DEPTH OF MODULAR INVARIANT RINGS

H.E.A. CAMPBELL, I.P. HUGHES, G. KEMPER, R.J. SHANK, AND D.L. WEHLAU

Abstract. It is well known that the ring of invariants associated to a non-modular representation of a finite group is Cohen-Macaulay and hence has depth equal to the dimension of the representation. For modular representations the ring of invariants usually fails to be Cohen-Macaulay and computing the depth is often very difficult. In this paper we obtain a simple formula for the depth of the ring of invariants for a family of modular representations. This family includes all modular representations of cyclic groups. In particular, we obtain an elementary proof of the celebrated theorem of Ellingsrud and Skjelbred [5]. We also prove that the ring of invariants of a p-group is Cohen-Macaulay if and only if it is Buchsbaum.

Introduction

We consider a faithful representation of a finite group G on a vector space V of dimension n over a field K. The representation is said to be modular if the characteristic of K divides the order of the group G. Otherwise it is called non-modular. The action of G on V induces an action of G on the dual V^{∗} given by σ(f) = f ∘ σ^{−1} for f ∈ V^{∗} and σ ∈ G. This action extends to an action by algebra automorphisms on the symmetric algebra R = S(V^{∗}). The ring of invariants is

R^{G} = {f ∈ S(V^{∗}) | σ(f) = f ∀σ ∈ G}.

Let R_{+}^{G} denote the augmentation ideal of R^{G}, i.e., the ideal generated by the homogeneous invariants of positive degree. A homogeneous sequence of elements a_{1}, a_{2}, …, a_{r} in R_{+}^{G} is regular on R^{G} if: for every i ≤ r and every m ∈ R^{G}, m · a_{i} ∈ (a_{1}, …, a_{i−1})R^{G} implies m ∈ (a_{1}, …, a_{i−1})R^{G}. Here (a_{1}, …, a_{i−1})R^{G} denotes the ideal in R^{G} generated by a_{1}, …, a_{i−1}. For an ideal I in R^{G} with I ⊆ R_{+}^{G}, we write depth_{I}(R^{G}) for the length of a maximal regular sequence on R^{G} all of whose elements lie in I. By depth(R^{G}) we mean depth_{R_{+}^{G}}(R^{G}). The ring R^{G} is Cohen-Macaulay if depth(R^{G}) = dim(V). A homogeneous sequence of elements a_{1}, a_{2}, …, a_{n} in R_{+}^{G} is a homogeneous system of parameters for R^{G} if R^{G} is a finitely generated module over the subalgebra generated by {a_{1}, …, a_{n}}. The ring R^{G} is Cohen-Macaulay if and only if every homogeneous system of parameters is a regular sequence. We write ht(I) for the height of an ideal I of R^{G}. We note that, since R^{G} is a finitely generated subalgebra of R, the height of I is the difference between the Krull dimension of R^{G} and the Krull dimension of R^{G}/I (see Bruns and Herzog [3, Theorem A.16]). We refer the reader to Bruns and Herzog [3] for standard results on height, depth and Cohen-Macaulay rings.

Date: December 8, 1997.

1991 Mathematics Subject Classification. 13A50.

Research partially supported by grants from ARP and NSERC.

For any non-modular representation the ring of invariants is a Cohen-Macaulay ring and hence has depth n. For almost all modular representations the ring of invariants is not Cohen-Macaulay (see Campbell et al. [4] and Kemper [6]). It is often difficult to compute the depth of R^{G} for a modular representation.

Throughout the sequel we will assume that K is a field of positive characteristic p. For σ ∈ G, we write V^{σ} for the set of vectors v ∈ V such that σ(v) = v. For a subgroup H ≤ G we define V^{H} = {v ∈ V | σv = v ∀σ ∈ H}.

Consider the subgroup G′_{p} ≤ G generated by the commutator subgroup and a Sylow p-subgroup of G. This is the smallest normal subgroup whose quotient group is abelian and of order coprime to p. We say that G is shallow if there exists a normal subgroup N of G with index p such that

{σ ∈ G′_{p} | dim(V^{σ}) > dim(V^{G′_{p}})} ⊆ N;

If N is a subgroup satisfying the required hypotheses, we say that G is shallow with respect to N.

