IWR-Preprint 97-38, Heidelberg 1997

On the Cohen-Macaulay Property of Modular Invariant Rings

Gregor Kemper^{∗}

IWR, Universität Heidelberg, Im Neuenheimer Feld 368 69 120 Heidelberg, Germany

                     email Gregor.Kemper@iwr.uni-heidelberg.de

October 31, 1997

Abstract If V is a faithful module for a finite group G over a field of characteristic p > 0, then the ring of invariants need not be Cohen-Macaulay if p divides the order of G. In this article the cohomology of G is used to study the question of Cohen-Macaulayness of the invariant ring.

Let R = S(V^{∗}) be the polynomial ring on which G acts. Then the main result can be stated as follow: If H^{r}(G, R) ≠ 0 for an r > 0 and if all σ ∈ G of order p have rank(σ − 1) ≥ r + 2, then the invariant ring R^{G} is not Cohen-Macaulay. A corollary is that if p divides the order of G, then the ring of vector invariants of sufficiently many copies of V is not Cohen-Macaulay. A further result is that if G is a p-group and R^{G} is Cohen-Macaulay, then G is a bireflection group, i.e., it is generated by elements σ with rank(σ − 1) ≤ 2.

Introduction

Let G ≤ GL(V) be a finite group acting on a vector space V of dimension n over a field K. Then G acts on the symmetric algebra R = S(V^{∗}) of the dual of V, which is a polynomial ring over K, and we consider the invariant ring R^{G}. By the Noether normalization lemma, there exist homogeneous f_{1}, …, f_{n} ∈ R^{G} such that R^{G} is finitely generated as a module over A = K[f_{1}, …, f_{n}]. R^{G} is called Cohen-Macaulay if it is a free module over A. This is independent of the choice of the set {f_{1}, …, f_{n}}. An equivalent condition is that f_{1}, …, f_{n} form an R^{G}-regular sequence (see the beginning of Section 1). R^{G} is always Cohen-Macaulay if the characteristic p of K does not divide the order of G. If, however, |G| is a multiple of p (which we call the modular case), then R^{G} is in general not Cohen-Macaulay. At the moment, the knowledge about which linear groups in the modular case have invariant rings which are Cohen-Macaulay and which ones do not is very sketchy, to say the least. For references relevant to this question, we refer the reader to the books by Smith [8] and Benson [3], which also provide introductory texts on invariant theory of finite groups.

In this paper we show that for large classes of modular linear groups, the invariant ring is not Cohen-Macaulay. As a technique to derive these results, the cohomology of G with values in R is used to obtain criteria for a sequence a_{1}, …, a_{m} ∈ R^{G} to be R^{G}-regular. In particular, if the annihilator in R^{G} of a nonzero element in a certain cohomology group is large enough, this implies that the invariant ring is not Cohen-Macaulay. This is the content of the first section of the paper. In Section 2, some geometric arguments are used to prove that the annihilator mentioned above is large enough in many cases. This leads to the main result (Theorem 2.3) and the corollary that in the modular case the ring of sufficiently large vector invariants is not Cohen-Macaulay. The latter statement confirms a conjecture made by the author in a talk given in April 1996. As a further application we get the result that for certain representations of symmetric groups, the invariant ring is not Cohen-Macaulay. These representations include the irreducible reflection representation of degree n − 2 of the symmetric group G = S_{n} on n letters, where p ≥ 5 divides n, and n > 5. It is also possible to use Theorem 2.3 to derive results on cohomology from the knowledge of invariant rings. For example, the fact that the symmetric and alternating groups S_{n} and A_{n} have no non-split central extension with kernel of order p ≥ 5 becomes a consequence of the well-known fact that the invariant rings of S_{n} and A_{n} (with the usual permutation representation) are Cohen-Macaulay (see Example 2.7(a)). In the third section we restrict our attention to the first cohomology group with values in K. This permits a more accurate analysis of the geometry involved, which leads to the second criterion for non-Cohen-Macaulayness (Theorem 3.6). The result that a p-group G is generated by bireflections if its invariant ring is Cohen-Macaulay arises as a corollary. This is remarkable since it yields a special case of a theorem by Kac and Watanabe [6], but under a much weaker hypothesis (see Remark 3.8). Refining the methods a little bit more, we recover one of the results in Nakajima [7], which consists of a further series of reflection groups whose invariant rings are not Cohen-Macaulay.

∗The author thanks Ian Hughes, Eddy Campbell, Jim Shank, and David Wehlau for their hospitality during his visit to Queen’s University in Kingston, Ontario, where this paper was prepared.

Apart from producing classes of groups whose invariant rings are not Cohen-Macaulay, the methods developed in this article provide a means to analyze the Cohen-Macaulay property of invariant rings. In fact, every example of a non-Cohen-Macaulay invariant ring known to the author can be understood in terms of these methods.

Smith [9] took an approach to the question of depth and Cohen-Macaulayness of modular invariant rings which uses cohomology of G with values in a certain Koszul complex. Although his paper makes heavy use of spectral sequences and this article does not, some of the methods are slightly similar. However, the results of both papers are almost disjoint.

This paper was written during a visit of the author to Queen’s University in Kingston, Ontario. I would like to express my thanks to Ian Hughes, Eddy Campbell, Jim Shank, and David Wehlau for many conversations which inspired this work, and for the stimulating atmosphere which they created. In particular, I am indebted to Jim Shank and Ian Hughes for sharing the ideas which lead to Proposition 3.4 and Example 3.10. I also thank David Benson, Kay Magaard and Jürgen Müller for very fruitful conversations.

1 Regular Sequences and Cohomology

In this section, let R be a Noetherian commutative ring with 1 and let G ≤ Aut(R) be a group of automorphisms of R. We write R^{G} for the invariant ring. A sequence a_{1}, …, a_{m} ∈ R is called R-regular if (a_{1}, …, a_{m}) ≠ R and a_{i} is not a zero divisor on R/(a_{1}, …, a_{i−1}), for i = 1, …, m. We have the corresponding definition of R^{G}-regularity, where the ideals have to be taken in R^{G}.

