THE COHEN-MACAULAY PROPERTY OF SEPARATING INVARIANTS OF FINITE GROUPS

EMILIE DUFRESNE, JONATHAN ELMER, AND MARTIN KOHLS

Abstract. In the case of finite groups, a separating algebra is a subalgebra of the ring of invariants which separates the orbits. Although separating algebras are often better behaved than the ring of invariants, we show that many of the criteria which imply the ring of invariants is non Cohen-Macaulay actually imply that no graded separating algebra is Cohen-Macaulay. For example, we show that, over a field of positive characteristic p, given sufficiently many copies of a faithful modular representation, no graded separating algebra is Cohen-Macaulay. Furthermore, we show that, for a p-group, the existence of a Cohen-Macaulay graded separating algebra implies the group is generated by bireflections. Our results lead us to conjecture that if one graded separating algebra is Cohen-Macaulay, then so is the invariant ring.

1. Introduction

Let G be a finite group, and let V be a finite dimensional representation of G over a field 𝕜 of characteristic p ≥ 0. We say V is a modular representation if p divides |G|. We write 𝕜[V] for the symmetric algebra S(V^{∗}) on the vector space dual of V. It is a polynomial ring with the standard grading. The action of G on V induces an action on 𝕜[V]: for f ∈ V^{∗} and v ∈ V, the action of σ ∈ G is given by (σ · f)(v) = f(σ^{−1} · v). The ring of invariants, denoted 𝕜[V]^{G}, is the ring formed by the elements of 𝕜[V] fixed by G. Since the group action preserves degree, 𝕜[V]^{G} is a graded subalgebra of 𝕜[V].

Let \overline{𝕜} be an algebraic closure of the field 𝕜, and let \overline{V} = \overline{𝕜} ⊗_{𝕜} V. As 𝕜[V] ⊆ \overline{𝕜}⊗_{𝕜}𝕜[V] ≅ \overline{𝕜}[\overline{V}], any f in 𝕜[V] can be considered as a function \overline{V} → \overline{𝕜}. The action of G on V extends to an action of G on \overline{V}, and so 𝕜[V]^{G} ⊆ \overline{𝕜}[\overline{V}]^{G}.

By definition, elements of 𝕜[V]^{G} are constant on G-orbits. Accordingly, if an invariant f takes distinct values on elements u, v ∈ \overline{V}, then these elements belong to distinct orbits, and we say f separates u and v. A geometric separating set is then a set of invariants which separates exactly the same points of \overline{V} as the whole ring of invariants. As G is finite, the ring of invariants separates orbits in \overline{V} [10, Lemma 2.1]. Hence, a geometric separating set is a set of invariants which separates the orbits of the G-action on \overline{V}.

Date: April 1, 2009.

The study of separating invariants, which has become quite popular in recent years, was initiated by Derksen and Kemper ([7, Section 2.3] and [20]). They defined a separating set as a set separating the same orbits in V as the whole ring of invariants [7, Definition 2.3.8]. For 𝕜 algebraically closed, geometric separating sets coincide with separating sets, but in general, a separating set is not always a geometric separating set (see Example 2.5). On the other hand, A ⊆ 𝕜[V]^{G} is a geometric separating algebra if and only if \overline{𝕜} ⊗_{𝕜} A ⊆ \overline{𝕜}[\overline{V}]^{G} is a separating algebra. Moreover, if A is graded, so is \overline{𝕜} ⊗_{𝕜} A, and the two rings have the same dimension and depth. Furthermore, the extension A ⊆ 𝕜[V]^{G} is integral if and only if \overline{𝕜} ⊗_{𝕜}A ⊆ \overline{𝕜}[\overline{V}]^{G} is integral. Thus, it often suffices to write proofs for separating sets over algebraically closed fields. Note that this works only because we are interested in geometric separating algebras.

Many defects of invariant rings disappear when one considers separating invariants. The ring of invariants is not always finitely generated (for non-reductive groups) [7, Example 2.1.4, due to Nagata], but there always exist finite geometric separating sets [7, Theorem 2.3.15]. Over algebraically closed fields, there is an upper bound on the size of separating sets, depending only on the dimension of the representation [11, Proposition 5.1.1]. Polarization, a classical method for obtaining vector invariants in characteristic zero, extends, for separating invariants, to all characteristics [10, 9]. For G finite, the Noether bound holds for separating invariants in all characteristics: although they may not generate the ring of invariants, the invariants of degree at most |G| always form a geometric separating set [7, Section 3.9].

Graded separating algebras are very closely related to the ring of invariants:

Proposition 1.1. If A ⊆ 𝕜[V]^{G} is a graded geometric separating algebra, then A ⊆ 𝕜[V]^{G} is an integral extension, and A is a finitely generated 𝕜-algebra.

Proof. We may assume that 𝕜 is algebraically closed. By [20, Lemma 1.3], the extension A ⊆ 𝕜[V]^{G} is integral. The finite generation of A as a 𝕜-algebra then follows of that of 𝕜[V]^{G} by Newstead [22, p. 52, (II)], or in the manner of [1, Proof of Theorem 1.3.1]. □

Proposition 1.2. Suppose p > 0. If A ⊆ 𝕜[V]^{G} is a graded subalgebra, then A is a geometric separating algebra if and only if 𝕜[V]^{G} is the purely inseparable closure of A in 𝕜[V], that is,

𝕜[V]^{G} = {f ∈ 𝕜[V] | for some m, f^{p^{m}} ∈ A}.

Proof. For 𝕜 algebraically closed, see [8, Remark 1.3]. The proof is an application of a result of van der Kallen [24, Sublemma A.5.1] (see the extended proof in [23]). A variation of the same argument first appeared in [15, Theorem 6]. The general case follows from a linear algebra argument. □ Remark 1.3. Propositions 1.1 and 1.2 hold for rational representations of reductive groups.

Kemper [20] exploited this close relationship to compute the invariants of reductive groups in positive characteristic. On the other hand, Dufresne [12] showed that the existence of polynomial or complete intersection separating algebras imposes strong conditions on the representation. The present paper is in the latter vein. We show that, in many instances, conditions which ensure that the ring of invariants is non Cohen-Macaulay, in fact imply that no graded geometric separating algebra is Cohen-Macaulay. We thus provide a (partial) negative answer to Kemper who asked if Cohen-Macaulay separating algebras should always exist [21]. Notably, we show: Theorem 1.4. If V is faithful and modular, then there exists r ≥ 1 such that, for all k, every graded geometric separating algebra in 𝕜[V^{⊕k}]^{G} has Cohen-Macaulay defect at least k − r − 1. In particular, for k > r + 1, no graded geometric separating algebra in 𝕜[V^{⊕k}]^{G} is Cohen-Macaulay.