The main goal of this paper is to prove the following theorem.

Theorem 1. Suppose that G is shallow and dim(V^{G′_{p}}) + 2 ≤ dim(V). Then

depth(R^{G}) = dim(V^{G′_{p}}) + 2.

Moreover, we will show that the following statement holds without the assumption of shallowness.

Theorem 2. If dim(V^{G′_{p}}) + 2 ≤ dim(V) then

depth(R^{G}) ≥ dim(V^{G′_{p}}) + 2.

On the other hand, if dim(V^{G′_{p}}) + 2 > dim(V), then depth(R^{G}) = dim(V).

We note that, since the representation of G is faithful, dim(V^{G′_{p}}) < n and if dim(V^{G′_{p}}) = n − 1, then Theorem 2 applies and depth(R^{G}) = n.

If G is a cyclic group of order divisible by p, then G′_{p} is the Sylow p-subgroup of G, and there is only one subgroup N of index p. In fact, in this case, each element of G′_{p} which does not generate G′_{p}, lies in N. Hence G is shallow. This proves the following corollary.

Corollary 3. Let G be a cyclic group and P its Sylow p-subgroup. Suppose that dim(V^{P}) + 2 ≤ dim(V). Then

depth(R^{G}) = dim(V^{P}) + 2.

The result of Ellingsrud and Skjelbred [5] on cyclic p-groups is a special case of the corollary.

Remark.

(a) The subset {σ ∈ G′_{p} | dim(V^{σ}) > dim(V^{G′_{p}})} ⊆ G′_{p} generates a normal subgroup of G′_{p}. Hence if G is a p-group, then G is shallow if and only if it is not generated by the elements σ ∈ G with dim(V^{σ}) > dim(V^{G}).

(b) Suppose that G is shallow and consider the diagonal action of G on the direct sum mV = ⊕_{i=1}^{m}V of m copies of V. Then G is also shallow as a linear group on mV.

Example 4. Landweber and Stong [7, Appendix] considered groups G with the property that every non-identity element is a transvection (see also Smith [9, Section 8.2]). For such groups there exists a basis of V such that all group elements are represented by matrices of the form

(\begin{matrix}1 & 0 & ⋯ & 0 & α_{1} \\ 0 & 1 & & 0 & α_{2} \\ ⋮ & & ... & & ⋮ \\ & & & 1 & α_{n−1} \\ 0 & ⋯ & & 0 & 1 \end{matrix}) with α_{1}, …, α_{n−1} ∈ K.

By Theorem 8.2.13 in Smith [9], the rings of invariants S(V^{∗})^{G} are isomorphic to polynomial rings. By the above remark, these groups are shallow and remain shallow if we consider their action on direct sums mV of m copies of V. For m ≥ 2, Theorem 1 yields the equation

depth(S((mV)^{∗})^{G}) = m(n − 1) + 2.

Note that for m ≤ 2, the ring of invariants is Cohen-Macaulay. However for m > 2, the depth is less than the dimension of the representation so the ring of invariants is not Cohen-Macaulay (compare with Corollary 16).

In Section 1 we investigate the geometry associated to ideals in R^{G}. In particular we describe the variety associated to the image of the relative transfer for a normal subgroup (Theorem 7). Section 2 is devoted to constructing regular sequences in R^{G}. The proof of Theorem 1 is contained in Section 3 and forms the heart of the paper. The results of Section 1 and Section 2 are used in this proof. In Section 4 we discuss the Buchsbaum property and prove that, if G is a p-group, then R^{G} is Cohen-Macaulay if and only if it is Buchsbaum (Theorem 21).

1. Geometric preliminaries

Let \overline{K} denote the algebraic closure of K. For a vector space, U, over K define \overline{U} = U ⊗_{K} \overline{K}. We identify U with the subset U ⊗_{K} K ⊆ \overline{U}. For any subset X of \overline{R} define V(X) = \{v ∈ \overline{V} | f(v) = 0 ∀f ∈ X\} ⊆ \overline{V} .

Proposition 5. If I is an ideal in R^{G}, then V(I) = V(\overline{I}).