The following proposition gives a cohomological criterion to decide whether a sequence a_{1}, …, a_{m} ∈ R^{G} which is R-regular is also R^{G}-regular. Before stating it, we recall the Koszul complex

0 ⟶ R ⟶^{∂_{m−1}} R^{m} ⟶^{∂_{m−2}} R^{\binom{m}{m−2}} ⟶^{∂_{m−3}} ⋯ ⟶^{∂_{3}} R^{\binom{m}{3}} ⟶^{∂_{2}} R^{\binom{m}{2}} ⟶^{∂_{1}} R^{m} ⟶^{∂_{0}} R  (1)

associated to a_{1}, …, a_{m}. If e_{1}, …, e_{m} is a basis for R^{m}, then ∂_{0} sends e_{i} to a_{i}, and if e_{i,j} for 1 ≤ i < j ≤ m is a basis for R^{\binom{m}{2}}, then ∂_{1}(e_{i,j}) = a_{j}e_{i}−a_{i}e_{j}. Furthermore, ∂_{m−1}(1) = ϵ_{1}a_{1}e_{1}+⋯+ϵ_{m}a_{m}e_{m} with ϵ_{i} ∈ {1, −1}.

Proposition 1.1. Let a_{1}, …, a_{m} ∈ R^{G} be an R-regular sequence. For k = 2, …, m, let M_{k} ⊆ R^{\binom{k}{2}} be the kernel of the map ∂_{1} of the Koszul complex associated to a_{1}, …, a_{k}. Then a_{1}, …, a_{m} is an R^{G}-regular sequence if and only if the maps

H^{1}(G, M_{k}) ⟶ H^{1}(G, R^{\binom{k}{2}})  (2)

induced by the embeddings M_{k} ⊆ R^{\binom{k}{2}} are injective for k = 2, …, m.

Proof. Since a_{1}, …, a_{m} is R-regular, the sequence (1) is exact (see, for example, Eisenbud [5, Corollary 17.5]). Also observe that all maps ∂_{i} are compatible with the G-action, since a_{i} ∈ R^{G}. By definition of M_{m}, we get an exact sequence

0 ⟶ M_{m} ⟶ R^{r} ⟶^{∂_{1}} R^{m} ⟶^{∂_{0}} R,  (3)

where r = \binom{m}{2}.

Let us assume that a_{1}, …, a_{m} is R^{G}-regular. By induction, it suffices to prove that

H^{1}(G, M_{m}) ⟶ H^{1}(G, R^{r})

is injective. So take a g ∈ H^{1}(G, M_{m}) which is mapped to 0. This means that there exists b ∈ R^{r} such that g_{σ} = (σ − 1)b for all σ ∈ G, where (g_{σ})_{σ∈G} ∈ Z^{1}(G, M_{m}) is a cocycle representing g. We must show that g = 0. Take u = ∂_{1}(b), then ∂_{0}(u) = 0 by the exactness of (3), i.e., we get a linear relation for the a_{i} with coefficients in R. Furthermore, (σ − 1)(u) = ∂_{1}(g_{σ}) = 0 for all σ ∈ G, since g_{σ} ∈ M_{m}, hence u lies in (R^{G})^{m}. Now by the R^{G}-regularity of a_{1}, …, a_{m}, the Koszul complex built over R^{G} is exact, hence there exists f ∈ (R^{G})^{r} with u = ∂_{1}(f). It follows that ∂_{1}(b − f) = 0, hence b − f = h with h ∈ M_{m} by the exactness of (3). But now

(σ − 1)h = (σ − 1)(b − f) = (σ − 1)b = g_{σ} for all σ ∈ G,

so g = 0 as an element of H^{1}(G, M_{m}).

For the converse, assume that the maps in (2) are injective. First we see that (a_{1}, …, a_{m})_{R^{G}} ≠ R^{G}, since by assumption 1 does not lie in the R-ideal spanned by the a_{i}. Suppose that u_{1}a_{1} + ⋯ + u_{m}a_{m} = 0 with u_{i} ∈ R^{G}, then we must show that u_{m} lies in the R^{G}-ideal spanned by a_{1}, …, a_{m−1}, and by induction this will also suffice. With u = u_{1}e_{1} + ⋯ + u_{m}e_{m}, the above equation becomes ∂_{0}(u) = 0, hence there exists b ∈ R^{r} such that u = ∂_{1}(b). The invariance of u implies ∂_{1}((σ − 1)b) = 0 for all σ ∈ G, so (σ − 1)b = g_{σ} with g_{σ} ∈ M_{m}. Hence (g_{σ})_{σ∈G} is a cocycle, and its class in H^{1}(G, M_{m}) lies in the kernel of the map H^{1}(G, M_{m}) ⟶ H^{1}(G, R^{r}). By assumption, there exists h ∈ M_{m} such that g_{σ} = (σ − 1)h for σ ∈ G. Now we set f = b − h, hence ∂_{1}(f) = ∂_{1}(b) = u, which in particular implies u_{m} = −(f_{1,m}a_{1} + ⋯ + f_{m−1,m}a_{m−1}), where f_{i,j} are the coefficients of f ∈ R^{r}. But

(σ − 1)f = (σ − 1)b − (σ − 1)h = g_{σ} − g_{σ} = 0 for all σ ∈ G,

which means that the f_{i,j} are invariants. This completes the proof.

We use the above proposition to prove

Theorem 1.2. Suppose that r ≥ 0 is an integer and assume that H^{i}(G, R) = 0 for 1 ≤ i < r. (This assumption is void if r ≤ 1.) Then any sequence in R^{G} of length ≤ r +1 which is R-regular is also R^{G}-regular. Furthermore, an R-regular sequence a_{1}, …, a_{r+2} ∈ R^{G} is R^{G}-regular if and only if the map

H^{r}(G, R) ⟶ H^{r}(G, R^{r+2})  (4)

induced by the multiplication with a_{1}, …, a_{r+2} is injective.

Proof. Let a_{1}, …, a_{m} ∈ R^{G} be R-regular, with 1 ≤ m ≤ r + 2. We first treat a few special cases. If m = 1, then the sequence is clearly also R^{G}-regular. If m = 2, then the module M_{m} from Proposition 1.1 is 0, hence the map (2) is injective and the sequence is R^{G}-regular. If also r = 0, then the map (4) is always injective, which establishes the claimed equivalence in that case. Furthermore, suppose m = 3 and r = 1. Then M_{m} is the image of R under the (injective) map ∂_{2} = ∂_{m−1} from (1), hence the map (4) is up to signs equal to the map (2). This reduces the theorem in this case to Proposition 1.1.

