EQUIVARIANT HILBERT SERIES

FRANK HIMSTEDT AND PETER SYMONDS

Abstract. Draft, 17th October 2007. We consider a finite group acting on a graded module and define an equivariant degree that generalizes the usual non-equivariant degree. The value of this degree is a module for the group, up to a rational multiple. We investigate how this behaves when the module is a ring and apply our results to reprove some results of Kuhn on the cohomology of groups.

1. Introduction

We consider a finitely generated graded module M over a graded ring R that is finitely generated over some base field k and such that R_{0} is finite dimensional over k. We suppose that there is a finite group G that acts on M, preserving the grading and commuting with R.

To this data we associate a formal Laurent series [M] in t in which the coefficient of t^{r} is the homogeneous part M_{r}, considered as a kG-module. The difficulty of the theory depends on whether we wish to keep track of these modules up to isomorphism (i.e. in the Green ring) or only up to composition factors (in the representation ring). We develop both cases.

This series [M] is shown to satisfy a form of the Hilbert-Serre Theorem (in particular it is a rational function, or at least a sum of them in the Green ring case). We define the equivariant degree deg_{G} M to be the coefficient of the leading term when we expand [M] as a Laurent series in 1 − t. This is a kG-module up to rational multiple, although there is sometimes a problem of whether it is well defined in the Green ring case. The dimension of this module agrees with the usual definition of the degree in the non-equivariant case.

We investigate various properties of the equivariant degree, in particular Theorem 6.4, which lists several equivalent characterizations.

In Section 7, we go on to consider the case of the homogeneous coordinate ring on a projective variety and show that in this case the degree is always defined and it is a permutation module that can be easily described in terms of the geometry. Finally, in Section 8, this theory is applied to the variety associated to the cohomology of a group to reprove a result of Nick Kuhn on the action of the outer automorphism group of a p-group G on the cohomology H^{∗}(G; 𝔽_{p}).

2. General Setup

Let R = ⨁_{j=0}^{∞} R_{j} be a commutative graded algebra over a field k. We suppose that R is a finitely generated k-algebra and that R_{0} is finite dimensional over k, so all the homogeneous components R_{j} are also finite dimensional vector spaces over k. Let G be a finite group and let M = ⨁_{i=N}^{∞} M_{i} a finitely generated graded left RG-module, where the action of G preserves the grading and each M_{i} is a finite dimensional k-vector space.

We recall some facts about the Hilbert series H(M, t) = ∑_{i=N}^{∞} dim_{k}(M_{i}) t^{i} of M. The graded version of Noether normalization (see Theorem 2.2.7 in [1]) guarantees the existence of homogeneous elements d_{1}, d_{2}, …, d_{n} of positive degrees in R that generate a polynomial subring k[d_{1}, …, d_{n}] of R and such that R is finitely generated as a k[d_{1}, …, d_{n}]-module. We write |d_{i}| := deg(d_{i}) for the degree of d_{i}. The number n is equal to the Krull dimension of R. By the Hilbert-Serre Theorem (see e.g. 2.1.1 in [1]) the Hilbert series H(M, t) is of the form

H(M, t) = \frac{f(M, t)}{∏_{i=1}^{n}(1 − t^{|d_{i}|})},

where f(M, t) is a Laurent polynomial with integer coefficients. As in [1, Section 2.4], for example, the rational number deg(M) is defined by the Laurent expansion of H(M, t) about t = 1:

(1)  H(M, t) = \frac{deg(M)}{(1 − t)^{n}} + O (\frac{1}{(1 − t)^{n−1}} ) .

Obviously the definition of the degree deg(M) ignores the action of G on M. In the next two sections, we shall define an equivariant analogue deg_{G}(M), which also incorporates the group action.

First, we define the degree of certain Laurent series. Let p(t) be a Laurent series of the form

p(t) = ∑_{i=N}^{∞} a_{i}t^{i} = \frac{g(t)}{∏_{i=1}^{n}(1 − t^{|d_{i}|})}

where the a_{i} are rational numbers and g(t) is a Laurent polynomial with rational coefficients. We define the rational number deg(p(t)) to be the coefficient of \frac{1}{(1−t)^{n}} in the Laurent expansion of p(t) about t = 1 and we call deg(p(t)) the degree of p(t). If we want to emphasize the dependency on n we write deg^{n}(p(t)) instead of deg(p(t)). In particular, we have deg(H(M, t)) = deg(M) with deg(M) as in (1).

3. Equivariant Degree over the Green Ring

As usual, the Green ring a(kG) is defined to be the ring with generators the isomorphism classes |V | of kG-modules, and relations |V | + |W | = |V ⊕ W |, |V | · |W | = |V ⊗_{k} W |. We set a(kG)_{ℚ} := ℚ ⊗_{ℤ} a(kG). The representation ring R(kG) is defined to be the quotient of a(kG) by the ideal generated by the elements |V_{2}|−|V_{1}|−|V_{3}|, where 0 → V_{1} → V_{2} → V_{3} → 0 is a short exact sequence of kG-modules. We set R(kG)_{ℚ} := ℚ ⊗_{ℤ} R(kG).

We will consider two versions of the equivariant degree: one is an element of a(kG)_{ℚ}, but is not always defined; the other is a weaker one, which is an element of R(kG)_{ℚ}, but it is always defined. The main tool used in the definition of the former is the following Weak Structure Theorem 3.1, so-called because it is a generalization of the Structure Theorem of [15].

Theorem 3.1. For any finitely generated graded k[d_{1}, …, d_{n}]G-module M,

M ≅ ⨁_{U∈Indecomp(M)} ⨁_{I⊆{1,...,n}} k[d_{I}] ⊗_{k} X_{U,I},

as a kG-module, where X_{U,I} is a finite dimensional graded kG-module that is a sum of U’s (ignoring grading) and k[d_{I}] = k[d_{i} | i ∈ I]. The map from right to left is given by multiplication.

Proof. The only difference between this theorem and Proposition 4.4 of [15] is that there Indecomp(M) is supposed to be finite. But the same proof works, although it is better to keep the different indecomposables separate by using the double summation, as in the statement above, rather than combining them as X̄_{I} = ⨁_{U∈Indecomp(M)} X_{U,I} as in [15]. □

Next we describe the definition of the degree with values in a(kG)_{ℚ}. For each i, the kG-module M_{i} defines an element |M_{i}| of a(kG).

