ^{p−t} + λ^{t})

−(λ^{p} + λ^{p−1} + … + λ^{p−t+2} + λ^{p−t+1}))P(R, λ)
= λ^{i}(λ^{p−t} + λ^{t} − λ^{p})P(R, λ).

Similarly for i = 2t−1 with 1 ≤ t ≤ (p−1)/2 the R-module A_{(2t−1,0)} is generated by the t + 1 elements: y_{1}^{i}y_{2}^{p−1}, y_{1}^{i}x_{2}y_{2}^{p−3}, y_{1}^{i}x_{2}^{2}y_{2}^{p−5}, …, y_{1}^{i}x_{2}^{t−1}y_{2}^{p−2t+1} and y_{1}^{i}x_{2}^{t}. These are the lead monomials of Tr^{Z/p}(y_{1}^{i}z_{2}^{p−1}), Tr^{Z/p}(y_{1}^{i}z_{2}^{p−2}), Tr^{Z/p}(y_{1}^{i}z_{2}^{p−3}), …, Tr^{Z/p}(y_{1}^{i}z_{2}^{p−t}) and w^{t−1}u respectively. Again by Corollary 5.4, we see that the Hilbert series of A_{(2t−1,0)} is

P(A_{(2t−1,0)}, λ) = λ^{i}(λ^{p−t} + λ^{t} − λ^{p})P(R, λ).

Summing the Hilbert series of all 2p homogeneous components, A_{(i,j)}, we obtain

P(A, λ) = ∑_{i=0}^{p−1} ∑_{j=0}^{1} P(A_{(i,j)}, λ)

= \frac{−λ^{(3p−1)/2} (λ^{3} + 3λ^{2} + 3λ + 1) + 2λ^{p} (λ^{2} + λ + 1) + λ^{2} + 1}{(λ^{3} − 1) (λ − 1)^{2} (λ^{2} − 1) (λ^{p} − 1)^{2}}.

Using the method of Hughes and Kemper [16], one can compute P(k[V_{2} ⊕ V_{3}]^{Z/p}, λ). Gregor Kemper has written a MAGMA script implementing this algorithm and using the output of this script we observe that P(k[V_{2} ⊕ V_{3}]^{Z/p}, λ) = P(A, λ). Since A is a subalgebra of LT(Q), P(A, λ) ≤ P(LT(Q), λ). By [21, Proposition 1.2], P(LT(Q), λ) = P(Q, λ). Furthermore Q ⊆ k[V_{2} ⊕ V_{3}]^{Z/p} implies that P(Q, λ) ≤ P(k[V_{2} ⊕ V_{3}]^{Z/p}, λ). The fact that P(k[V_{2} ⊕ V_{3}]^{Z/p}, λ) = P(A, λ), means that all of these Hilbert series are equal. Thus A = LT(Q), Q = k[V_{2} ⊕ V_{3}]^{Z/p} and C is a SAGBI basis for k[V_{2} ⊕ V_{3}]^{Z/p}. This completes the proof of Theorem 5.1.

Now we show how Corollary 5.2 follows from Theorem 5.1.

Proof. Suppose, by way of contradiction that, Tr^{Z/p}(y_{1}^{p−1}y_{2}z_{2}^{p−1}) = f_{1}h_{1} + f_{2}h_{2} + ⋯ + f_{m}h_{m} with f_{ℓ}, h_{ℓ} ∈ k[V_{2} ⊕ V_{3}]_{+}^{Z/p} for ℓ = 1, 2, …, m. We may assume LM(f_{ℓ}h_{ℓ}) ≥ LM(f_{ℓ+1}h_{ℓ+1}). Clearly, either LM(f_{1}h_{1}) = LM(Tr^{Z/p}(y_{1}^{p−1}y_{2}z_{2}^{p−1})) = y_{1}^{p−1}y_{2}^{p} or LM(f_{1}h_{1}) = LM(f_{2}h_{2}) > y_{1}^{p−1}y_{2}^{p}. The elements in LM(C) of the form y_{1}^{i}y_{2}^{j} are {y_{2}^{2}, y_{1}^{p}, y_{2}^{p}} ∪ {y_{1}^{i}y_{2}^{p−1}, y_{1}^{i}y_{2}^{p} | 1 ≤ i ≤ p − 1}. Thus y_{1}^{p−1}y_{2}^{p} is an indecomposable element of A = LT(k[V_{2} ⊕ V_{3}]^{Z/p}). Therefore LM(f_{1}h_{1}) = LM(f_{2}h_{2}) > y_{1}^{p−1}y_{2}^{p}. Note that k[V_{2}⊕V_{3}] ≅ k[V_{2}]⊗k[V_{3}] is bi-graded, the action of Z/p respects this grading, and all of our generators are homogeneous with respect to this grading. However, there are no monomials with bi-degree (p − 1, p) which are greater than y_{1}^{p−1}y_{2}^{p} and contained in A. Thus Tr^{Z/p}(y_{1}^{p−1}y_{2}z_{2}^{p−1}) is indecomposable.