Now we assume that r > 1. Then by assumption H^{1}(G, R) = 0, so the injectivity conditions in Proposition 1.1 are satisfied if and only if H^{1}(G, M_{k}) = 0 for k = 2, …, m. Hence we have to show that H^{1}(G, M_{m}) = 0 for 2 ≤ m ≤ r + 1 and that H^{1}(G, M_{r+2}) = 0 if and only if the map (4) is injective. We first prove by induction on k that for 1 ≤ k ≤ min{r − 1, m − 1}, H^{1}(G, M_{m}) is isomorphic to H^{k}(G, ker(∂_{k})), where the ∂_{k} are the maps from the Koszul complex (1). In fact, from (1) we get the short exact sequence

0 ⟶ ker(∂_{k}) ⟶ R^{\binom{m}{k+1}} ⟶^{∂_{k}} ker(∂_{k−1}) ⟶ 0,

which gives rise to the exact sequence

0 = H^{k−1}(G, R^{\binom{m}{k+1}}) ⟶ H^{k−1}(G, ker(∂_{k−1})) ⟶ H^{k}(G, ker(∂_{k})) ⟶ H^{k}(G, R^{\binom{m}{k+1}}) = 0

of cohomology, which proves the claim.

Now if m ≤ r, then we have shown that H^{1}(G, M_{m}) ≅ H^{m−1}(G, ker(∂_{m−1})), but ker(∂_{m−1}) = 0. Hence H^{1}(G, M_{k}) = 0 in this case. If m = r + 1, then H^{1}(G, M_{m}) ≅ H^{r−1}(G, ker(∂_{m−2})) ≅ H^{r−1}(G, R) = 0. Finally, if m = r + 2, then H^{1}(G, M_{m}) ≅ H^{r−1}(G, ker(∂_{m−3})), and the short exact sequence

0 ⟶ R ⟶^{∂_{m−1}} R^{m} ⟶^{∂_{m−2}} ker(∂_{m−3}) ⟶ 0

gives rise to the exact sequence

0 = H^{r−1}(G, R^{m}) ⟶ H^{r−1}(G, ker(∂_{m−3})) ⟶ H^{r}(G, R) ⟶^{φ} H^{r}(G, R^{m}),

where φ is up to signs induced by multiplication with a_{1}, …, a_{m}. Hence for m = r+2, H^{1}(G, M_{m}) ≅ H^{r−1}(G, ker(∂_{m−3})) is 0 if and only if the map (4) is injective, which was to be shown.

We now change our point of view by fixing an element from H^{r}(G, R) and considering its annihilator, which is an ideal in R^{G}. We need some more terminology and a few facts from commutative algebra. For an ideal I ◁ R the maximal length of an R^{G}-regular sequence whose elements lie in I is denoted by depth_{R}(I), and ht(I) denotes the height of the ideal, which is the minimal height of a prime ideal containing I. Furthermore, a sequence a_{1}, …, a_{m} ∈ R is said to be a partial system of parameters if (a_{1}, …, a_{m}) ≠ R and ht(a_{1}, …, a_{k}) = k for k = 1, …, m.

Lemma 1.3. Let a_{1}, …, a_{m} ∈ R such that (a_{1}, …, a_{m}) ≠ R. Then the following statements hold:

(a) The sequence a_{1}, …, a_{m} is a partial system of parameters if and only if a_{i} lies in none of the associated prime ideals 𝔭 ◁ R of (a_{1}, …, a_{i−1}) with ht(𝔭) = i − 1, for i = 1, …, m.

(b) The sequence a_{1}, …, a_{m} is R-regular if and only if a_{i} lies in none of the associated prime ideals of (a_{1}, …, a_{i−1}), for i = 1, …, m. In particular, if a_{1}, …, a_{m} is R-regular, then it is a partial system of parameters.

(c) If R is Cohen-Macaulay and a_{1}, …, a_{m} is a partial system of parameters, then it is R-regular.

(d) If I ◁ R is an ideal of height m, then there exist a_{1}, …, a_{m} ∈ I which are a partial system of parameters.

(e) If R ⊆ S is an integral extension of rings and I ◁ R, then ht((I)_{S}) = ht(I), where (I)_{S} denotes the ideal in S generated by I. In particular, if a_{1}, …, a_{m} is a partial system of parameters in R, it is also one in S.

Proof. Clearly if a_{i} ∈ 𝔭 for an associated prime ideal 𝔭 of (a_{1}, …, a_{i−1}) of height i − 1, then ht(a_{1}, …, a_{i}) ≤ i − 1. Conversely, if ht(a_{1}, …, a_{i}) = ht(a_{1}, …, a_{i−1}) = i − 1 for some i, then there exists a prime ideal of height i − 1 containing (a_{1}, …, a_{i}). This prime must then be a minimal prime containing (a_{1}, …, a_{i−1}) and is hence an associated prime of (a_{1}, …, a_{i−1}) (see Eisenbud [5, Theorem 3.1]). This proves (a). The same theorem in [loc. cit.] says that the set of zero divisors of R/(a_{1}, …, a_{i−1}) is the union of the associated primes of (a_{1}, …, a_{i−1}), from which (b) follows immediately. Now (c) follows from the unmixedness theorem (see [loc. cit., Corollary 18.14]). To prove (d), we assume that a_{1}, …, a_{i−1} ∈ I with ht(a_{1}, …, a_{i−1}) have already been found. Then there exists a_{i} ∈ I which lies in none of the associated primes of (a_{1}, …, a_{i−1}) of height i − 1, since otherwise I would be contained in one of these prime ideals by the prime avoidance lemma (see [loc. cit., Lemma 3.3]), and hence ht(I) ≤ i − 1. By (a), this leads to a partial system of parameters.

To prove (e), let 𝔭 ◁ R be a prime of minimal height m containing I, and let 𝔭_{0} ⫋ … ⫋ 𝔭_{m} = 𝔭 be an ascending chain of primes. By the going-up theorem (see [loc. cit, Proposition 4.15]), there exists a chain 𝔮_{0} ⊆ … ⊆ 𝔮_{m} of primes 𝔮_{i} ◁ S with 𝔮_{i} ∩ R = 𝔭_{i}, and by [loc. cit., Corollary 4.18], this chain cannot be refined. Since 𝔮_{m} contains (I)_{S}, ht((I)_{S}) ≤ m. For the reverse inequality, let 𝔮_{0} ⫋ … ⫋ 𝔮_{r} be an ascending chain of primes in S with (I)_{S} ⊆ 𝔮_{r}, r = ht((I)_{S}), and set 𝔭_{i} = 𝔮_{i} ∩R. Then by [loc. cit., Corollary 4.18], 𝔭_{0} ⫋ … ⫋ 𝔭_{r}, and I ⊆ (I)_{S} ∩ R ⊆ 𝔭_{r}. This shows ht(I) ≤ r.

With these facts, we can now deduce the following corollary from Theorem 1.2.