Definition 3.2. We call the Laurent series

(2)  [M] := ∑_{i=N}^{∞} |M_{i}| t^{i}

with coefficients in a(kG) the equivariant Hilbert series of M with coefficients in the Green ring.

Clearly, if G = {1} is the trivial group, we can identify |M_{i}| with the dimension of M_{i} as a k-vector space. So in this situation [M] coincides with the usual Hilbert series of M. The equivariant Hilbert series has the following basic properties:

Lemma 3.3. Suppose M′ = ⨁_{i=N′}^{∞} M′_{i} is another finitely generated graded left RG-module, such that the action of G preserves the grading and every M′_{i} is a finite dimensional k-vector space. Then

[M ⊕ M′] = [M] + [M′] and [M ⊗_{k} M′] = [M] · [M′].

Proof. Clear. □

Besides the Hilbert series H(M, t) we can consider a Hilbert series that counts the multiplicity of some isomorphism class of indecomposable summands. Let Indecomp(M) be a set of representatives for the isomorphism classes of all indecomposable kG-modules which occur as a direct summand of some M_{i} and let m_{U,i} be the multiplicity of U ∈ Indecomp(M) as a direct summand of M_{i}. We set H_{U}(M, t) := ∑_{i=N}^{∞} m_{U,i} t^{i}. The Laurent series H_{U}(M, t) can be written as a rational function too.

Proposition 3.4. For each U ∈ Indecomp(M) the Laurent series H_{U}(M, t) can be written as

H_{U}(M, t) = \frac{f_{U}(M, t)}{∏_{i=1}^{n}(1 − t^{|d_{i}|})},

where f_{U}(M, t) is a Laurent polynomial in t with integer coefficients.

Proof. This is a consequence of the Weak Structure Theorem 3.1. □

Let F be an arbitrary finite subset of Indecomp(M). We consider the Laurent series with integer coefficients q(t) := H(M, t) − ∑_{U∈F} dim_{k}(U)H_{U}(M, t). By definition of the Hilbert series, all the coefficients of q(t) are non-negative integers, and q(t) is of the form

q(t) = \frac{g(t)}{∏_{i=1}^{n}(1 − t^{|d_{i}|})}

for some Laurent polynomial g(t) with integer coefficients since something similar holds for H(M, t) and H_{U}(M, t) by Proposition 3.4. So we can take degrees and obtain

(3)  deg(M) = (∑_{U∈F} dim_{k}(U) deg(H_{U}(M, t))) + deg(q(t)).

It turns out that all the degrees occurring in (3) are non-negative with bounded denominators by the following result.

Lemma 3.5. Suppose that

p(t) = \frac{h(t)}{∏_{i=1}^{n}(1 − t^{|d_{i}|})} = ∑_{i=N}^{∞} a_{i} t^{i}

where h(t) is a Laurent polynomial with rational coefficients and the a_{i}’s are non-negative integers. Then deg(p(t)) ≥ 0. If all the coefficients of h(t) are integers then deg(p(t)) is of the form deg(p(t)) = _{\frac{d}{∏_{i=1}^{n} |d_{i}|}} for some non-negative integer d.

Proof. We compute:

deg(p(t)) = lim_{t→1}(1 − t)^{n}p(t) = lim_{t→1} \frac{h(t)}{∏_{i=1}^{n}(1 + t + ⋯ + t^{|d_{i}|−1})} = \frac{h(1)}{∏_{i=1}^{n} |d_{i}|} .

We still have to show that deg(p(t)) ≥ 0. Since multiplication with ∏_{i=1}^{n}(1 + t + ⋯ + t^{|d_{i}|−1}) and a suitable power of t does not affect the sign of the degree or the sign of the a_{i}’s we may assume that p(t) is a Laurent polynomial in 1 − t with rational coefficients, that is that

p(t) = \frac{b_{−n}}{(1 − t)^{n}} + \frac{b_{1−n}}{(1 − t)^{n−1}} + ⋯ + b_{m−1}(1 − t)^{m−1} + b_{m}(1 − t)^{m}

for some rational numbers b_{i} and a non-negative integer m. In particular, b_{−n} = deg(p(t)). Expanding the \frac{1}{(1−t)^{j}} ’s as power series in t and comparing the coefficients of t^{i} we see that there exists a polynomial r(i) in i of degree at most n − 2 (or r(i) = 0 if n = 1) with coefficients depending on n and the b_{j}’s such that a_{i} = \frac{1}{(n−1)!} b_{n} i^{n−1} + r(i) for all large enough i. So the condition a_{i} ≥ 0 implies that deg(p(t)) = b_{n} ≥ 0. □

Corollary 3.6. There are only finitely many U ∈ Indecomp(M) with deg(H_{U}(M, t)) ≠ 0 and we have

∑_{U} dim_{k}(U) deg(H_{U}(M, t)) ≤ deg(M)

where the sum means the sum over all those U ∈ Indecomp(M) with deg(H_{U}(M, t)) ≠ 0.

Proof. This follows from Equation (3) and Lemma 3.5. □

We can now define the equivariant degree with values in the Green ring.

Definition 3.7. We say that deg_{G}(M) is defined (over the Green ring) if

∑_{U} dim_{k}(U) deg(H_{U}(M, t)) = deg(M).

In this case we call deg_{G}(M) := ∑_{U} deg(H_{U}(M, t)) |U| ∈ a(kG)_{ℚ} the equivariant degree of M (in the Green ring). If we want to emphasize the dependency on n we write deg_{G}^{n}(M) instead of deg_{G}(M).

The existence of the degree in the Green ring can be characterized as follows.

Lemma 3.8. For R, G, M as above the following statements are equivalent.

(1) deg_{G}(M) is defined in the Green ring.

(2) There is a finite set F of indecomposable kG-modules such that

∑_{U∈F} dim_{k}(U) deg(H_{U}(M, t)) = deg(M).

(3) There is a finite set F of indecomposable kG-modules such that

deg (∑_{U∉F} dim_{k}(U)H_{U}(M, t)) = 0.

Here we have set ∑_{U∉F} dim_{k}(U)H_{U}(M, t) := H(M, t) − ∑_{U∈F} dim_{k}(U)H_{U}(M, t).