An element σ of G acts as a bireflection on V if its fixed space is of codimension at most 2 in V. Theorem 1.5. Let G be a p-group. If there exists a graded geometric separating algebra in 𝕜[V]^{G} which is Cohen-Macaulay, then G is generated by elements acting as bireflections.

Theorem 1.5 fits well with [12, Theorem 1.3]. In the important special case of p-groups, we obtain that G is generated by bireflections from the existence of a Cohen-Macaulay rather than a complete intersection graded geometric separating algebra, a much weaker hypothesis. This mirrors the situation for invariant rings ([18, Corollary 3.7] and [17, Theorem A]). Example 2.10 shows that not only the converse of Theorem 1.5, but also the converses of [12, Theorem 1.1 and Theorem 1.3], are not true.

In Section 2, we extend the methods introduced in [18] to prove our main results. Section 3 concentrates on the alternating group A_{4}. We conclude in Section 4 with a discussion of the general situation and a conjecture. Acknowledgements. This paper was prepared during visits of the first and second authors to TU München, and of the second and third authors to Universität Heidelberg. We wish to thank Gregor Kemper, B. Heinrich Matzat, and MATCH for making these visits possible.

2. The Cohen-Macaulay Defect of Separating Algebras

Let A ⊆ 𝕜[V] be a finitely generated graded subalgebra, and let A_{+} denote its maximal homogeneous ideal. Homogeneous elements a_{1}, …, a_{k} in A_{+} form a partial homogeneous system of parameters (phsop) if they generate an ideal of height k in A. If additionally k = dim A, then they form a homogeneous system of parameters (hsop). Noether’s normalization theorem guarantees that an hsop always exists. If, for i = 1, …, k, the element a_{i} is not a zero divisor in A/(a_{1}, …, a_{i−1})A, then the elements a_{1}, …, a_{k} form a regular sequence. Every regular sequence is a phsop. We say A is Cohen-Macaulay when every phsop is a regular sequence. The depth of a homogeneous ideal I ⊆ A_{+}, written depth_{A}(I), is the maximal length of a regular sequence in I. Note that the height of I, ht(I), is equal to the maximal length of a phsop in I. We write depth(A) := depth_{A}(A_{+}), and define the Cohen-Macaulay defect of A to be cmdef(A) := dim A − depth(A). Thus, A is Cohen-Macaulay precisely when cmdef A = 0.

In Theorem 2.1, we relate the Cohen-Macaulay defect of graded geometric separating algebras to the n-th cohomology group H^{n}(G, \overline{𝕜}[\overline{V}]). For the theory of these groups for arbitrary n, we refer to [2, 25]. Since in most applications of this theorem, we have n = 1, we give the construction of this group in the following.

Let W be a representation of the group G over the field 𝕜. A 1-cocycle is a map g : G → W, σ ↦ g_{σ} such that g_{στ} = σg_{τ} + g_{σ}, for all σ, τ ∈ G. We write Z^{1}(G, W) for the additive group of all 1-cocycles. For each w ∈ W, the map given by σ ↦ (σ − 1)w := σw − w is a 1-cocycle, which is called a 1-coboundary. The coboundaries form the subgroup B^{1}(G, W) of Z^{1}(G, W). The first cohomology group of G with coefficients in W is the quotient group H^{1}(G, W) := Z^{1}(G, W)/B^{1}(G, W). A cocycle g is nontrivial if and only if its cohomology class g+B^{1}(G, W) is nonzero in H^{1}(G, W). Similar definitions hold for H^{n}(G, W). We will sometimes abuse notation by using the same symbols for cocycles and cohomology classes.

The group H^{n}(G, 𝕜[V]) has a natural graded 𝕜[V]^{G}-module structure. For a homogeneous g ∈ H^{n}(G, 𝕜[V]), its annihilator in 𝕜[V]^{G},

Ann_{𝕜[V]^{G}}(g) := {a ∈ 𝕜[V]^{G} : ag = 0 ∈ H^{n}(G, 𝕜[V])},

is a homogeneous ideal. If p > 0, the m-fold Frobenius homomorphism 𝕜[V] → 𝕜[V], f ↦ f^{p^{m}}, induces a map H^{n}(G, 𝕜[V]) → H^{n}(G, 𝕜[V]). We write g^{p^{m}} for the image of an element g ∈ H^{n}(G, 𝕜[V]) under this map. In particular, for g ∈ H^{1}(G, 𝕜[V]), the cohomology class g^{p^{m}} ∈ H^{1}(G, 𝕜[V]) is given by the cocycle σ ↦ (g_{σ})^{p^{m}}.

Over fields of characteristic zero, and in the non-modular case in general, the ring of invariants is always Cohen-Macaulay [16]. In particular, there will always be a Cohen-Macaulay geometric separating algebra. Accordingly, from now on we assume V is modular.

Our most general statement generalizes [18, Corollary 1.6] (see also [19, Proposition 6]) to the case of separating algebras:

Theorem 2.1. Let n ≥ 1 be the smallest integer such that there exists a homogeneous g ∈ H^{n}(G, \overline{𝕜}[\overline{V}]) such that g^{p^{m}} is nonzero for every m ≥ 0. If A is a graded geometric separating algebra in 𝕜[V]^{G}, then I := Ann_{A}(g), the annihilation ideal of g in A, has depth at most n+1. Furthermore, if Ann_{\overline{𝕜}[\overline{V}]^{G}}(g) has height k, then A has Cohen-Macaulay defect at least k − n − 1.

Since the case n = 1 suffices for most of our applications, we give an additional more elementary proof of the first part. Without loss of generality, we assume 𝕜 = \overline{𝕜} in both arguments.

Proof of the case n = 1. By Proposition 1.2, there exists a p-power q such that (𝕜[V]^{G})^{q} ⊆ A. Suppose, for a contradiction, that depth_{A}(I) is at least 3. Hence, there exists an A-regular sequence a_{1}, a_{2}, a_{3} in I. Since a_{i}g = 0, there are b_{1}, b_{2}, b_{3} ∈ 𝕜[V] such that

a_{i}g_{σ} = (σ − 1)b_{i}, for all σ ∈ G, i = 1, 2, 3.