Corollary 1.4. Assume that R is Cohen-Macaulay and G is finite, and that H^{i}(G, R) = 0 for 1 ≤ i < r, where r > 0 is an integer. Let g ∈ H^{r}(G, R) be nonzero. Then for

I = Ann_{R^{G}}(g) := {a ∈ R^{G} | a · g = 0} ◁ R^{G}

we have

depth_{R^{G}}(I) = min{r + 1, ht(I)}.

In particular, R^{G} is not Cohen-Macaulay if ht(I) > r + 1.

Proof. Assume that there exist a_{1}, …, a_{r+2} ∈ I which form an R^{G}-regular sequence. By Lemma 1.3(b), the a_{i} are a partial system of parameters in R^{G}. So by (e) and the finiteness of G, they are also a partial system of parameters in R, hence the a_{i} form an R-regular sequence by (c). But since the a_{i} lie in I, g lies in the kernel of the map (4) from Theorem 1.2. Since g ≠ 0, it follows that a_{1}, …, a_{r+2} is in fact not R^{G}-regular. This proves that depth_{R^{G}}(I) ≤ r + 1. Also clearly depth_{R^{G}}(I) ≤ ht(I).

By Lemma 1.3(d), there exists a partial system of parameters a_{1}, …, a_{m} of length m := ht(I) with a_{i} ∈ I. Let m′ = min{r + 1, m}. Then a_{1}, …, a_{m′} is R-regular, and by Theorem 1.2, it is also R^{G}-regular. Hence depth_{R^{G}}(I) ≥ m′. If m > r + 1, then a_{1}, …, a_{r+2} is a partial system of parameters which is not an R^{G}-regular sequence, hence R^{G} is not Cohen-Macaulay by Lemma 1.3(c).

In the above corollary the cohomology group H^{r}(G, R) is regarded as a module over R^{G}, and a non-empty statement can be made if I ≠ 0, i.e., if the element g ∈ H^{r}(G, R) under consideration is a torsion element. If R is an integral domain with field of fractions Quot(R), then the kernel of the map

H^{r}(G, R) ⟶ H^{r}(G, Quot(R))

consists exactly of the torsion elements. But it is well known that H^{r}(G, Quot(R)) = 0. In fact, by the normal basis theorem Quot(R) is isomorphic to the regular module over Quot(R^{G}). Hence H^{r}(G, R) is a torsion module. We will make use of this in the next section. However, we will need more precise information on the annihilators than is provided by the above argument.

Lemma 1.5. Suppose that U is a finitely generated KG-module and let g ∈ H^{r}(G, U) with r > 0. Let W = KG be the regular module and a = ∑_{σ∈G} σ ∈ W^{G}. Then a ⊗ g = 0 as an element of H^{r}(G, W ⊗ U).

Proof. We first observe that H^{r}(G, W) = 0. This can be seen by Shapiro’s lemma, for example. It follows that H^{r}(G, P) = 0 for any projective module P. But W ⊗ U is the tensor product of a projective module and another module, hence it is projective (see, for example, Alperin [1, Section 7, Lemma 4]). So H^{r}(G, W ⊗ U) = 0.

2 Invariant Rings

In this section, we specialize the assumptions by looking at the standard situation of invariant theory of finite groups: K is a field, V is a finite dimensional vector space over K, and R = S(V^{∗}) is the symmetric algebra of the dual of V, which is isomorphic to a multivariate polynomial ring. Furthermore, G ≤ GL(V) is a finite linear group on V, which has a natural action on R. As in Section 1, we write R^{G} for the invariant ring. Furthermore, let p be the characteristic of K, which may be 0.

In order to use Lemma 1.5 for finding elements a ∈ R^{G} which annihilate a given g ∈ H^{r}(G, R), we have to recover (copies of) the regular module in R. This is done in the next lemma, where we assume that K is algebraically closed, which allows us to view the elements of R as functions on V. We write V^{σ} ≤ V for the fixed space of a σ ∈ G, and Stab_{G}(v) for the stabilizer of a v ∈ V.

Lemma 2.1. Assume that K is algebraically closed, let m ∈ {1, …, dim(V)} be an integer and suppose that every element σ ∈ G of order p has rank(σ − 1) ≥ m. (This assumption is void if p = char(K) = 0.) Then there exist m embeddings

φ_{i}: KG ↪ R (i = 1, …, m)

of the regular KG-module into R such that the polynomials

a_{i} = φ_{i} (∑_{σ∈G} σ) (i = 1, …, m)

form a partial system of parameters in R^{G}.

Proof. Suppose by induction that φ_{1}, …, φ_{k−1} have already been constructed for a k ∈ {1, …, m}. By assumption, the set

X = \{v ∈ V | Stab_{G}(v) has an order divisible by p\} = ⋃_{\begin{matrix}σ∈G, \\ ord(σ)=p \end{matrix}} V^{σ}

has dimension ≤ n − m, where n = dim_{K}(V). But every associated prime 𝔭 ◁ R of (a_{1}, …, a_{k−1}) has height k − 1 and Krull dimension n − k + 1, which is greater than n − m. Hence there exists a point w_{𝔭} ∈ V_{V} (𝔭) \ X ⊆ V for every such 𝔭, where V_{V} (𝔭) denotes the variety in V given by 𝔭, and the w_{𝔭} can be chosen such that w_{𝔭} ≠ σ(w_{𝔭′}) for 𝔭 ≠ 𝔭′ and σ ∈ G. Furthermore, we can choose a point v_{0} ∈ V such that the set

{σ(v_{0}) | σ ∈ G} ∪ {w_{𝔭} | 𝔭 ∈ Ass(a_{1}, …, a_{k−1})} ∪ {0}

has exactly |G| + | Ass(a_{1}, …, a_{k−1})| + 1 (distinct) elements. In fact, v_{0} has to avoid the points 0 and σ(w_{𝔭}) for σ ∈ G, and the finite union ∪_{σ∈G\{id}}V^{σ} of proper subspaces. Now there exists a polynomial g ∈ R with the following properties (where δ is the Kronecker-delta):

(i) g(σ(v_{0})) = δ_{σ,id},

(ii) g(σ(w_{𝔭})) = δ_{σ(w_{𝔭}),w_{𝔭}},

(iii) g(0) = 0.

We define φ_{k}: KG → R by setting φ_{k}(σ) = σ(g). This is clearly a G-map. To prove that it is injective, suppose that

∑_{σ∈G} α_{σ} · σ(g) = 0

with α_{σ} ∈ K. For τ ∈ G, evaluation at τ(v_{0}) yields

0 = ∑_{σ∈G} α_{σ} · σ(g)(τ(v_{0})) = ∑_{σ∈G} α_{σ} · g(σ^{−1}τv_{0}) = α_{τ} .