Similarly, since LM(Tr^{Z/p}(y_{1}^{p−1}z_{2}^{p−1})) = y_{1}^{p−1}y_{2}^{p−1} we see that Tr^{Z/p}(y_{1}^{p−1}z_{2}^{p−1}) is also indecomposable.

By [10] or [6] we know that k[V_{2} ⊕ V_{3}]^{Z/p} is not Cohen–Macaulay. Here we will show explicitly that the partial homogeneous system of parameters x_{1}, x_{2}, d in k[V_{2} ⊕ V_{3}]^{Z/p} is not a regular sequence.

Since LT(Tr^{Z/p}(z_{2}^{p−1})) = −y_{2}^{p−1} = LT(−d^{(p−1)/2}), there exists f ∈ k[V_{3}]^{Z/p} such that d^{(p−1)/2} = − Tr^{Z/p}(z_{2}^{p−1}) + x_{2}f. Thus

ud^{(p−1)/2} + (Tr^{Z/p}(y_{1}z_{2}^{p−1}) − fu)x_{2} = u(x_{2}f − Tr^{Z/p}(z_{2}^{p−1}))
+(Tr^{Z/p}(y_{1}z_{2}^{p−1}) − fu)x_{2}
= Tr^{Z/p}(−uz_{2}^{p−1} + y_{1}z_{2}^{p−1}x_{2})
= Tr^{Z/p}((x_{2}y_{1} − u)z_{2}^{p−1})
= Tr^{Z/p}(x_{1}y_{2}z_{2}^{p−1})
= x_{1} Tr^{Z/p}(y_{2}z_{2}^{p−1}) ∈ (x_{1})k[V_{2} ⊕ V_{3}]^{Z/p}.

Note that {x_{1}, x_{2}} is a Gröbner basis for (x_{1}, x_{2})k[V_{2}⊕V_{3}]^{Z/p} and LM(ud^{(p−3)/2}) = x_{2}y_{1}y_{2}^{p−3}. Therefore, using the SAGBI basis C, ud^{(p−3)/2} ∉ (x_{1}, x_{2})k[V_{2} ⊕ V_{3}]^{Z/p}. Thus x_{1}, x_{2}, d is not a regular sequence in k[V_{2} ⊕ V_{3}]^{Z/p}.

6. The 2p − 3 Conjecture

As a consequence of Theorem 4.2, β(V_{p}) is an upper bound for the Noether number of every indecomposable representation of Z/p. Since V_{p} is a permutation representation, Göbel [15] gives us β(V_{p}) ≤ max\{2, \binom{p}{2}\}. For p > 3, this has been improved to \binom{p−1}{2} + 1 by Fleischmann [13, Proposition 12.3] and \frac{(p+3)(p−1)}{4} + 1 by Hughes and Kemper [16, Corollary 2.15].

Conjecture 6.1. β(V_{p}) = 2p − 3.

Direct calculation of F_{p}[V_{p}]^{Z/p} using MAGMA [4] confirms the conjecture for p ≤ 7. A less direct method, outlined below, confirms the conjecture for the primes 11 and 13.

Let z be a distinguished variable for V_{p} and let N be the norm of z. Let I^{Z/p} denote the image of Tr^{Z/p} : k[V] → k[V]^{Z/p}. Thus I^{Z/p} is an ideal in k[V]^{Z/p}. It is well known that k[V_{p}]^{Z/p}/I^{Z/p} = k[N] which has Noether number p (see, for example, [13, Example 12.1] or [16, Lemma 2.12]). Therefore to prove the conjecture it is sufficient to study indecomposable invariants in I^{Z/p}.

Theorem 6.2. β(V_{p}) ≥ 2p − 3.