Set u_{ij} := a_{i}b_{j} − a_{j}b_{i}, for 1 ≤ i < j ≤ 3. For all i, j, u_{ij} is invariant, and so u_{ij}^{q} belongs to A. Since a_{1}^{q}, a_{2}^{q}, a_{3}^{q} forms an A-regular sequence [13, Corollary 17.8 (a)], and since

a_{1}^{q}u_{23}^{q} − a_{2}^{q}u_{13}^{q} + a_{3}^{q}u_{12}^{q} = | \begin{matrix}a_{1} & a_{2} & a_{3} \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \end{matrix} |^{q} = 0,

it follows that u_{12}^{q} ∈ (a_{1}^{q}, a_{2}^{q})A. Thus, there exist f_{1}, f_{2} ∈ A such that

u_{12}^{q} = a_{1}^{q}b_{2}^{q} − a_{2}^{q}b_{1}^{q} = f_{1}a_{1}^{q} + f_{2}a_{2}^{q}.

As a_{1}, a_{2} is a phsop in A, it is also a phsop in its integral extension 𝕜[V], and thus a_{1}, a_{2} are coprime in 𝕜[V]. From a_{1}^{q}(b_{2}^{q} − f_{1}) = a_{2}^{q}(f_{2} + b_{1}^{q}), it follows that a_{1}^{q} divides f_{2} + b_{1}^{q} in 𝕜[V]. Therefore, there exists h ∈ 𝕜[V] such that a_{1}^{q}h = f_{2} + b_{1}^{q}. Hence, for every σ ∈ G, we have

a_{1}^{q}(σ − 1)h = ((σ − 1)b_{1})^{q} = (a_{1}g_{σ})^{q} = a_{1}^{q}g_{σ}^{q},

that is, g_{σ}^{q} = (σ − 1)h, a contradiction since g^{q} is nonzero. Thus, depth_{A}(I) ≤ 2. □

Proof of the general case. For some m ≥ 0, we have (𝕜[V]^{G})^{p^{m}} ⊆ A. For each i, H^{i}(G, 𝕜[V]) is finitely generated as a 𝕜[V]^{G}-module. Therefore, for each 0 < i < n, there exists some m(i) such that α^{p^{m(i)}} = 0 for all α ∈ H^{i}(G, 𝕜[V]). Set q := p^{m′}, where m′ is the maximum of m, m(1), m(2), …, m(n − 1). Assume, for a contradiction, that depth_{A}(I) ≥ n + 2, and let a_{1}, a_{2}, …, a_{n+2} be an A-regular sequence in I. Consider the bar resolution of ℤ as a ℤG-module:

… →^{∂_{n+1}} X_{n} →^{∂_{n}} X_{n−1} →^{∂_{n−1}} … →^{∂_{1}} X_{0} →^{∂_{0}} ℤ → 0,

where X_{0} = ℤG, X_{−1} := ℤ, and ∂_{0} is the augmentation map. The cohomology class g is represented by a u_{{1}} ∈ Hom_{ℤG}(X_{n}, 𝕜[V]) such that u_{{1}}∂_{n+1} = 0, that is, u_{{1}} ∈ Z^{n}(G, 𝕜[V]) (the notation will become clear later). For each i, a_{i}g = 0, so there is an element h_{{i}} ∈ Hom_{ℤG}(X_{n−1}, 𝕜[V]) such that a_{i}u_{{1}} = h_{{i}}∂_{n}. We define, for 2 ≤ r ≤ n + 1 and for each ordered r-subset J := {j(1), j(2), …, j(r)} of {1, …, n+2}, a map u_{J} ∈ Hom_{ℤG}(X_{n−r+1}, 𝕜[V]) such that u_{J}∂_{n−r+2} = 0, that is, u_{J} ∈ Z^{n−r+1}(G, 𝕜[V]). For 2 ≤ r ≤ n, the definition of q implies u_{J}^{q} ∈ B^{n−r+1}(G, 𝕜[V]), so there is a map h_{J} ∈ Hom_{ℤG}(X_{n−r}, 𝕜[V]) satisfying u_{J}^{q} = h_{J}∂_{n−r+1}. We now define u_{J}, and thus h_{J}, by induction on r:

(1)  u_{J} := ∑_{i=1}^{r}(−1)^{i+1}a_{j(i)}^{q^{r−2}}h_{J\{j(i)}}.

Next, we show that u_{J}∂_{n−r+2} = 0 for 2 ≤ r ≤ n + 1. For r = 2, we get

u_{{j(1),j(2)}}∂_{n} = a_{j(1)}h_{{j(2)}}∂_{n} − a_{j(2)}h_{{j(1)}}∂_{n}
= a_{j(1)}a_{j(2)}u_{{1}} − a_{j(2)}a_{j(1)}u_{{1}} = 0.

For 2 < r ≤ n + 1, we obtain similarly:

(2)  u_{J}∂_{n−r+2} = ∑_{i=1}^{r}(−1)^{i+1}a_{j(i)}^{q^{r−2}}u_{J\{j(i)}}^{q} = 0,

since the middle term is

∑_{i=1}^{r}(−1)^{i+1}a_{j(i)}^{q^{r−2}} (∑_{k=1}^{i−1}(−1)^{k+1}a_{j(k)}^{q^{r−2}}h_{J\{j(i),j(k)}}^{q}

+ ∑_{k=i+1}^{r}(−1)^{k}a_{j(k)}^{q^{r−2}}h_{J\{j(i),j(k)}}^{q}) =

∑_{1≤k<i≤r} (−1)^{i+k}a_{j(i)}^{q^{r−2}}a_{j(k)}^{q^{r−2}}h_{J\{j(i),j(k)}}^{q} + ∑_{1≤i<k≤r} (−1)^{i+k+1}a_{j(i)}^{q^{r−2}}a_{j(k)}^{q^{r−2}}h_{J\{j(i),j(k)}}^{q},

which equals zero. When r = n + 1, u_{J} ∈ Z^{0}(G, 𝕜[V]). It follows that u_{J}(ι) ∈ 𝕜[V]^{G}, which implies u_{J}(ι)^{q} ∈ A (ι ∈ G is the neutral element). Therefore, for each 1 ≤ i ≤ n + 2, we have u_{{1,2,...n+2}\{i}}^{q}(ι) ∈ A. The second equality in (2) is also valid for J = {1, 2, …, n + 2} (r = n + 2), that is,

∑_{i=1}^{n+2}(−1)^{i+1}a_{i}^{q^{n}}u_{{1,2,...,n+2}\{i}}^{q}(ι) = 0.