Hence the σ(g) are linearly independent.

The polynomial a_{k} defined in the statement of the lemma is clearly an invariant, and 1 ∉ (a_{1}, …, a_{k}) by the property (iii) of h. Evaluating a_{k} at w_{𝔭} yields

a_{k}(w_{𝔭}) = ∑_{σ∈G} σ(g)(w_{𝔭}) = ∑_{σ∈G} g(σ^{−1}(w_{𝔭})) = | Stab_{G}(w_{𝔭})| ≠ 0,

since w_{𝔭} ∉ X. Because w_{𝔭} ∈ V_{V} (𝔭), this means that a_{k} lies in none of the associated prime ideals of (a_{1}, …, a_{k−1}), which by Lemma 1.3(a) shows that a_{1}, …, a_{k} is a partial parameter system. This completes the proof.

Remark 2.2. If |G| is a multiple of p, then by Benson [2, Theorem 4.1.3] H^{r}(G, K) ≠ 0 for some r > 0. In fact, we see from the proof in [2] that if σ ∈ G is an element of order p and if the index of ⟨σ⟩ in its normalizer is p^{a}h with p ∤ h, then r can be chosen as 2(p − 1)p^{a}. Since K occurs as the direct summand S^{0}(V^{∗}) in R, it follows that H^{r}(G, R) ≠ 0.

Putting the various strands together, we obtain

Theorem 2.3. Suppose that H^{r}(G, R) ≠ 0 for an integer r > 0 and that every element σ ∈ G of order p has rank(σ − 1) ≥ r + 2. Then R^{G} is not Cohen-Macaulay.

Proof. We may assume that r > 0 is minimal with H^{r}(G, R) ≠ 0. Furthermore, since H^{r}(G, R) ≠ 0, p must divide the order of G, hence there exist elements σ ∈ G of order p. By the assumption it follows that n := dim(V) ≥ r + 2. Assume that R^{G} is Cohen-Macaulay. Then R^{G} is a free module over the algebra K[a_{1}, …, a_{n}] generated by a homogeneous parameter system. If K̄ is the algebraic closure of K, then it follows that K̄ ⊗_{K} R^{G} is free over K̄[a_{1}, …, a_{n}], hence K̄ ⊗_{K} R^{G} is also Cohen-Macaulay. So we can assume that K is algebraically closed. Then by Lemma 2.1 there are m := r + 2 embeddings φ_{1}, …, φ_{m} of the regular module KG into R, and the images contain invariants a_{i} ∈ R^{G} which form a partial system of parameters of length m. Now take a nonzero g ∈ H^{r}(G, R). Then by Lemma 1.5, a_{i} ⊗ g = 0 as elements in H^{r}(G, R ⊗ R). Applying the map H^{r}(G, R ⊗ R) → H^{r}(G, R) induced by R ⊗ R → R, f ⊗ g ↦ fg yields that a_{i}g = 0 in H^{r}(G, R), hence the a_{i} lie in the annihilator I of g. It follows that ht(I) ≥ m > r + 1, so R^{G} is not Cohen-Macaulay by Corollary 1.4.

We obtain the following result on vector invariants.

Corollary 2.4. Suppose that |G| is a multiple of p. Then there exists an m ∈ ℕ such that S((V^{k})^{∗})^{G} is not Cohen-Macaulay for k ≥ m. Here V^{k} denotes the direct sum of k copies of V, and S((V^{k})^{∗}) is the symmetric algebra of its dual. In particular, there exists a KG-module W such that S(W^{∗})^{G} is not Cohen-Macaulay.

Proof. By Remark 2.2, there exists an r > 0, such that H^{r}(G, K) ≠ 0. Then H^{r}(G, S((V^{k})^{∗})) ≠ 0 for all k ∈ ℕ. Now if k ≥ r + 2, then and every σ ∈ G with σ ≠ id acts on V^{k} with rank_{V^{k}}(σ − 1) ≥ r + 2. So the assertion follows from Theorem 2.3.

Remark. As we see by the above proof, one can take m = 3 if G contains a normal subgroup of index p, since this implies the existence of a nonzero additive character G → K, or, equivalently, a nonzero element in H^{1}(G, K). This generalizes one of the results in Campbell et al. [4].

Corollary 2.5. Suppose that p = char(K) ≥ 5 and that G acts as a transitive permutation group on a basis e_{1}, …, e_{n} of a vector space W over K, where n is a multiple of p. If V is the quotient module W/K · (e_{1} + ⋯ + e_{n}), then R^{G} = S(V^{∗})^{G} is not Cohen-Macaulay.

Furthermore, if G contains a transitive cyclic subgroup, n > 5, and V_{0} is the kernel of the trace map

π: V → K, ∑_{i=1}^{n} α_{i}e_{i} + K · (e_{1} + ⋯ + e_{n}) ↦ ∑_{i=1}^{n} α_{i},

then also S(V_{0}^{∗})^{G} is not Cohen-Macaulay.

Proof. Consider the exact sequence

0 ⟶ K ⟶ W ⟶ V ⟶ 0.

By the transitivity of G, a G-map from W into K must assign the same value to all e_{i}. Composing this with the map K → W yields the zero-map K → K. Hence the sequence is non-split. Dualizing gives a non-split extension of K by V^{∗}, which shows that H^{1}(G, V^{∗}) ≠ 0, hence H^{1}(G, R) ≠ 0. Now consider the exact sequence

0 ⟶ K ⟶ W_{0} ⟶ V_{0} ⟶ 0,

where W_{0} is the kernel of the trace map, and assume there exists a σ_{0} ∈ G with σ_{0}(e_{i}) = e_{i+1}, where the indices are taken modulo n. Then a G-map W_{0} → K must take the same value α on all e_{i} − e_{i+1}, hence the vector e_{1} + ⋯ + e_{n} = ∑_{i=1}^{n} i · (e_{i} − e_{i+1}) is mapped to \binom{n+1}{2}α = 0. As above, the sequence is non-split, and we obtain H^{1}(G, V_{0}^{∗}) ≠ 0.