Proof. This is clear from the definition of deg_{G}(M). □

Certainly the equivariant degree deg_{G}(M) is defined if M has only finitely many isomorphism types of indecomposable summands. For example, this is the case if k is a finite field, M a polynomial ring in n variables over k, G a finite group acting on this polynomial ring by homogeneous linear substitutions and R = M^{G} the ring of invariants (see Theorem 17.1 in [9]). The following example shows that there are situations where deg_{G}(M) is not defined:

Example (see Example 4.4 in [10]). Let k be a field of two elements and R = k[x, y] a polynomial ring in two variables over k. The Klein four group G = ⟨α, β⟩ ≅ ℤ_{2} × ℤ_{2} acts on M = k[x, y]⟨1, z⟩ by α : z ↦ z + x and β : z ↦ z + y. We can regard M as a subset of k[x, y, z] or as a free R-module of rank two.

If we attach a grading to R and the module M by assigning x, y and z grading 1, then M is the direct sum M = ⨁_{i=0}^{∞} M_{i}. It is shown in [10] that M_{i} ≅ Ω^{i}k as kG-modules where Ω^{i}k is the i-th Heller translate of the trivial kG-module k. In particular, the M_{i}’s are indecomposable and pairwise non-isomorphic.

We have n = 2, Indecomp(M) = {Ω^{i}k | i ∈ ℕ_{0}} and H_{Ω^{i}k}(M, t) = t^{i}. So we obtain deg(H_{Ω^{i}k}(M, t)) = 0 for all i. On the other hand we have

H(M, t) = ∑_{i=0}^{∞} dim_{k}(M_{i}) t^{i} = ∑_{i=0}^{∞}(2i + 1)t^{i} = \frac{2}{(1 − t)^{2}} − \frac{1}{1 − t}

and thus deg(M) = 2. So deg_{G}(M) is not defined in this example.

4. Equivariant Degree in the Representation Ring

One way to construct an equivariant degree which is defined for every module M (satisfying the assumptions in Section 2) is to work over the representation ring. In this section we will define the equivariant degree with values in the representation ring.

The first steps are very similar to those for the Green ring. Let R, G and M be as in Section 2. For each i, the kG-module M_{i} defines an element |M_{i}| of R(kG).

Definition 4.1. We call the Laurent series [M] := ∑_{i=N}^{∞} |M_{i}| t^{i} with coefficients in R(kG) the equivariant Hilbert series of M with coefficients in the representation ring.

Clearly Lemma 3.3 carries over to equivariant Hilbert series with coefficients in the representation ring.

For each irreducible kG-module V let m_{V,i} be the multiplicity of V as a composition factor of M_{i}. We set H_{V} (M, t) := ∑_{i=N}^{∞} m_{V,i} t^{i}. We choose a polynomial subring k[d_{1}, …, d_{n}] of R as in Section 2. In fact the Laurent series H_{V} (M, t) can be written as a rational function.

Lemma 4.2. For each irreducible kG-module V the Laurent series H_{V} (M, t) can be written as

H_{V} (M, t) = \frac{f_{V} (M, t)}{∏_{i=1}^{n}(1 − t^{|d_{i}|})},

where f_{V} (M, t) is a Laurent polynomial in t with rational coefficients. If k is a splitting field for V then all the coefficients of f_{V} (M, t) are integers.

Proof. Let P_{V} be a projective cover of V. The graded k[d_{1}, …, d_{n}]-module Hom_{kG}(P_{V} , M) is a direct summand of the graded k[d_{1}, …, d_{n}]-module Hom_{kG}(kG, M) ≅ M. This implies that Hom_{kG}(P_{V} , M) is finitely generated as k[d_{1}, …, d_{n}]-module. So, by the Hilbert-Serre Theorem 2.1.1 in [1], the Hilbert series H(Hom_{kG}(P_{V} , M), t) has the form \frac{f̃_{V} (t)}{∏_{i=1}^{n}(1−t^{|d_{i}|})} for

some Laurent polynomial f̃_{V} (t) with integer coefficients. Since dim_{k}(Hom_{kG}(P_{V} , M_{i})) = dim_{k}(End_{kG}(V)) · m_{V,i} we get

H_{V} (M, t) = \frac{1}{dim_{k}(End_{kG}(V))}H(Hom_{kG}(P_{V} , M), t) = \frac{1}{dim_{k}(End_{kG}(V))} · \frac{f̃_{V} (t)}{∏_{i=1}^{n}(1 − t^{|d_{i}|})}.

If k is a splitting field for V then dim_{k}(End_{kG}(V)) = 1. □

Corollary 4.3. The equivariant Hilbert series [M] with coefficients in the representation ring is of the form

[M] = \frac{[f](M, t)}{∏_{i=1}^{n}(1 − t^{|d_{i}|})}

where [f](M, t) is a Laurent polynomial with coefficients in R(kG)_{ℚ}. If k is a splitting field for G then all the coefficients of [f](M, t) are elements of R(kG).

Proof. This follows from Lemma 4.2. □

Now we can define the equivariant degree with values in the representation ring:

Definition 4.4. We call deg_{G}(M) := ∑_{V} deg(H_{V} (M, t)) |V | ∈ R(kG)_{ℚ} the equivariant degree of M (in the representation ring). Here the sum varies over a set of representatives for the isomorphism classes of irreducible kG-modules. If we want to emphasize the dependency on n we also write deg_{G}^{n}(M) instead of deg_{G}(M).

We use the same notation for the two degrees, specifying the ring in which the values lie explicitly when necessary. In any case the two versions are compatible in the following sense. Let π : a(kG)_{ℚ} ↠ R(kG)_{ℚ} denote the canonical map.

Proposition 4.5. The map π takes the equivariant degree of M in the Green ring to the equivariant degree of M in the representation ring whenever the former is defined.

Proof. This follows from an easy calculation based on Definitions 3.7 and 4.4. □

5. Basic properties of the equivariant degree

In this section we collect some of the basic properties of the equivariant degree. We always assume that R, G and M are as in Section 2 and that M′ and M′′ are finitely generated graded left RG-modules, where the action of G preserves the grading and every homogeneous component is finite dimensional as k-vector space. We choose a polynomial subring k[d_{1}, …, d_{n}] of R as in Section 2.

We begin with a trivial observation showing that the equivariant degree coincides with the usual degree if there is “no group action”:

Lemma 5.1. If G = {1} is the trivial group then deg_{G} M is defined and deg_{G} M = deg(M)|k| where k is the trivial kG-module.