As a_{1}^{q^{n}}, …, a_{n+2}^{q^{n}} is still A-regular, u_{{1,...,n+1}}^{q}(ι) ∈ (a_{1}^{q^{n}}, …, a_{n+1}^{q^{n}})A. Thus there exist f_{1}, f_{2}, …, f_{n+1} ∈ Hom_{ℤG}(X_{0}, A) (in particular, f_{i}∂_{1} = 0) such that u_{{1,...,n+1}}^{q} = ∑_{i=1}^{n+1} a_{i}^{q^{n}}f_{i}. Substituting (1) for u_{{1,...,n+1}} yields

∑_{i=1}^{n+1}(a_{i}^{q^{n}})((−1)^{i+1}h_{{1,...,n+1}\{i}}^{q} − f_{i}) = 0.

Since a_{1}^{q^{n}}, …, a_{n+1}^{q^{n}} is a phsop in A, it is also a phsop in 𝕜[V], and so ((−1)^{n}h_{{1,...,n}}^{q} − f_{n+1})(ι) ∈ (a_{1}^{q^{n}}, a_{2}^{q^{n}}, …, a_{n}^{q^{n}})𝕜[V]. It follows that

h_{{1,...,n}}^{q} − (−1)^{n}f_{n+1} = ∑_{i=1}^{n} a_{i}^{q^{n}}l_{i}

for some l_{1}, l_{2}, …, l_{n} ∈ Hom_{ℤG}(X_{0}, 𝕜[V]) ≅ 𝕜[V]. We next apply ∂_{1} to this expression. For n = 1, we have a_{1}^{q}u_{{1}}^{q} = a_{1}^{q}l_{1}∂_{1}, or alternatively u_{{1}}^{q} = l_{1}∂_{1} ∈ B^{1}(G, 𝕜[V]), a contradiction to g^{q} ≠ 0. Now assume that n ≥ 2. Applying ∂_{1} leads to

(3)  u_{{1,...,n}}^{q^{2}} = ∑_{i=1}^{n} a_{i}^{q^{n}}l_{i}∂_{1}.

For 2 ≤ r ≤ n, we prove by reverse induction that there exist elements l_{1}, l_{2}, …, l_{r} ∈ Hom_{ℤG}(X_{n−r}, 𝕜[V]) such that

(4)  u_{{1,...,r}}^{q^{n+2−r}} = ∑_{i=1}^{r} a_{i}^{q^{n}}l_{i}∂_{n−r+1}.

The case r = n is covered by (3). Suppose u_{{1,...,r+1}}^{q^{n+1−r}} = ∑_{i=1}^{r+1} a_{i}^{q^{n}}l′_{i}∂_{n−r} for some l′_{1}, l′_{2}, … l′_{r+1} ∈ Hom_{ℤG}(X_{n−r−1}, 𝕜[V]). Using (1), we obtain

∑_{i=1}^{r+1}(−1)^{i+1}a_{i}^{q^{n}}h_{{1,...,r+1}\{i}}^{q^{n+1−r}} = ∑_{i=1}^{r+1} a_{i}^{q^{n}}l′_{i}∂_{n−r},

and rearranging yields

∑_{i=1}^{r+1} a_{i}^{q^{n}}((−1)^{i+1}h_{{1,...,r+1}\{i}}^{q^{n+1−r}} − l′_{i}∂_{n−r}) = 0.

Since a_{1}^{q^{n}}, …, a_{r+1}^{q^{n}} is a phsop for 𝕜[V], we have

h_{{1,...,r}}^{q^{n+1−r}} − (−1)^{r}l′_{r+1}∂_{n−r} = ∑_{i=1}^{r} a_{i}^{q^{n}}l_{i}

for some l_{1}, l_{2}, …, l_{r} ∈ Hom_{ℤG}(X_{n−r}, 𝕜[V]). Here we have used that X_{n−r} is a free ℤG-module. Applying ∂_{n−r+1} to this expression gives us

h_{{1,...,r}}^{q^{n+1−r}}∂_{n−r+1} = ∑_{i=1}^{r} a_{i}^{q^{n}}l_{i}∂_{n−r+1},

which implies (4), as required.

When r = 2, Equation (4) reads u_{{1,2}}^{q^{n}} = a_{1}^{q^{n}}l_{1}∂_{n−1} + a_{2}^{q^{n}}l_{2}∂_{n−1}, where l_{1}, l_{2} ∈ Hom_{ℤG}(X_{n−2}, 𝕜[V]). Substituting (1) for u_{{1,2}}, we obtain a_{1}^{q^{n}}(h_{{2}}^{q^{n}} − l_{1}∂_{n−1}) = a_{2}^{q^{n}}(h_{{1}}^{q^{n}} + l_{2}∂_{n−1}). For the same reasons as before, we have h_{{1}}^{q^{n}}+l_{2}∂_{n−1} = a_{1}^{q^{n}}l for some l ∈ Hom_{ℤG}(X_{n−1}, 𝕜[V]). Applying ∂_{n} gives us a_{1}^{q^{n}}u_{{1}}^{q^{n}} = a_{1}^{q^{n}}l∂_{n}. Hence, u_{{1}}^{q^{n}} = l∂_{n} ∈ B^{n}(G, 𝕜[V]), a contradiction to g^{q^{n}} ≠ 0. Therefore, depth_{A}(I) ≤ n + 1.

Finally, if c_{1}, …, c_{k} ∈ Ann_{𝕜[V]^{G}}(g) forms a phsop in 𝕜[V]^{G}, then c_{1}^{q}, …, c_{k}^{q} ∈ I forms a phsop in A, and so I has height at least k. The graded analogue of [4, Exercise 1.2.23] implies the Cohen-Macaulay defect of A is at least ht_{A}(I) − depth_{A}(I) ≥ k − n − 1. □

Remark 2.2. If g ∈ H^{n}(G, 𝕜) is nonzero, then g^{p^{m}} is nonzero for all m ≥ 0. For n = 1, this is clear since elements of H^{1}(G, 𝕜) are group homomorphisms G → (𝕜, +). Assume g ∈ Z^{n}(G, 𝕜) is nontrivial. In the notation of the proof of Theorem 2.1, we have g ∈ Hom_{ℤG}(X_{n}, 𝕜) and g∂_{n+1} = 0. If g^{p^{m}} is trivial for some m, then g^{p^{m}} = h∂_{n} for some h ∈ Hom_{ℤG}(X_{n−1}, 𝕜). The finiteness of G implies that, in fact, g and h take values in some 𝔽_{p^{l}} ⊆ 𝕜. In particular, for some large enough k, we have g = (g^{p^{m}})^{p^{kl−m}} = h^{p^{kl−m}}∂_{n}, so g is trivial, a contradiction.

Remark 2.3. For n > 1, Theorem 2.1 is new even in the case A = 𝕜[V]^{G}.