The proof is complete if we can show that rank_{V} (σ − 1) and rank_{V_{0}}(σ − 1) are at least 3 for any element σ ∈ G of order p. We can assume that the disjoint cycle representation of σ contains the cycle (1, 2, …, p), and will show that (σ − 1)(e_{1}), (σ − 1)(e_{2}), (σ − 1)(e_{3}) are linearly independent in V. Indeed, taking a linear combination with coefficients α_{1}, α_{2}, α_{3} ∈ K, we obtain

α_{1}(e_{2} − e_{1}) + α_{2}(e_{3} − e_{2}) + α_{3}(e_{4} − e_{3}) = α(e_{1} + ⋯ + e_{n}),

from which α = α_{3} = α_{2} = α_{1} = 0. Next we show that (σ−1)(e_{2}−e_{1}), (σ−1)(e_{3}−e_{2}), (σ−1)(e_{4}−e_{3}) are linearly independent in V_{0} if n > 5. Here we obtain

α_{1}(e_{3} − 2e_{2} + e_{1}) + α_{2}(e_{4} − 2e_{3} + e_{2}) + α_{3}(e_{5} − 2e_{4} + e_{3}) = α(e_{1} + ⋯ + e_{n}),

so again all α_{i} are zero.

Example 2.6. If n is a multiple of p and p ≥ 5, then the symmetric group S_{n} is an example of the type dealt with in Corollary 2.5. With the notation from the corollary, we get the result that S(V^{∗})^{S_{n}} is not Cohen-Macaulay, and neither is S(V_{0}^{∗})^{S_{n}} if n > 5. S_{n} acts on both V and V_{0} as a reflection group. Thus we have found an infinite series of finite reflection groups whose invariant rings are not even Cohen-Macaulay. Another such series, which consists of abelian pgroups, was given by Nakajima [7] (see Example 3.10 below). In our example, the action of S_{n} on V_{0} is irreducible for n > 5. What is quite surprising is that although S(V^{∗})^{S_{n}} is not Cohen-Macaulay, the invariant ring S(V)^{S_{n}} of the dual representation is a polynomial ring. In fact it is easily seen that S(V)^{S_{n}} is generated by the images of the elementary symmetric polynomials s_{2}(e_{1}, …, e_{n}), …, s_{n}(e_{1}, …, e_{n}) ∈ S(W) in S(V).

It is sometimes possible to read Theorem 2.3 “backwards” to obtain lower bounds on r > 0 such that H^{r}(G, R) ≠ 0. This leads to an example where easy facts from invariant theory can be used to obtain non-trivial statements of group theory.

Example 2.7. Suppose that G = S_{n} or G = A_{n} is the symmetric or alternating group on n letters. We look at several permutation representations of G.

(a) First, let V be the natural permutation module, and p = char(K) ≥ 3. (We do not assume that p divides n.) The invariant ring R^{G} is Cohen-Macaulay. In fact, it is isomorphic to a polynomial ring if G = S_{n}, and a hypersurface of G = A_{n}. For an element σ ∈ G of order p we have rank(σ − 1) ≥ p − 1. It now follows by Theorem 2.3 that H^{r}(G, R) = 0 for 0 < r ≤ p − 3. In particular, H^{r}(G, K) = 0 for such r. Thus the fact that S_{n} and A_{n} have no non-split central extension with kernel of order p ≥ 5 can easily be derived from Theorem 2.3.

(b) Now suppose that V is a direct sum of m copies of the natural permutation module of G (m ∈ ℕ). In order to calculate the cohomology of R, we look at a decomposition of R into a direct sum of KG-modules, which will yield a decomposition of H^{r}(G, R). Such a decomposition is given by taking the submodules of R spanned by G-orbits of monomials in the variables x_{i,j} (1 ≤ i ≤ m, 1 ≤ j ≤ n), which are a basis of V^{∗} on which G acts by σ(x_{i,j}) = x_{i,σ(j)}. Each of these modules is induced from the trivial module over KH, where H is the stabilizer of a monomial. So by Shapiro’s lemma, the cohomology of G with values in the span of a monomial-orbit is equal to the cohomology of the stabilizer H of the monomial with values in K. But we see that such a stabilizer is either a direct product of symmetric groups (possibly on fewer letters) or the subgroup of even permutations contained in this product, so it has no normal subgroup of index p except for the case G = A_{3}, hence H^{1}(H, K) = 0 in all other cases. It follows that H^{1}(G, R) = 0 if G ≠ A_{3}. In fact, one can combine the arguments from parts (a) and (b) of this example to show that H^{r}(G, R) = 0 for 0 < r ≤ p − 3.

(c) Let V be as in (b) and assume that G = S_{p} or A_{p}. Take an element σ ∈ G of order p, then ⟨σ⟩ has an index divisible by p − 1 in its normalizer. It follows by Remark 2.2 that the first r > 0 with H^{r}(G, R) ≠ 0 is bounded from above by 2(p − 1). On the other hand, rank(σ − 1) = m(p − 1), so by Theorem 2.3, R^{G} is not Cohen-Macaulay if m ≥ 3. In view of (b), this yields an example where the higher cohomology modules are indeed needed.

3 The Geometric Picture

In this section we restrict our point of view drastically by only considering H^{1}(G, R) and most of the time only cocycles with values in K. Using H^{1}(G, R) means that we are looking for partial systems of parameters of length 3 which are not R^{G}-regular sequences. It is surprising how much can be said in spite of this narrowing of possibilities. The benefit of the restriction lies in a more accurate geometric description of the ideal I = Ann_{R^{G}}(g) occurring in Corollary 1.4.

We adopt the same notation as in the previous section, so V is a finite dimensional vector space over a field K of characteristic p, and G ≤ GL(V) is a finite linear group on V with the natural action on the symmetric algebra R = S(V^{∗}) of the dual. Furthermore, if X ⊆ V is a set of point, we write I_{R}(X) and I_{R^{G}}(X) for the ideals of all polynomials or invariants, respectively, which vanish on all points of X. If I ⊆ R is a set of polynomials, we write V_{V} (I) for the set of points in V where all f ∈ I vanish.

Proposition 3.1. Let g ∈ H^{1}(G, R) be nonzero, (g_{σ})_{σ∈G} ∈ Z^{1}(G, R) a cocycle representing g, and let

X = ⋃_{σ∈G} (V^{σ} \ V_{V} (g_{σ})) ⊆ V.

Then Ann_{R^{G}}(g) ⊆ I_{R^{G}}(X).

Proof. Take f ∈ I := Ann_{R^{G}}(g). Then there exists an h ∈ R such that f ·g_{σ} = (σ−1)h for all σ ∈ G. Hence if a point v ∈ V lies in V^{σ} \V_{V} (g_{σ}) for some σ, we obtain f(v)·g_{σ}(v) = h(σ^{−1}(v))−h(v) = 0, hence f(v) = 0. This shows that f ∈ I_{R^{G}}(X).