Proof. This is clear from the definition of the degree. □

From now on G is again an arbitrary finite group. The next lemma holds both for the equivariant degree taking values in the Green ring as well as for the degree taking values in the representation ring.

Lemma 5.2. If deg_{G} M is defined (which is of course always the case over the representation ring), then there is a positive integer c such that c · deg_{G} M is a genuine module, i.e. it is of the form |V | for some kG-module V.

Proof. By the definition of deg_{G} M and Lemma 3.5 we can take c := ∏_{i=1}^{n} |d_{i}| if deg_{G} M is defined over the Green ring. In the case of the representation ring, c := (∏_{V} dim_{k}(End_{kG}(V)) · (∏_{i=1}^{n} |d_{i}|) does the job (where V runs through a set of representatives for the isomorphism classes of irreducible kG-modules). □

Lemma 5.3. If M′ ↪ M ↠ M′′ is a short exact sequence of finitely generated graded RG-modules that is split over kG, then deg_{G} M is defined if and only if both deg_{G} M′ and deg_{G} M′′ are defined. If this is the case then deg_{G} M = deg_{G} M′ + deg_{G} M′′.

This formula holds for any short exact sequence when the degree takes values in the representation ring.

Proof. Let U ∈ Indecomp(M). The splitting implies H_{U}(M, t) = H_{U}(M′, t)+H_{U}(M′′, t). So ∑_{U∈F} dim_{k}(U) deg(H_{U}(M, t)) = ∑_{U∈F} dim_{k}(U) deg(H_{U}(M′, t)) + ∑_{U∈F} dim_{k}(U) deg(H_{U}(M′′, t))

with F as in Lemma 3.8. By additivity of the non-equivariant degree we have deg(M) = deg(M′) + deg(M′′). Since all these degrees are non-negative by Lemma 3.5 we get that deg_{G} M is defined if and only if both deg_{G} M′ and deg_{G} M′′ are defined and in this case we

get: deg_{G}(M) = ∑_{U∈F} deg(H_{U}(M, t))|U| = ∑_{U∈F} deg(H_{U}(M′, t))|U| + ∑_{U∈F} deg(H_{U}(M′′, t))|U| =

deg_{G}(M′) + deg_{G}(M′′). The statement about the degree over the representation ring follows from H_{V} (M, t) = H_{V} (M′, t) + H_{V} (M′′, t) for every irreducible kG-module V. □

For W, W′ ∈ R(kG)_{ℚ} we write W ≤ W′ if W′ − W is a linear combination of isomorphism classes of kG-modules with non-negative rational coefficients. We write W ≥ W′ if W′ ≤ W.

Corollary 5.4. For a finitely generated graded RG-module M, as at the beginning of this section, the following properties hold for the degree with values in the representation ring.

(1) If M′ is a graded RG-submodule of M then deg_{G}(M′) ≤ deg_{G}(M).

(2) If M′ is a graded RG-epimorphic image of M then deg_{G}(M) ≥ deg_{G}(M′).

Proof. This follows from Lemmas 5.2 and 5.3. □

For an integer d we write M[d] for M with a degree shift of d, so that M[d]_{i} = M_{i+d}. For a positive integer q let R^{[q]} be the graded k-algebra obtained from R by multiplying all degrees by q, that is (R^{[q]})_{iq} = R_{i} and (R^{[q]})_{i} = 0 for all i not divisible by q. Analogously, we can construct a graded R^{[q]}G-module M^{[q]} with G-action from M by multiplying all degrees by q, that is (M^{[q]})_{iq} = M_{i} and (M^{[q]})_{i} = 0 for all i not divisible by q.

Lemma 5.5. With the above notation, deg_{G} M has the following properties.

(1) If the Krull dimension of M is at most n − 1 then deg_{G} M is defined and equal to 0.

(2) deg_{G}(M[d]) is defined if and only if deg_{G} M is defined. If this is the case then deg_{G}(M[d]) = deg_{G} M.

(3) deg_{G}(M^{[q]}) is defined if and only if deg_{G} M is defined. If this is the case then deg_{G}(M^{[q]}) = q^{−n} deg_{G} M.

Proof. (1) follows from the corresponding property of the non-equivariant degree, (Lemma 2.4.1 in [1]). (2) and (3) are clear. □

Sometimes it is convenient to add an element z in degree 1 to R. Then R[z] ⊗_{k} M is finitely generated over R[z], which has dimension n + 1.

Lemma 5.6. The degree deg_{G}^{n+1}(R[z] ⊗_{R} M) is defined if and only if deg_{G}^{n} M is defined, and if this is the case then they are both equal.

Proof. Clear. □

Sometimes it is convenient to change the field k. We then write deg_{kG} M to denote the degree with value in the Green ring of kG.

Lemma 5.7. Let ℓ be a field extension of k.

(1) If deg_{kG} M is defined then so is deg_{ℓG}(ℓ ⊗_{k} M) and deg_{ℓG}(ℓ ⊗_{k} M) = ℓ ⊗_{k} deg_{kG} M.

(2) If ℓ/k is finite and L is a finitely generated graded (ℓ⊗_{k}RG)-module such that deg_{ℓG} L is defined then deg_{kG}(L ↓_{k}^{ℓ}) = (deg_{ℓG} L) ↓_{k}^{ℓ}.

(3) If ℓ/k is finite and if deg_{ℓG}(ℓ ⊗_{k} M) is defined then so is deg_{kG} M and deg_{kG} M = |ℓ : k|^{−1}(deg_{ℓG}(ℓ ⊗_{k} M)) ↓_{k}^{ℓ}.

Proof. Only (3) needs any comment. Since (ℓ ⊗_{k} M) ↓_{k}^{ℓ}≅ M^{|ℓ/k|}, we obtain (deg_{ℓG}(ℓ ⊗_{k} M)) ↓_{k}^{ℓ}= deg_{G}(M^{|ℓ/k|}) by (2). But then deg_{G} M is defined and the formula holds, by 5.3. □

6. Further Results

In this section R = k[d_{1}, …, d_{n}] is a graded polynomial ring with generators in positive degrees. Unless otherwise stated the degree will always take values in the Green ring.

We say that a map of R-modules dominates when the cokernel has dimension strictly less than n. This is not consistent with the customary use of dominant in algebraic geometry, but it is very convenient for us here.