Example 2.4. Let G ⊆ (𝕜, +) be a finite nontrivial subgroup. Consider the threefold sum of the 2-dimensional representation V of G over 𝕜

given by σ ↦ ( \begin{matrix}1 & 0 \\ −σ & 1 \end{matrix} ), and write 𝕜[V^{⊕3}] = 𝕜[x_{1}, y_{1}, x_{2}, y_{2}, x_{3}, y_{3}]. The

map g : G → 𝕜, σ ↦ σ yields a nonzero element in H^{1}(G, 𝕜). For all σ ∈ G, we have x_{i}g_{σ} = (σ − 1)y_{i}, that is, x_{i}g is trivial. Since x_{1}, x_{2}, x_{3} form a phsop in 𝕜[V^{⊕3}], Theorem 2.1 implies that no graded geometric separating algebra in 𝕜[V^{⊕3}]^{G} is Cohen-Macaulay. ◁

The following example shows that Theorem 2.1 applies only to graded geometric separating algebras:

Example 2.5. Let V be the permutation representation of the cyclic group C_{4} = ⟨σ⟩ of order 4 over the field 𝔽_{2}. Consider the C_{4}-invariants c_{1} := x_{1}+x_{2}+x_{3}+x_{4}, c_{2} := x_{1}x_{3}+x_{2}x_{4}, c_{3} := x_{1}x_{2}+x_{2}x_{3}+x_{3}x_{4}+x_{1}x_{4}, and c_{4} := x_{1}x_{2}x_{3}x_{4}. The action of C_{4} on V partitions its 16 elements into 5 orbits, which one can check are separated by c_{1}, c_{2}, c_{3}, and c_{4}. As c_{1}, c_{2}, c_{3}, c_{4} form a hsop in 𝔽_{2}[V], the subalgebra 𝔽_{2}[c_{1}, c_{2}, c_{3}, c_{4}] is a polynomial graded (non geometric) separating algebra. In particular, it is Cohen-Macaulay.

On the other hand, if g : C_{4} → 𝔽_{2} is the nontrivial cocycle given by σ^{i} ↦ i mod 2, then c_{1}g_{σ^{i}} = (σ^{i} − 1)(x_{1} + x_{3}), c_{2}g_{σ^{i}} = (σ^{i} − 1)(x_{1}x_{3}), and c_{3}g_{σ^{i}} = (σ^{i} − 1)(x_{1}x_{4} + x_{2}x_{3}), that is, c_{1}, c_{2}, c_{3} ∈ Ann_{\overline{𝔽}_{2}[\overline{V}]^{C_{4}}}(g). Since c_{1}, c_{2}, c_{3}, c_{4} form a hsop in 𝔽_{2}[V], by Theorem 2.1, no graded geometric separating algebra in 𝔽_{2}[V]^{C_{4}} is Cohen-Macaulay. ◁

Using Theorem 2.1, we can quickly generalize several results which were consequences of its analogue [18, Corollary 1.6]. A first consequence is the following generalization of [5, Corollary 21]:

Corollary 2.6. Assume G contains a normal subgroup N of index p (for example, G is a p-group). Then for any faithful representation V, every graded geometric separating algebra in 𝕜[V^{⊕k}]^{G} has Cohen-Macaulay defect at least k − 2.

Proof. Since G/N ≅ (𝔽_{p}, +), there is a nonzero element in H^{1}(G, 𝕜). As V is faithful, the fixed subspaces of nonidentity elements in G have codimension at least k in V^{⊕k}. The result follows from Lemma 2.7. □

Lemma 2.7. Suppose V is faithful. If the fixed subspace of every element of order p in G has codimension at least k in V, then for any homogeneous g ∈ H^{n}(G, \overline{𝕜}[\overline{V}]), the ideal Ann_{\overline{𝕜}[\overline{V}]^{G}}(g) has height at least k.

Therefore, if there exists a homogeneous g ∈ H^{n}(G, \overline{𝕜}[\overline{V}]) satisfying the hypotheses of Theorem 2.1, then every graded geometric separating algebra in 𝕜[V]^{G} has Cohen-Macaulay defect at least k − n − 1.

Proof. We may assume 𝕜 = \overline{𝕜}. By Kemper [18, Lemma 2.1] there exist (in general, non-homogeneous) elements a_{1}, …, a_{k} ∈ 𝕜[V]_{+}^{G} such that a_{i}H^{n}(G, 𝕜[V]) = 0, for all i, and ht(a_{1}, …, a_{k}) = k. It follows that Ann_{𝕜[V]^{G}}(g) has height at least k. □

Proof of Theorem 1.4. By [3, Theorem 4.1.3], there is a number r such that H^{r}(G, 𝕜) ≠ 0. Thus by Remark 2.2, for any k, there is a minimal number n ≤ r such that the hypotheses of Theorem 2.1 are satisfied for V^{⊕k}. The same argument as in Corollary 2.6 shows that A has Cohen-Macaulay defect at least k − n − 1 ≥ k − r − 1. □

Next, we generalize three results of [18]. Note that since, for us, elements acting trivially are bireflections, we do not need to assume that V is faithful.

Corollary 2.8. If G has a normal subgroup N of index p which contains all elements acting as bireflections on V, then no graded geometric separating algebra in 𝕜[V]^{G} is Cohen-Macaulay.

Proof. The proof of [18, Theorem 3.6] shows that the hypotheses of Theorem 2.1 are fulfilled with n = 1 and k = 3. □

Proof of Theorem 1.5. For p-groups, if the elements acting as bireflections generate a proper subgroup, then this subgroup lies in a normal subgroup of index p. □

Proposition 2.9. Suppose G has a normal subgroup N with factor group an elementary abelian p-group. If there exists σ ∈ G \ N whose fixed space in V is not contained in the fixed space of any bireflection in G \ N, then no graded geometric separating algebra in 𝕜[V]^{G} is Cohen-Macaulay.