We are going to prove the reverse inclusion for the special case that the cocycle (g_{σ}) takes values in K. Before doing so, we present the following cautionary example.

Example 3.2. Suppose that G = ⟨σ⟩ is a cyclic group and we are interested in computing the ideal I ◁ R^{G} consisting of all (σ − 1)h ∈ R^{G} with h ∈ R. If v ∈ V^{σ}, then ((σ − 1)h)(v) = h(σ^{−1}(v)) − h(v) = 0. If on the other hand v lies in V \ V^{σ}, then there exists an h ∈ R which takes different values on v and on σ^{−1}(v), hence ((σ − 1)h)(v) ≠ 0. So one might be tempted to conclude that the radical ideal of I is exactly I_{R^{G}}(V^{σ}). But if K is of characteristic 0, then I must be the zero ideal, since (σ − 1)h = g ∈ R^{G} implies σ^{i}(h) = h + i · g, hence g = 0 or G would be infinite. So the conclusion \sqrt{I} = I_{R^{G}}(V^{σ}) is in general quite wrong.

It is surprising that in the situation of Proposition 3.5 we will obtain exactly the result that turned out to be false in the above example. In order to move on safe ground, we prove

Lemma 3.3. Suppose that K is algebraically closed and let A be a subalgebra of R such that R is finitely generated as a module over A. Then for an ideal I ⊴ A we have

\sqrt{I} = I_{A}(V_{V} (I)).

Proof. If f ∈ \sqrt{I}, then f^{k} ∈ I for some k ∈ ℕ, so for v ∈ V_{V} (I) we have f^{k}(v) = 0, hence f ∈ I_{A}(V_{V} (I)).

Conversely, suppose that f ∈ I_{A}(V_{V} (I)). Then f lies in all maximal ideals 𝔪 ◁ R in R containing I, since K is algebraically closed. Let 𝔭 ◁ A be a prime ideal in A containing I. Then by the goingup theorem, there exists a prime ideal 𝔮 ◁ R such that 𝔮 ∩ A = 𝔭. By Hilbert’s Nullstellensatz, 𝔮 is equal to the intersection of all maximal ideals in R containing 𝔮. But f lies in each of these maximal ideals, hence f ∈ 𝔮 and then also f ∈ 𝔭, since f is an invariant. We have shown that f lies in every prime ideal in A containing I, and by Krull’s intersection theorem, the intersection of all these prime ideals is the radical of I.

If g ∈ H^{1}(G, K), then the cocycle (g_{σ}) representing g is uniquely determined and is in fact a homomorphism from G into the additive group of K. Hence we can look at its kernel.

Proposition 3.4. Assume that K is algebraically closed and let g ∈ H^{1}(G, K) be nonzero with kernel N ◁ G. Let J ◁ R^{G} be the image of the relative transfer

Tr_{N}^{G} : R^{N} → R^{G}, f ↦ ∑_{i=1}^{r} σ_{i}(f),

where σ_{1}, …, σ_{r} is a system of coset representatives of N in G. Then

\sqrt{J} = I_{R^{G}}(X) with X = ⋃_{σ∈G\N} V^{σ}.

Proof. In view of Lemma 3.3, we must show that X = V_{V} (J). So take a point v ∈ V^{σ} for some σ ∈ G \ N. We have σ^{p} ∈ N, since g_{σ^{p}} = p · g_{σ} = 0. Let H ≤ G be the subgroup generated by N and σ, then for h ∈ R^{N} we have

Tr_{N}^{H}(h)(v) = ∑_{i=0}^{p−1} h(σ^{−i}(v)) = ∑_{i=0}^{p−1} h(v) = 0,

hence Tr_{N}^{G}(h)(v) = 0. This shows that v ∈ V_{V} (J).

Now suppose that v ∈ V \ X. An easy calculation shows that this implies that the N-orbits of σ_{i}(v) for i = 1, …, r are pairwise disjoint. Hence there exists an h ∈ R^{N} such that h(σ_{i}^{−1}(v)) = δ_{1,i}. It follows that Tr_{N}^{G}(h)(v) = 1, hence v ∉ V_{V} (J).

I owe the idea of the preceding proof to a conversation with Jim Shank. We now put Proposition 3.1 and Proposition 3.4 together.

Proposition 3.5. Assume that K is algebraically closed and let g ∈ H^{1}(G, K) be nonzero with kernel N ◁ G. Moreover, let I = Ann_{R^{G}}(g) be its annihilator. Then

\sqrt{I} = I_{R^{G}}(X) with X = ⋃_{σ∈G\N} V^{σ}.

Proof. The inclusion \sqrt{I} ⊆ I_{R^{G}}(X) was already shown in Proposition 3.1. So suppose that f ∈ I_{R^{G}}(X). Then by Proposition 3.4, f^{k} = Tr_{N}^{G}(h) with h ∈ R^{N} and k ∈ ℕ. Now G/N is embedded in K, so it must be an elementary abelian p-group. Take σ_{1}, …, σ_{m} ∈ G to be generators for this group, then

f^{k} = ∑_{i_{1},... ,i_{m}=0}^{p−1} σ_{1}^{i_{1}} ⋯ σ_{m}^{i_{m}}(h) = (σ_{1} − 1)^{p−1} ⋯ (σ_{m} − 1)^{p−1}(h),

where we used the polynomial identity 1 + X + ⋯ + X^{p−1} = (X − 1)^{p−1} over K. Form

h̃ = ∑_{i=1}^{m} g_{σ_{i}} · ((σ_{1} − 1)^{p−1} ⋯ (σ_{i−1} − 1)^{p−1}(σ_{i} − 1)^{p−2}(σ_{i+1} − 1)^{p−1} ⋯ (σ_{m} − 1)^{p−1}(h)) .

Since (σ_{i} − 1)^{p} yields zero when applied to an invariant under N, it follows that (σ_{i} − 1)h̃ = g_{σ_{i}} · f^{k}, and from that (σ − 1)h̃ = g_{σ} · f^{k} for any σ ∈ G. Hence f^{k} lies in I = Ann_{R^{G}}(g) and so f ∈ \sqrt{I}.

An automorphism σ ≠ id of a vector space is called a bireflection if rank(σ − 1) ≤ 2. In the case r = 1 of Theorem 2.3, the hypothesis is that G contains no bireflection of order p. With the help of Proposition 3.5, we can now weaken this hypothesis.

Theorem 3.6. Assume that G has a normal subgroup N of index p, which contains all bireflections in G. Then R^{G} is not Cohen-Macaulay.