Proposition 6.1. The degree deg_{G} M of a finitely generated graded RG-module M is defined if and only if there is a finite dimensional graded kG-submodule X ⊆ M such that the multiplication map R ⊗_{k} X → M is injective and dominant and the image is a summand over kG. If this holds then deg_{G} M = deg R · |X|.

Proof. Suppose that such an X exists; then deg_{G}(M/(R ⊗_{k} X)) is defined and equal to 0 by hypothesis (take F = ∅). We claim that deg_{G}(R ⊗_{k} X) = deg R · |X|.

It is easy to see that H(R ⊗ X, t) = H(R, t)H(X, t), so

deg_{G}(R ⊗_{k} X) = lim_{t→1} H(R, t)H(X, t) = deg R · |X|.

By additivity (Lemma 5.3), deg_{G} M is defined and is equal to deg R · |X|.

Conversely, suppose that deg_{G} M is defined using a finite set F ⊆ Indecomp M. Then, using the notation of the Weak Structure Theorem 3.1, we must have

deg ( ⨁_{U∉F} ⨁_{I⊆{1,...,n}} k[d_{I}] ⊗_{k} X_{U,I}) = 0.

Thus we can take X = ⨁_{U∈F} X_{U,{1,...,n}}. □

A lot of our work is made easier by the next easy, but surprising, result.

Proposition 6.2. If M is a finitely generated graded RG-module and X is a finite dimensional graded kG-submodule such that the multiplication map R ⊗_{k} X → M is injective and dominates then the image is a summand over kG, so in particular deg_{G} M is defined and is equal to deg R · |X|.

Proof. There is a homogeneous element z ∈ R that annihilates the cokernel. Consider the composition of maps

R ⊗_{k} X → M →^{z} zM ⊆ R ⊗_{k} X.

The image is zR ⊗_{k} X, and since zR is a k-summand of R it follows that the image is a kG-summand of R ⊗_{k} X. Thus the image of R ⊗_{k} X in M is also a summand. □

Given a graded commutative ring S, let Q(S) denote the graded ring of fractions, where we invert all the homogeneous elements. It is a ℤ-graded ring and Q(S)_{0} is a field. Q(S) = Q(S)_{0}[z, z^{−1}], where z is an element of Q(S) of least positive degree. Q(S) is flat over S.

Notice that if M is a finitely generated graded RG-module then Q(R) ⊗_{R} M is a finitely generated Q(R)-module and in each degree it is a finite dimensional vector space over Q(R)_{0}. In addition, Q(R) ⊗_{R} M = 0 if and only if dim M < dim R.

Proposition 6.3. Let M be a finitely generated graded RG-module. Then deg_{G} M is defined if and only if there is a finite dimensional graded kG-submodule X ⊆ Q(R) ⊗_{R} M such that Q(R) ⊗_{R} M = Q(R) ⊗_{k} X. If this is the case then deg_{G} M = deg R · |X|.

Proof. If deg_{G} M is defined then we have a short exact sequence R⊗_{k}X ↪ M ↠ M/(R⊗_{k}X) with dim(M/(R ⊗_{k} X)) < dim R, by Proposition 6.1. If we tensor this with Q(R) we obtain Q(R) ⊗_{k} X ↪ Q(R) ⊗_{R} M ↠ Q(R) ⊗_{R} (M/(R ⊗_{k} X)). But the last term must be 0.

Conversely, suppose that we have an X satisfying the conditions of the statement of the proposition. Let \{x_{i}\} be a k-basis for X and write x_{i} = ∑_{j} \frac{a_{i,j}}{b_{i,j}} m_{j}, where a_{i,j}, b_{i,j} ∈ R and m_{j} ∈ M, all homogeneous. Let b̄ be the product of all the b_{i,j}. Then b̄X ⊆ M, and we have a short exact sequence R ⊗_{k} b̄X ↪ M ↠ M/(R ⊗_{k} b̄X). But when we tensor with Q(R) the first arrow becomes an isomorphism, so we must have Q(R) ⊗_{R} (M/(R ⊗_{k} b̄X)) = 0 and thus dim(M/(R ⊗_{k} b̄X)) < dim R, as required by 6.2. □

We now summarize the equivalent characterizations of the equivariant degree.

Theorem 6.4. Let M be a finitely generated graded RG-module. The following conditions on M are equivalent.

(1) deg_{G} M is defined in the Green ring.

(2) There is a finite dimensional graded kG-submodule X ⊆ M such that the multiplication map R ⊗_{k} X → M dominates and is split injective over kG.

(3) There is a finite dimensional graded kG-submodule Y ⊆ M such that the multiplication map R ⊗_{k} Y → M dominates and is injective.

(4) There is a finite dimensional graded kG-submodule Z ⊆ Q(R) ⊗_{R} M such that

Q(R) ⊗_{R} M = Q(R) ⊗_{k} Z. When these conditions hold we have |X| = |Y | = |Z| = \frac{1}{deg R} deg_{G} M.

Proof. Just combine 6.1, 6.2 and 6.3. □

Lemma 6.5. Let R and R′ be polynomial rings in n and n′ variables respectively and let M and M′ be finitely generated graded RG- and R′G-modules respectively. Let L be a finitely generated graded RH-module and let H be a subgroup of G. The degree commutes with the following operations (when the quantity on the right hand side is defined):

(1) tensor product, deg_{G}^{n+n′}(M ⊗_{k} M′) = deg_{G}^{n}(M) · deg_{G}^{n′}(M′);

(2) restriction, deg_{H}(M ↓_{H}^{G}) = (deg_{G} M) ↓_{H}^{G};

(3) induction, deg_{G}(L ↑_{H}^{G}) = (deg_{H} L) ↑_{H}^{G};

(4) fixed points, when H is a normal subgroup of G, deg_{G/H} M^{H} = (deg_{G} M)^{H}.

Proof. These all follow easily from property 6.4 (3). □

In the remaining part of this section we consider how Theorem 6.4 and Lemma 6.5 can be reformulated for the degree with values in the representation ring. Clearly if one of the conditions in Theorem 6.4 is satisfied then Proposition 4.5 implies that |X| = |Y | = |Z| = \frac{1}{deg R} deg_{G} M also holds for the degree over the representation ring. The analogue of Lemma 6.5 is the following

Lemma 6.6. With the same hypotheses as in the previous lemma, the degree with values in the representation ring commutes with the following operations:

(1) tensor product, deg_{G}^{n+n′}(M ⊗_{k} M′) = deg_{G}^{n}(M) · deg_{G}^{n′}(M′);

(2) restriction, deg_{H}(M ↓_{H}^{G}) = (deg_{G} M) ↓_{H}^{G},

(3) induction, deg_{G}(L ↑_{H}^{G}) = (deg_{H} L) ↑_{H}^{G}.