Proof. Without loss of generality, assume 𝕜 = \overline{𝕜}. As G/N is an elementary abelian p-group, there is a g ∈ H^{1}(G, 𝕜) with kernel N. The proof of [18, Theorem 3.9] provides a phsop a_{1}, a_{2}, a_{3} in 𝕜[V]^{G}, and a homogeneous invariant h ∉ \sqrt{Ann_{𝕜[V]^{G}}(g)} such that ha_{i} ∈ \sqrt{Ann_{𝕜[V]^{G}}(g)}, for i = 1, 2, 3. Hence, for some k ≥ 0, the invariants a_{i}^{k} annihilate g′ := h^{k}g ∈ H^{1}(G, 𝕜[V]). Thus, the annihilator ideal Ann_{𝕜[V]^{G}}(g′) has height at least 3. By Theorem 2.1, it now suffices to show that g^{′p^{m}} is nonzero for all m ≥ 0. By [18, Proposition 3.5], we have \sqrt{Ann_{𝕜[V]^{G}}(g^{p^{m}})} = \sqrt{Ann_{𝕜[V]^{G}}(g)}. Therefore, if g^{′p^{m}} is zero, then h^{kp^{m}} annihilates g^{p^{m}}, and so h ∈ \sqrt{Ann_{𝕜[V]^{G}}(g)}, a contradiction. □

Example 2.10. Let 𝕜 be a finite field. For m ≥ 3, set V = 𝕜^{2m+1}, and consider the group G ≤ GL(V) formed by the (2m + 1) × (2m + 1) matrices of the form

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

where α_{0}, …, α_{m} ∈ 𝕜, and I_{m} denotes the m × m identity matrix. The group G is a p-group, and is generated by reflections, that is, by elements whose fixed space has codimension at most 1 in V. Example 3.10 in [18] shows that the hypotheses of Proposition 2.9 are satisfied, with N the subgroup formed by the elements such that α_{m} = 0, and σ the element such that α_{i} = 1 for all i. Hence, no graded geometric separating algebra in 𝕜[V]^{G} is Cohen-Macaulay. ◁

We end this section with a generalization of [18, Theorem 2.7].

Theorem 2.11. Let V_{reg} be the regular representation of G over 𝕜. If p divides |G|, then every graded geometric separating algebra in 𝕜[V_{reg}]^{G} has Cohen-Macaulay defect at least |G|\frac{p−1}{p} − 2. For |G| ≥ 5, this number is at least one.

Proof. By [18, Lemma 2.6], there is a nonzero g ∈ H^{1}(G, 𝕜[V_{reg}]). Lemma 2.12 implies all powers g^{p^{m}} are also nonzero. Since the fixed subspaces of elements of G of order p have codimension |G|(p − 1)/p, the hypotheses of Lemma 2.7 are satisfied with k = |G|(p − 1)/p. □

The regular representation of C_{4} was studied in Example 2.5. For G = C_{2}, C_{3}, and C_{2}×C_{2}, the invariant ring 𝕜[V_{reg}]^{G} is Cohen-Macaulay [18, Theorem 2.7]. Thus, these are the only groups such that there exists a graded Cohen-Macaulay geometric separating algebra in 𝕜[V_{reg}]^{G}.

Lemma 2.12. Let V be a permutation representation of G. If g in H^{1}(G, 𝕜[V]) is nonzero, then g^{p^{m}} is nonzero for all m ≥ 0. If, in addition, V is faithful, then every graded geometric separating algebra in 𝕜[V^{⊕k}]^{G} has Cohen-Macaulay defect at least k − 2.

Proof. As V is a permutation representation, there is a set of monomials M ⊆ 𝕜[V] such that 𝕜[V] = ⨁_{h∈M}⟨Gh⟩. Thus, if g ∈ H^{1}(G, 𝕜[V]), there is a (finite) decomposition g = ∑_{h∈M} g_{h}, where each g_{h} is in H^{1}(G, ⟨Gh⟩). For m ≥ 0, we have the decomposition g^{p^{m}} = ∑_{h∈M} g_{h}^{p^{m}}, where g_{h}^{p^{m}} is in H^{1}(G, ⟨Gh′⟩), and h′ is the unique element of Gh^{p^{m}} ∩M. If g is nonzero, then g_{h} is nonzero for some h. As ⟨Gh⟩ and ⟨Gh^{p^{m}}⟩ are isomorphic permutation representations of G, the element g_{h}^{p^{m}} is also nonzero. The additional statement follows by Lemma 2.7. □

3. The Alternating Group A_{4}

In this section we concentrate on representations of the alternating group A_{4} over an algebraically closed field 𝕜 of characteristic 2. The group A_{4} is the smallest modular group G for which the cohomology H^{1}(G, 𝕜) is trivial. In order to apply Theorem 2.1, we must look outside the direct summand H^{1}(G, 𝕜) of H^{1}(G, 𝕜[V]).

Pick χ ∈ A_{4}, an element of order 2, and τ, a 3-cycle, so that A_{4} is generated by χ and τ. The unique Sylow 2-subgroup P in A_{4} is generated by χ and τ^{−1}χτ. Let ω be a fixed primitive third root of unity in 𝕜. Define^{ω}𝕜 and^{ω^{2}}𝕜 to be the one-dimensional representations on which P acts trivially, and τ acts via multiplication by ω and ω^{2}, respectively. Formally, ^{1}𝕜 denotes the trivial representation.

Lemma 3.1. For i = 1, 2, there exists a nonzero g ∈ H^{1}(A_{4},^{ω^{i}}𝕜).

Proof. Assume i = 1. Let V := ⟨v_{1}, v_{2}⟩ be the representation given by

χ ↦ ( \begin{matrix}1 & ω^{2} \\ 0 & 1 \end{matrix} ) , and τ ↦ ( \begin{matrix}ω & 0 \\ 0 & 1 \end{matrix} ) .

This is an indecomposable representation [14, Theorem 7.0.3], with a submodule W := ⟨v_{1}⟩ isomorphic to ^{ω}𝕜. Furthermore, A_{4} acts trivially on the quotient V/W, and so we have a nonsplit exact sequence

0 →^{ω}𝕜 → V → 𝕜 → 0.

By [19, section 2], it follows that H^{1}(A_{4},^{ω} 𝕜) ≠ 0. In particular, the cocycle given by χ ↦ ω^{2}, τ ↦ 0 is nontrivial. Similarly, for i = 2, the cocycle in Z^{1}(A_{4},^{ω^{2}} 𝕜) given by χ ↦ ω, τ ↦ 0 is nontrivial. □

Corollary 3.2. If V :=^{ω^{i}}𝕜⊕W^{⊕k}, where W is a faithful representation and i = 1, 2, then every graded geometric separating algebra in 𝕜[V]^{A_{4}} has Cohen-Macaulay defect at least k − 2.

Proof. Without loss of generality, i = 1. Since V^{∗} ≅^{ω^{2}}𝕜⊕(W^{∗})^{⊕k}, there is a direct summand of 𝕜[V] isomorphic to S(^{ω^{2}}𝕜). Since S^{2^{m}}(^{ω^{2}}𝕜) is isomorphic to ^{ω^{2}}𝕜 or ^{ω}𝕜, Lemma 3.1 implies that the cohomology class g_{m} ∈ H^{1}(A_{4}, S^{2^{m}}(^{ω^{2}}𝕜)), given by the cocyle χ ↦ ω^{2^{m}}, τ ↦ 0, is nonzero for all m. As g_{m} = g_{0}^{2^{m}}, the result follows by Lemma 2.7. □

Exploiting the classification of the finite dimensional representations of A_{4} [6] (with the notation of [14, Chapter 7]), we obtain a much stronger result:

Theorem 3.3. Suppose V is a faithful, indecomposable finite dimensional representation of A_{4}. If 𝕜[V]^{A_{4}} is non Cohen-Macaulay, then no graded geometric separating algebra in 𝕜[V]^{A_{4}} is Cohen-Macaulay.