Proof. Let y := Δ(z). We will show that f := Tr^{Z/p}(z^{p−1}y^{p−2}) is indecomposable. We use the graded reverse lexicographic monomial order with Δ^{i}(z) > Δ^{i+1}(z). Suppose, by way of contradiction, that f = f_{1}h_{1} + f_{2}h_{2} + ... + f_{m}h_{m} with f_{i}, h_{i} ∈ k[V_{p}]_{+}^{Z/p} and LM(f_{i}h_{i}) ≥ LM(f_{i+1}h_{i+1}). From [21, Theorem 3.3], LM(f) = y^{2p−3}. Thus either LM(f_{1}h_{1}) = y^{2p−3} or LM(f_{1}h_{1}) = LM(f_{2}h_{2}) > y^{2p−3}. In either case LM(f_{1}h_{1}) is a monomial in z and y with degree 2p − 3. Thus LM(f_{1}) = z^{j}y^{k} and LM(h_{1}) = z^{r}y^{s} with j + k + r + s = 2p − 3.

Since we are using a triangular basis, deg_{z}(f) ≤ deg_{z}(z^{p−1}y^{p−2}) = p − 1. Thus f ∈ k[V_{p}]^{♭} and, by the uniqueness of the norm decomposition, we may take f_{1} and h_{1} to be elements of k[V_{p}]^{♭}. Therefore f_{1} and h_{1} lie in I^{Z/p}. Consider the Z/p-equivariant surjection ρ : k[V_{p}] → k[V_{2}] = k[z̃, ỹ] given by ρ(z) = z̃. The map ρ commutes with the transfer. Thus f̃_{1} := ρ(f_{1}) and h̃_{1} := ρ(h_{1}) are in the image of the transfer. Furthermore, it is clear that LM(f̃_{1}) = z̃^{j}ỹ^{k} and LM(h̃_{1}) = z̃^{r}ỹ^{s}. However the image of the transfer in k[V_{2}]^{Z/p} is the principal ideal generated by ỹ^{p−1} (see [7, Corollary 9.7]). Thus k ≥ p − 1 and s ≥ p − 1. Therefore j + k + r + s ≥ 2p − 2 > 2p − 3, giving the required contradiction.

For a finite dimensional representation V of a finite group G, k[V] is a finite module over k[V]^{G}. Define a homomorphism of k[V]^{G}-modules, Tr^{G} : k[V] → k[V]^{G}, by Tr^{G}(f) := ∑_{g∈G} g(f). Applying Tr^{G} to a set of module generators gives a generating set for the image of Tr^{G}, the ideal I^{G}. Let H := k[V]_{+}^{G}k[V] denote the Hilbert ideal of k[V]. A homogeneous basis for the finite dimensional algebra k[V]/H lifts to a set of k[V]^{G}-modules generators for k[V]. Thus a basis element of largest degree in k[V]/H lifts to a module generator of largest degree in k[V]. For a homogeneous ideal J ⊂ k[V] of height dim(V), the quotient, k[V]/J, is a finite dimensional graded algebra. Let td(k[V]/J) denote the top degree of k[V]/J, that is, the largest degree in which k[V]/J is non-zero.

Proposition 6.3. Suppose J ⊂ k[V_{n}] is an ideal of height n. If J is contained in the Hilbert ideal, H = k[V_{n}]_{+}^{G}k[V_{n}], then max{p, td(k[V_{n}]/J)} is an upper bound for β(V_{n}).

Proof. Using the notation of Section 2, recall that k[V_{n}] = (N)k[V_{n}] ⊕ k[V_{n}]^{♭} and that k[V_{n}]_{d}^{♭} is a free Z/p-module if d ≥ p−n+1. Since invariants associated to free Z/p-modules are in the image of the transfer, we conclude that all indecomposable invariants in degrees greater than p are in the image of the transfer. The proposition follows from the fact that td(k[V_{n}]/J) is an upper bound for td(k[V_{n}]/H) which in turn is an upper bound for the degree of an indecomposable in I^{Z/p}.

Proposition 6.3 was used to verify Conjecture 6.1 for the primes 11 and 13. In each case it was possible to construct an ideal J, satisfying the hypotheses of the proposition, with td(k[V_{p}]/J) = 2p − 3. The calculations were performed using MAGMA [4].

Acknowledgements. We would like to thank Gregor Kemper for providing us with a closed form for the Hilbert series, P(k[V_{2} ⊕ V_{3}]^{Z/p}, λ), which we required for our proof of Theorem 5.1. We would also like to thank the referee for several helpful suggestions.

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Institute of Mathematics & Statistics, University of Kent at Canterbury, CT2 7NF, UK

E-mail address: R.J.Shank@ukc.ac.uk

Department of Mathematics & Computer Science, Royal Military College, Kingston, Ontario, Canada K7K 7B4 E-mail address: wehlau@rmc.ca