Proof. This is straightforward and left to the reader. □

7. Rings

Throughout this section, S will be a graded ring in non-negative degrees that is finitely generated over the field k and such that S_{0} is finite dimensional over k. We suppose that a finite group G acts on S by graded k-algebra automorphisms.

Geometrically, G acts as a group of automorphisms of the projective variety V = Proj(S), defined over k. Conversely, S could be the homogeneous coordinate ring of a variety over k on which G acts.

The invariant subring S^{G} is necessarily noetherian and S is finitely generated over S^{G} ([1] 1.3.1). By Noether normalization, we can find a graded polynomial subring R ≤ S^{G} such that S^{G} is finitely generated over R ([1] 2.2.7). Thus S is finitely generated over R, and S and R have the same dimension. We need this ring R to exist in order for the preceding theory to apply, but it does not matter which ring R we choose.

Proposition 7.1. If S is an integral domain and G acts faithfully, then deg_{G} S is defined and

deg_{G} S = \frac{deg S}{|G|} · kG.

Proof. In [14], a graded submodule F ≤ M is produced such that F ≅ kG and such that the multiplication map S^{G} ⊗_{k} F ↪ S dominates and is split over kG. It follows from the Additivity Lemma 5.3 that deg_{G} S = deg_{G}(S^{G} ⊗ F) = deg S^{G} · kG.

There is an alternative proof that we sketch here. By Lemma 5.6, we may assume that R contains an element z of degree 1. But S is an integral domain, so it injects into Q(S), thus G acts faithfully on Q(S). Since Q(S) = Q(S)_{0}[z, z^{−1}] and G acts trivially on z, G must act faithfully on Q(S)_{0}. By the Normal Basis Theorem there is a basis {x_{g}}_{g∈G} for Q(S)_{0} over Q(S)_{0}^{G} that is freely permuted by G.

But Q(S)_{0} is a finite dimensional vector space over Q(R); let {y_{i}} be a basis. If we let X be the k-span of the set {y_{i}x_{g}}, then this is the module that we require. □

Let P_{0} denote the (finite) set of prime ideals in S of height 0.

Lemma 7.2. The natural map S → ⨁_{𝔭∈P_{0}} S/𝔭 dominates and has rad S as kernel.

Proof. The radical is equal to the intersection of all the prime ideals, which is equal to the intersection of the minimal ones.

The claim of domination we prove by labeling the distinct prime ideals of height 0 as 𝔭_{1}, …, 𝔭_{m} and showing by induction on r that the map S → ⨁_{i=1}^{r} S/𝔭_{i} dominates.

This is clearly true when r = 1, and the induction step follows from considering the following diagram with exact rows and columns.

S/ ∩_{i=1}^{r+1} 𝔭_{i} ⟶ S/ ∩_{i=1}^{r} 𝔭_{i} ⊕ S/𝔭_{r+1} ⟶ S/(∩_{i=1}^{r}𝔭_{i} + 𝔭_{r+1})
∥  ↓  ↓
S/ ∩_{i=1}^{r+1} 𝔭_{i} ⟶  ⨁_{i=1}^{r+1} S/𝔭_{i}  ⟶  X
↓  ↓
Y  Y

The induction hypothesis applied to the middle column shows that dim Y < dim S, and dim S/(∩_{i=1}^{r}𝔭_{i} + 𝔭_{r+1}) < dim S by construction. Thus dim X < dim S and the middle row yields the next stage in the induction. □

Given a prime 𝔭 < S, let G_{𝔭} denote the stabilizer in G of 𝔭 and let Ḡ_{𝔭} be the pointwise stabilzer of S/𝔭. We can now state a decomposition theorem for the degree of S.

Theorem 7.3. If S contains no nilpotent elements then deg_{G} S is defined and

deg_{G} S = ∑_{\begin{matrix}𝔭∈P_{0}/G \\ dim S/𝔭=dim S \end{matrix}} \frac{deg S/𝔭}{|G_{𝔭}/Ḡ_{𝔭}|} · k[G/Ḡ_{𝔭}].

Proof. In view of Proposition 7.1, Lemma 7.2 and Theorem 6.5, all we need to do is to show that deg_{G}(⨁_{𝔭∈P_{0}} S/𝔭) is equal to the expression shown.

But

⨁_{𝔭∈P_{0}} S/𝔭 ≅ ⨁_{𝔭∈P_{0}/G} ⨁_{𝔮∼_{G}𝔭}S/𝔮 ≅ ⨁_{𝔭∈P_{0}/G}Ind_{G_{𝔭}}^{G} S/𝔭.

So

deg_{G}(⨁_{𝔭∈P_{0}} S/𝔭) ≅ ⨁_{𝔭∈P_{0}/G}deg_{G} Ind_{G_{𝔭}}^{G} S/𝔭

≅ ⨁_{𝔭∈P_{0}/G}Ind_{G_{𝔭}}^{G} deg_{G_{𝔭}} S/𝔭 by Lemma 6.5 (3)

≅ ⨁_{𝔭∈P_{0}/G}Ind_{G_{𝔭}}^{G} \frac{deg S/𝔭}{|G_{𝔭}/Ḡ_{𝔭}|} · k[G_{𝔭}/Ḡ_{𝔭}] by Proposition 7.1

≅ ⨁_{𝔭∈P_{0}/G} \frac{deg S/𝔭}{|G_{𝔭}/Ḡ_{𝔭}|} · k[G/Ḡ_{𝔭}].

We can omit from the sum the primes 𝔭 for which dim S/𝔭 ≠ dim S, since for these deg S/𝔭 = 0. □

Geometrically, the permutation modules that occur in the statement of the theorem correspond to the way that the group permutes the irreducible components of maximum dimension of the projective variety Proj(S).