Proof. As in the proof of Theorem 2.3, we can assume that K is algebraically closed. There is an element g ∈ H^{1}(G, K) with kernel N. Since all bireflections of G are contained in N, the codimension of all V^{σ} for σ ∈ G\N is at least 3. Hence for X = ∪_{σ∈G\N}V^{σ} we have ht(I_{R^{G}}(X)) ≥ 3. But by Proposition 3.5, I_{R^{G}}(X) is the radical of I = Ann_{R^{G}}(g), hence ht(I) ≥ 3, and the theorem follows from Corollary 1.4.

If G is not generated by bireflections, then the bireflections in G generate a proper normal subgroup. If G is a p-group, then this can be extended to a normal subgroup of index p. So we obtain

Corollary 3.7. If G is a p-group and R^{G} is Cohen-Macaulay, then G is generated by bireflections.

Remark 3.8. Kac and Watanabe proved in [6] that if the invariant ring of a finite linear group G is a complete intersection, then G is generated by bireflections. Since the complete intersection property implies the Cohen-Macaulay property (see Stanley [10]), we have recovered their result for the special case of p-groups. The remarkable thing is that in this case the much weaker hypothesis of Cohen-Macaulayness of the invariant ring suffices.

We can do better than Theorem 3.6 if we widen our point of view just very slightly by multiplying a 1-cocycle with values in K, as considered in Theorem 3.6, by an invariant from R^{G}. This leads to the following improvement.

Theorem 3.9. Suppose that G has a normal subgroup N with factor group an elementary abelian p-group, and suppose that there is a σ_{0} ∈ G \ N, σ_{0} not a bireflection, such that for all bireflections σ ∈ G \ N we have

V^{σ_{0}} ⊈ V^{σ}.

Then R^{G} is not Cohen-Macaulay.

Proof. As before, we can assume that K is algebraically closed. Write

X = ⋃_{σ∈G\N} V^{σ} and X′ = ⋃_{σ∈G\N, σ bireflection} V^{σ}.

Then the hypothesis says that X′ ⫋ X. Since X and X′ are G-stable, there exists an invariant h ∈ I_{R^{G}}(X′) \ I_{R^{G}}(X). Let I ◁ R be the ideal of the invariants which vanish on all fixed spaces V^{σ} for σ ∈ G \ N not a bireflection. Then ht(I) ≥ 3, hence by Lemma 1.3(d) there exist a_{1}, a_{2}, a_{3} ∈ I which form a partial system of parameters. We have h · a_{i} ∈ I_{R^{G}}(X).

There exists a g ∈ H^{1}(G, K) with kernel N. By Proposition 3.5, the h · a_{i} lie in the radical of the annihilator of g, hence h^{k} · a_{i}^{k} ∈ Ann_{R^{G}}(g) for some k ∈ ℕ. It follows that a_{i}^{k} ∈ Ann_{R^{G}}(g′) with g′ = h^{k} · g (i = 1, 2, 3). The proof is complete by Corollary 1.4 if we can show that g′ is nonzero. But that is equivalent to h^{k} ∉ Ann_{R^{G}}(g), which is true since h ∉ I_{R^{G}}(X) = \sqrt{Ann_{R^{G}}(g)}.

Clearly Theorem 3.6 cannot be used to show the non-Cohen-Macaulayness of R^{G} in the case that G is generated by bireflections. However, in the following example G is even generated by reflections, and we are able to prove that R^{G} is not Cohen-Macaulay by using Theorem 3.9.

Example 3.10. In [7], Nakajima gave the following groups as an example of reflection groups whose invariant rings are not Cohen-Macaulay. Let K be a finite field, m ≥ 3, n = 2m + 1, and consider the group G consisting of the n × n-matrices

( \begin{matrix}1 \\ & ... & & & & 0 \\ & & 1 \\ & & & 1 \\ α_{0} & & & α_{m} & 1 \\ & ... & & ⋮ & & ... \\ & & α_{m−1} & α_{m} & & & 1 \end{matrix} )

with α_{0}, …, α_{m} ∈ K. Our goal is to recover Nakajima’s result that R^{G} is not Cohen-Macaulay. Let N ◁ G be the subgroup consisting of all matrices with α_{m} = 0. For any bireflection σ ∈ G \ N, α_{m} must be nonzero, and at most one other α_{i} can be nonzero, since m ≥ 3. Hence the (m + 1)-st coordinate of a vector in V^{σ} must be zero. Now let σ_{0} be the matrix with α_{0} = … = α_{m} = 1. Then σ_{0} ∈ G \ N is not a bireflection, and V^{σ_{0}} contains a vector whose (m + 1)-st coordinate is nonzero. This means that V^{σ_{0}} ⊈ V^{σ} for all bireflections σ ∈ G \ N. Hence the result follows by Theorem 3.9.

The argument in the above example is based on a more elementary proof which was shown to me by Ian Hughes.

References

[1] J. P. Alperin, Local Representation Theory, Cambridge Studies in Advanced Mathematics 11, Cambridge University Press, Cambridge 1986.

[2] David J. Benson, Representations and Cohomology II, Cambridge studies in advanced mathematics 31, Cambridge Univ. Press, Cambridge 1991.

[3] David J. Benson, Polynomial Invariants of Finite Groups, Lond. Math. Soc. Lecture Note Ser. 190, Cambridge Univ. Press, Cambridge 1993.

[4] H. E. A. Campbell, A. V. Geramita, I. P. Hughes, R. J. Shank, D. L. Wehlau, Non-Cohen-Macaulay Vector Invariants and a Noether Bound for a Gorenstein Ring of Invariants, Preprint, Kingston, Ontario 1997.

[5] David Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Springer-Verlag, New York 1995.

[6] Victor Kac, Kei-Ichi Watanabe, Finite Linear Groups whose Ring of Invariants is a Complete Intersection, Bull. Amer. Math. Soc. 6 (1982), 221–223.

[7] Haruhisa Nakajima, Invariants of Finite Abelian Groups Generated by Transvections, Tokyo J. Math. 3 (1980), 201–214.

[8] Larry Smith, Polynomial Invariants of Finite Groups, A. K. Peters, Wellesley, Mass. 1995.

[9] Larry Smith, Homological Codimension of Modular Rings of Invariants and the Koszul Complex, Preprint, Toulouse, March 1997.

[10] Richard P. Stanley, Invariants of Finite Groups and their Applications to Combinatorics, Bull. Amer. Math. Soc. 1(3) (1979), 475–511.