Remark 3.4. The indecomposable representations V of A_{4} such that 𝕜[V]^{A_{4}} is Cohen-Macaulay are listed in [14, Corollary 5.2.16]. In particular, when dim_{𝕜}(V) ≥ 7, 𝕜[V]^{A_{4}} is non Cohen-Macaulay.

Proof. Whenever 𝕜[V]^{A_{4}} is non Cohen-Macaulay, the fixed space of any nonidentity element in P has codimension at least 3 in V. Thus, by Lemma 2.7, it suffices to find g ∈ H^{1}(A_{4}, 𝕜[V]) such that g^{2^{m}} is nonzero for all m ≥ 0. We use the classification to separate our argument in two cases.

First, we suppose V is of the form W_{s}(ω^{e}), where s ∈ ℤ (s may be negative only if it is odd), and 0 ≤ e ≤ 2. The dimension of V is n := |s|. Let {v_{1}, v_{2}, …, v_{n}} be a basis of V such that A_{4} acts via the matrices given in [14, Theorem 7.0.3]. In particular, the action of A_{4} is upper triangular, and τ acts diagonally via multiplication by third roots of unity. If 𝕜[V]^{A_{4}} is non Cohen-Macaulay, the subset {v_{1}, v_{2}, …, v_{l}} is contained in V^{P}, for some l ≥ 2. Moreover for some r ≤ l, we have τv_{r} = ω^{−j}v_{r}, where j = 1 or 2. Let {x_{1}, x_{2}, …, x_{n}} be the dual basis, so that 𝕜[V] = 𝕜[x_{1}, x_{2}, … x_{n}]. Define u := ∏_{σ∈P} (σx_{r}). Then

τu = ∏_{σ∈P}(τστ^{−1}(τx_{r})) = ∏_{σ∈P} ω^{j}(σx_{r}) = ω^{j}u.

Since v_{r} ∈ V^{P}, the set {x_{1}, x_{2}, …, x_{r−1}, x_{r+1}, …, x_{n}} spans a 𝕜P - submodule of V^{∗}, thus 𝕜[V] = 𝕜[x_{1}, x_{2}, …, x_{r−1}, x_{r+1}, …, x_{n}][x_{r}] as 𝕜P -modules. Therefore, by [14, Proof of Lemma 1.3.2], if M is a 𝕜P - module direct summand of S(V^{∗}), so is u · M, and by induction so is u^{i} · M for any integer i. In particular, as 1 ∈ S^{0}(V^{∗}), we have a sequence ⟨1⟩, ⟨u⟩, ⟨u^{2}⟩, … of 𝕜P -direct summands. Each ⟨u^{i}⟩ is a 𝕜A_{4}- direct summand, and ⟨u^{i}⟩ ≅^{ω^{ij}} 𝕜. The element g_{m} ∈ H^{1}(A_{4}, ⟨u^{2^{m}}⟩) given by the cocyle χ ↦ ω^{−2^{m}j}u^{2^{m}}, τ ↦ 0, is nonzero by Lemma 3.1. As g_{m} = g_{0}^{2^{m}}, we are done.

Second, suppose V takes the form \overline{W}_{6d,λ}, where λ ∈ 𝕜 ∪ \{∞\}, not a third root of unity, and d ≥ 1, an integer, are such that if d = 1, then λ is not 0, 1, or ∞ (see [14, Theorem 7.0.3] for notation). As a 𝕜P -module, V^{∗} ≅ V_{2d,α} ⊕ V_{2d,β} ⊕ V_{2d,γ} [2, Section 4.3], where α, β, and γ are elements of 𝕜 ∪ {∞} depending on λ. The element τ acts on V by permuting the three 𝕜P -module summands. As a 𝕜P - module, S(V^{∗}) ≅ S(V_{2d,α}) ⊗ S(V_{2d,β}) ⊗ S(V_{2d,γ}), and therefore, S(V^{∗}) has direct summands isomorphic to S(V_{2d,α}), S(V_{2d,β}), and S(V_{2d,γ}), respectively. These summands are permuted by the action of τ. Let {x_{1}, …, x_{6d}} be a basis of V^{∗} such that the action of P is given in block form; that is, x_{1}, …, x_{2d} is a basis for the summand isomorphic to V_{2d,α}, and so on. Define u_{α} := ∏_{σ∈P} σx_{1}, and u_{β} := ∏_{σ∈P} σx_{2d+1}, and u_{γ} := ∏_{σ∈P} σx_{4d+1}. By the same argument as before, for each integer i, ⟨u_{α}^{i}⟩ is a direct summand (with trivial P -action) of the 𝕜P - module S(V_{2d,α}), and hence of the 𝕜P -module S(V^{∗}). The analogue holds for u_{β} and u_{γ}. Furthermore, we have

τu_{α}^{i} = τ(∏_{σ∈P} σx_{1})^{i} = (∏_{σ∈P}(τστ^{−1})x_{2d+1})^{i} = u_{β}^{i}.

Similarly, τu_{β}^{i} = u_{γ}^{i} and τu_{γ}^{i} = u_{α}^{i}. Thus, U^{i} := ⟨u_{α}^{i}, u_{β}^{i}, u_{γ}^{i}⟩ is a 𝕜A_{4}- direct summand of S(V^{∗}) on which P acts trivially. Since [A_{4} : P] is odd, τ acts diagonally with respect to some basis of U^{i}. A short calculation shows that u := u_{γ} + ωu_{β} + ω^{2}u_{α} spans a direct summand of U^{1} on which τ acts via multiplication by ω, in other words, ⟨u⟩ ≅^{ω}𝕜. For each m, we have that u^{2^{m}} = u_{γ}^{2^{m}} + ω^{2^{m}}u_{β}^{2^{m}} + ω^{2^{m+1}}u_{α}^{2^{m}} spans a direct summand of U^{2^{m}} isomorphic to ^{ω^{2^{m}}}𝕜.