Now suppose that the action of G on S can be written over a finite field 𝔽_{q}. Recall from Lemma 5.5 that the operation of multiplying all degrees by q gives us a new ring S^{[q]} with G-action and deg_{G} S^{[q]} = q^{−n} deg_{G} S. Let S^{q} < S denote the subring of qth powers. There is a surjection S^{[q]} → S^{q} and this is an isomorphism if rad S = 0.

Lemma 7.4. In the representation ring we have deg_{G} S^{q} ≤ q^{−n} deg_{G} S and if S contains no nilpotents then deg_{G} S^{q} = q^{−n} deg_{G} S in the Green ring.

Proof. This follows from the the preceding remarks and the Additivity Lemma 5.3. □

8. Group Cohomology

In this section we apply some of the theory that we have developed to a problem in group cohomology considered by Nick Kuhn [11]. We fix a prime p and a finite group P (we do not yet require P to be a p-group). Then G = Aut(P) acts on the graded commutative ring H^{∗}(P) = H^{∗}(P; 𝔽_{p}).

By the Evens-Venkov theorem (e.g. [2] 3.10, 4.2), H^{∗}(P) is noetherian, hence so is H^{∗}(P)^{G}, thus H^{∗}(P) is certainly finitely generated over some commutative polynomial ring R such that the action of G commutes with that of R; we can assume that dim R = dim H^{∗}(P).

Given a p-group P and a simple G-module V, Kuhn asked whether the dimension of V as a composition factor of H^{∗}(P) is equal to dim H^{∗}(P). It was already known from [5], [7] and [13] that V does occur in H^{∗}(P).

Theorem 8.1. (Kuhn [11]) For p odd the dimension of V as a composition factor of H^{∗}(P) is equal to the dimension of H^{∗}(P).

The case of p = 2 is still undecided. Kuhn’s methods used the nilpotent filtration of the category of unstable modules over the Steenrod algebra. We will show how this theorem can be proved using the equivariant degree. Clearly what we need to do is to show that V occurs as a composition factor of deg_{G} H^{∗}(P).

For any finite elementary abelian p-group E, let F^{∗}(E) = H^{∗}(E)/ rad, which is just the symmetric algebra 𝔽_{p}[E] = S^{∗}(E^{∗}), where E^{∗} = Hom(E, 𝔽_{p}) is in degree 2 (or degree 1 if p = 2).

In general, let

F^{∗}(P) = \underleftarrow{lim}_{E∈A_{p}(P)} F^{∗}(E),

where A_{p}(P) denotes the category with objects the elementary abelian subgroups of P and morphisms the inclusions between them. G acts naturally on this.

Quillen in [12] (see [2] 5.6) showed that the natural map induced by restrictions, r : H^{∗}(P) → F^{∗}(P)^{P} is a purely inseparable isogeny (or uniform F-isomorphism): that is that the kernel is nilpotent and there is an integer N such that (F^{∗}(P)^{P})^{p^{N}} ⊆ Im(r). From this he deduced that dim H^{∗}(P) is equal to the p-rank of P, which we will denote by n.

Consider what this means for the degree with values in the representation ring. We have deg_{G} H^{∗}(P) ≥ deg_{G} Im(r) ≥ deg_{G}((F^{∗}(P)^{P})^{p^{N}}), using Lemma 5.3. By Lemma 7.4 we have deg_{G}((F^{∗}(P)^{P})^{p^{N}}) = \frac{1}{p^{Nn}} deg_{G} F^{∗}(P)^{P}, since F^{∗}(P) contains no nilpotent elements.

Now we work in the Green ring to see that deg_{G} F^{∗}(P)^{P} = deg_{G}(F^{∗}(P))^{P}, by Lemma 6.5 (4).

We conclude that it is sufficient to show that deg_{G} F^{∗}(P) contains every simple G-module as a submodule.

But the Decomposition Theorem 7.3 tells us that

deg_{G} F^{∗}(P) = ∑_{\begin{matrix}E∈A_{p}(P)/G \\ rank E=n \end{matrix}} \frac{deg S/𝔭_{E}}{|N_{G}(E)/C_{G}(E)|} · 𝔽_{p}[G/C_{G}(E)],

where 𝔭_{E} denotes the ideal corresponding to E. Since each E in the sum has maximal rank, deg(S/𝔭_{E}) ≠ 0.

Suppose that some C_{G}(E) is a p-group. Then

Hom_{G}(V, 𝔽_{p}[G/C_{G}(E)]) ≅ Hom_{C_{G}(E)}(V, 𝔽_{p}) ≠ 0,

so V does occur in deg_{G} F^{∗}(P) and we are done. That this always happens when p is odd is the content of the next lemma, which appears as [11] 2.3, although we first learnt it from Benson [3]. We include the proof for the convenience of the reader.

Lemma 8.2. If p is odd and E is maximal then C_{G}(E) is a p-group.

Proof. Consider the composition of homomorphisms C_{G}(E) →^{α} Aut(C_{P} (E)) →^{β} Aut(E).

The composition is trivial, so it suffices to prove that the kernel of each map is a p-group. For β we use the result ([6] 5.3.10) that if p is odd and Q is a p-group then the kernel of the map Aut(Q) → Aut(Ω_{1}(Q)) is a p-group. (This is the only place in this section where the argument requires p to be odd.)

For α we use Thompson’s A × B Lemma ([6] 5.3.4), which states that for any p-group P, if A × B ⊆ Aut(P) with A a p′-group and B a p-group such that A acts trivially on C_{P} (B), then A = 1. We apply this with A some p′-subgroup of Ker(α) and B the image of E in G. □

9. Further results on the degree with values in the representation ring

We assume that k is a splitting field for the group G, but we do not need R to be polynomial.

Let V be a simple kG-module and let M be a finitely generated graded RG-module. Let M_{V} denote the part of M that is generated by submodules isomorphic to V.

Lemma 9.1. Hom_{kG}(V, M) ⊗_{k} V ≅ M_{V} by the map f ⊗ v ↦ f(v).

Proof. Since Hom_{kG}(V, M) ≅ Hom_{kG}(V, M_{V}) we may assume that M = M_{V}. But now the claimed isomorphism is additive in M_{V}, and M_{V} is just a direct sum of submodules isomorphic to V, so we are reduced to the case where M_{V} = V. But now it holds by the assumption that k is a splitting field, so End_{kG}(V) ≅ k ([4] 7.14). □

The next result is an equivariant analogue of [8] I 7.4.

Proposition 9.2. Let M be a finitely generated graded RG-module. Then M has a finite filtration 0 = M_{0} ≤ M_{1} ≤ ⋯ ≤ M_{m} = M by graded RG-submodules such that M_{i}/M_{i−1} ≅ R/𝔭_{i}[ℓ_{i}] ⊗_{k} V_{i}, where 𝔭_{i} is a homogeneous prime ideal of R and V_{i} is a simple kG-module.

(1) The minimal elements among the 𝔭_{i} occurring are the minimal primes for M.

(2) For each minimal prime 𝔭 of M, let k(𝔭) denote the quotient field of R/𝔭. For each simple kG-module V, the number of times that R/𝔭 ⊗_{k} V occurs as a composition factor of the filtration is equal to the number of times that the simple R_{𝔭}G-module k(𝔭) ⊗_{k} V occurs as a composition factor of the localisation M_{𝔭}, hence is independent of the filtration.

Proof. Let 𝔭 be an associated prime of Hom_{RG}(V, M), so it is the annihilator of some ϕ : V → M. Thus we have an injection of graded RG-modules R/𝔭 ↪ Hom_{RG}(V, M), r ↦ rϕ and hence an injection R/𝔭 ⊗_{k} V ↪ Hom_{RG}(V, M) ⊗ V.

By Lemma 9.1 this leads to an injection R/𝔭 ⊗_{k} V ↪ M; denote its image by M_{1}.

Now repeat the process with M/M_{1}, and let M_{2} be the inverse image in M of the resulting submodule. In this way we obtain an ascending sequence of graded RG-submodules of M, which must terminate since M is noetherian.

Notice that this filtration can be refined to a non-equivariant one by filtering the V. Thus (1) follows from the non-equivariant case.

For (2), let 𝔮 be a minimal prime and consider what happens when we localize at 𝔮. If 𝔭_{i} ≠ 𝔮 then (R/𝔭_{i})_{𝔮} = 0, since 𝔮 is minimal in {𝔭_{1}, …, 𝔭_{m}}. If 𝔭_{i} = 𝔮 then (R/𝔮)_{𝔮} = k(𝔮) and (R/𝔮 ⊗_{k} V)_{𝔮} = k(𝔮) ⊗_{k} V. This is a simple S_{𝔮}G-module since k is a splitting field. □

Write m(𝔭, V, M) for the number of times that R/𝔭 ⊗_{k} V occurs as a factor in a filtration of M of the type considered in the Theorem above.

Corollary 9.3.

deg_{G} M = ∑_{dim R/𝔭=dim M} m(𝔭, V, M) deg(R/𝔭) · |V |.

There are some straightforward reduction methods for calculating the degree with values in the representation ring.

Lemma 9.4. Let f ∈ R be homogeneous and let M be a finitely generated graded RG-module of dimension m. Suppose that the dimension of the kernel of the multiplication map ϕ_{f} : M → M, m ↦ fm, has dimension at most m − 2. Then deg_{G}^{m−1}(M/fM) = |f| deg_{G}^{m} M.

Proof. There is a short exact sequence ker(ϕ_{f}) → M →^{ϕ_{f} [|f|]} fM[|f|], where [|f|] denotes the degree shift needed to make all the maps degree preserving). Thus [fM[|f|]] = [M] + O(\frac{1}{(1−t)^{m−2}} ) as a Laurent series in 1 − t and so [fM] = t^{|f|}[M] + O(\frac{1}{(1−t)^{m−2}} ).

There is also a short exact sequence fM → M → M/fM, so [M/fM] = [M] − [fM]. Combining, we find that [M/fM] = [M] − t^{|f|}[M] + O(\frac{1}{(1−t)^{m−2}} ). Thus deg_{G}^{m−1}[M/fM] =

lim_{t→1}(1 − t)^{m−1} · (1 − t^{|f|})[M] = lim_{t→1} \frac{(1−t^{|f|})}{1−t} · (1 − t)^{m}[M] = |f| deg_{G}^{m} M. □

Our last result follows by repeated use of this lemma.

Proposition 9.5. Let M be a finitely generated graded RG-module of dimension m and suppose that f_{1}, …, f_{r} ∈ R is an M-regular sequence of homogeneous elements. Then

deg_{G}^{m} M = ∏ |f_{i}| · deg_{G}^{m−r}(M/(f_{1}, …, f_{r})M).

References

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[2] D.J. Benson, Representations and Cohomology II, Cambridge studies in advanced mathematics 30 (1995), Cambridge Univ. Press, Cambridge.

[3] D.J. Benson, private communication, 1996.

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[5] T. Diethelm, U. Stammbach, On the module structure of the mod p cohomology of a p-group, Arch. Math. (Basel) 43 (1984), 488–492.

[6] D. Gorenstein, Finite Groups, Harper & Row, New York-London 1968.

[7] J. Harris, N. Kuhn, Stable decompositions of classifying spaces of finite abelian p-groups, Math. Proc. Cambridge Philos. Soc. 103 (1988), 427–449.

[8] R. Hartshorne, Algebraic Geometry, Springer, New York 1977.

[9] D. Karagueuzian, P. Symonds, The module structure of a group action on a polynomial ring: a finiteness theorem, Journal Amer. Math. Soc. 20 (2007), 931-967.

[10] D. Karagueuzian, P. Symonds, The module structure of a group action on a polynomial ring: examples, generalizations and applications, CRM Proceedings and Lecture Notes 35 (2004) 139–158.

[11] N. Kuhn, The nilpotent filtration and the action of automorphisms on the cohomology of finite p-groups, to appear in Math. Proc. Cambridge Philos. Soc..

[12] D. Quillen, The spectrum of an equivariant cohomology ring. I, II. Ann. of Math. (2) 94 (1971), 549–572; ibid. (2) 94 (1971), 573–602.

[13] P. Symonds, The action of automorphisms on the cohomology of a p-group, Math. Proc. Cambridge Philos. Soc. 127 (1999), 495–496.

[14] P. Symonds, Group actions on polynomial and power series rings, Pacific J. Math. 195 (2000), 225–230.

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Technische Universität München, Zentrum Mathematik - M11, Boltzmannstr. 3, 85748 Garching, Germany

 E-mail address: himstedt@ma.tum.de

School of Mathematics, University of Manchester, Manchester M13 9PL, United Kingdom

 E-mail address: Peter.Symonds@manchester.ac.uk