The element g_{m} ∈ H^{1}(A_{4}, ⟨u^{2^{m}}⟩) given by the cocyle χ ↦ ω^{2^{m+1}}u^{2^{m}}, τ ↦ 0 is nonzero for each m ≥ 0. As g_{m} = g_{0}^{2^{m}}, the result now follows by Lemma 2.7. □

4. Concluding Remarks

Many of our results arose in our unsuccessful search for an example of a Cohen-Macaulay graded geometric separating algebra in a non Cohen-Macaulay ring of invariants. Given this accumulated evidence, we make the following conjecture:

Conjecture 4.1. If there exists a Cohen-Macaulay graded geometric separating algebra, then the ring of invariants is also Cohen-Macaulay.

A more ambitious conjecture would be that the depth of graded geometric separating algebras is bounded above by the depth of the ring of invariants. Even in the nonmodular case, however, they are not always equal:

Example 4.2. Let G = C_{4} be the cyclic group of order 4, and let 𝕜 be a field of odd characteristic containing a primitive fourth root of unity ζ. Consider the 2-dimensional representation V of C_{4} = ⟨σ⟩ given by

σ ↦ ( \begin{matrix}ζ & 0 \\ 0 & ζ \end{matrix} ). If 𝕜[V] = 𝕜[x, y], then 𝕜[V]^{C_{4}} = 𝕜[x^{4}, x^{3}y, x^{2}y^{2}, xy^{3}, y^{4}].

On points where x^{4} is zero, the function x^{2}y^{2} also takes the value zero; on the other points v, we have x^{2}y^{2}(v) = \frac{(x^{3}y(v))^{2}}{x^{4}(v)} . Thus, the value of x^{2}y^{2} as a function is entirely determined by the value of x^{4} and x^{3}y. Therefore x^{4}, x^{3}y, xy^{3}, y^{4} form a geometric separating set and A := 𝕜[x^{4}, x^{3}y, xy^{3}, y^{4}] is a geometric separating algebra. Note that A is not Cohen-Macaulay, since the hsop x^{4}, y^{4} does not form a regular sequence (despite being coprime) [4, Exercise 2.1.18].

On points where x^{4} is zero, xy^{3} is zero, and on other points v, xy^{3}(v) = \frac{(x^{3}y(v))^{3}}{(x^{4}(v))^{2}} . Thus, the hypersurface 𝕜[x^{4}, x^{3}y, y^{4}] is also a graded geometric separating algebra. Therefore, we have a Cohen-Macaulay graded geometric separating algebra, inside a non Cohen-Macaulay graded geometric separating algebra, inside a Cohen-Macaulay invariant ring. ◁

The close relationship between graded geometric separating algebras and the ring of invariants also exists for rational representations of reductive groups. Hence, some of our results, and maybe also our conjecture, might hold in a more general context.

References

[1] D. J. Benson. Polynomial invariants of finite groups, volume 190 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1993.

[2] D. J. Benson. Representations and cohomology. I, volume 30 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 1998. Basic representation theory of finite groups and associative algebras.

[3] D. J. Benson. Representations and cohomology. II, volume 31 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 1998. Cohomology of groups and modules.

[4] Winfried Bruns and Jürgen Herzog. Cohen-Macaulay rings, volume 39 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1993.

[5] H. E. A. Campbell, I. P. Hughes, G. Kemper, R. J. Shank, and D. L. Wehlau. Depth of modular invariant rings. Transform. Groups, 5(1):21–34, 2000.

[6] S. B. Conlon. Certain representation algebras. J. Aust. Math. Soc, 5:83–99, 1965.

[7] Harm Derksen and Gregor Kemper. Computational invariant theory. Invariant Theory and Algebraic Transformation Groups, I. Springer-Verlag, Berlin, 2002. Encyclopaedia of Mathematical Sciences, 130.

[8] Harm Derksen and Gregor Kemper. Computing invariants of algebraic groups in arbitrary characteristic. Adv. Math., 217(5):2089–2129, 2008.

[9] M. Domokos. Typical separating invariants. Transform. Groups, 12(1):49–63, 2007.

[10] Jan Draisma, Gregor Kemper, and David Wehlau. Polarization of separating invariants. Canad. J. Math., 60(3):556–571, 2008.

[11] Emilie Dufresne. Separating invariants. Ph. D. thesis, Kingston Ontario, 2008.

http://hdl.handle.net/1974/1407.

[12] Emilie Dufresne. Separating invariants and finite reflection groups. Adv. Math. (to appear), 2009. arXiv:0805.2605v3.

[13] David Eisenbud. Commutative algebra with a view toward algebraic geometry, volume 150 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995.

[14] Jonathan Elmer. Symmetric algebras and the depth of in-

variant rings. Ph.D thesis,  University of Kent,  2007.

http://www.maths.abdn.ac.uk/^{∼}bensondj/papers/e/elmer/klein4new.dvi.

[15] F. D. Grosshans. Vector invariants in arbitrary characteristic. Transform. Groups, 12(3):499–514, 2007.

[16] M. Hochster and John A. Eagon. Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci. Amer. J. Math., 93:1020–1058, 1971.

[17] Victor Kac and Kei-Ichi Watanabe. Finite Linear Groups whose Ring of Invariants is a Complete Intersection. Bull. (New Series) of the AMS, 6(2).

[18] Gregor Kemper. On the Cohen-Macaulay property of modular invariant rings. J. Algebra, 215(1):330–351, 1999.

[19] Gregor Kemper. A characterization of linearly reductive groups by their invariants. Transform. Groups, 5(1):85–92, 2000.

[20] Gregor Kemper. Computing invariants of reductive groups in positive characteristic. Transform. Groups, 8(2):159–176, 2003.

[21] Gregor Kemper. Separating invariants. Journal of Symbolic Computations, 2009. doi:10.1016/j.jsc.2008.02.012.

[22] P. E. Newstead. Introduction to moduli problems and orbit spaces, volume 51 of Tata Institute of Fundamental Research Lectures on Mathematics and Physics. Tata Institute of Fundamental Research, Bombay, 1978.

[23] Wilberd van der Kallen. http://www.math.uu.nl/people/vdkallen/errbmod.

[24] Wilberd van der Kallen. Lectures on Frobenius splittings and B-modules. Published for the Tata Institute of Fundamental Research, Bombay, 1993. Notes by S. P. Inamdar.

[25] Charles A. Weibel. An introduction to homological algebra, volume 38 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1994.

Mathematics Center Heidelberg (MATCH), Ruprecht-Karls Universität Heidelberg, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany

  E-mail address: emilie.dufresne@iwr.uni-heidelberg.de

Lehrstuhl A für Mathematik, RWTH Aachen, 52062 Aachen, Germany

  E-mail address: jonathan.elmer@mathA.rwth-aachen.de

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

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