Proof. Since I ⊆ \overline{I}, we have V(\overline{I}) ⊆ V(I). On the other hand, any f ∈ \overline{I} can be written as a linear combination of elements from I with coefficients from \overline{K}. Thus if v ∈ V(I), it follows that f(v) = 0. Therefore V(I) ⊆ V(\overline{I}).

Proposition 6. Suppose I and J are ideals in R^{G}. Then V(I) = V(J) if and only if \sqrt{I} = \sqrt{J}.

Proof. This is essentially the Hilbert Nullstellensatz. For a finite group G, the affine variety associated to R^{G} is the orbit space \overline{V} /G. Corresponding to each ideal, I, in R^{G} is a subvariety of \overline{V} /G. The set V(I) is the preimage in \overline{V} of the subvariety of \overline{V} /G associated to I. By the Hilbert Nullstellensatz, \sqrt{I} = \sqrt{J} if and only if the corresponding varieties are equal. We refer the reader to Zariski and Samuel [12, Ch. VII, §3] and Serre [8, Ch. III, §12] for details.

Suppose H is a subgroup of G and let R be a set of left coset representatives for H in G. The relative transfer, Tr_{H}^{G}(f) = ∑_{ρ∈R} ρ(f), is an R^{G}-module homomorphism from R^{H} to R^{G}. Therefore the image of the relative transfer, I_{H}^{G}, is an ideal in R^{G}. If x is an element of a G-set then the isotropy subgroup of x is G_{x} = {σ ∈ G | σ(x) = x} and the orbit of x is the set Gx = {σ(x) | σ ∈ G}.

Theorem 7. If H is a normal subgroup of G, then

V(I_{H}^{G}) = \{v ∈ \overline{V} | p divides [G_{v} : H_{v}]\}.

Proof. Extend the action of G on V to a linear action on \overline{V}. Using this action, the image of T r_{H}^{G} applied to \overline{R}^{H} is \overline{I_{H}^{G}}. Thus it follows from Proposition 5 that we may assume that K is algebraically closed.

First suppose that p divides [G_{v} : H_{v}] for a point v ∈ V. Then there exists τ ∈ G_{v} such that τH has order p in G/H. Let ⟨H, τ⟩ denote the subgroup generated by H and τ. For any f ∈ R^{H} we have

(Tr_{H}^{⟨H,τ⟩}(f)) (v) = ∑_{i=0}^{p−1} f(τ^{−i}(v)) = ∑_{i=0}^{p−1} f(v) = 0,

and, since Tr_{H}^{G} = Tr_{⟨H,τ⟩}^{G} ∘ Tr_{H}^{⟨H,τ⟩}, it follows that (Tr_{H}^{G}(f)) (v) = 0. Thus v ∈ V(I_{H}^{G}).

Conversely, take v ∈ V with p ∤ [G_{v} : H_{v}] and choose f ∈ R such that for τ ∈ G, f(τ(v)) = 1 if τ(v) lies in Hv, and f(τ(v)) = 0 otherwise. Set g = ∏_{ρ∈H} ρ(f) ∈ R^{H}. Then for τ ∈ G we have

g(τ(v)) = \begin{cases}1 & if τ(v) ∈ Hv, \\ 0 & otherwise. \end{cases}

However τ(v) ∈ Hv if and only if τ ∈ H · G_{v} = G_{v} · H. Hence

(Tr_{H}^{G}(g)) (v) = |{τH ∈ G/H | τH ⊆ G_{v}H}| = |G_{v}/H_{v}| ≠ 0,

so v ∉ V(I_{H}^{G}), which completes the proof.

We include the following example to illustrate the fact that in the preceding theorem the hypothesis that H is a normal is necessary.

Example 8. Let G be the symmetric group Σ_{3} and consider the usual three dimensional permutation representation. Let H be the subgroup generated by the transposition (1, 2), take p = 2 and let v = (0, 1, 1). Since [G : H] is coprime to p, the relative transfer is surjective. In particular σ_{2} = x_{1}x_{2} +x_{1}x_{3} +x_{2}x_{3} = Tr_{H}^{G}(\frac{1}{3}σ_{2}) ∈ I_{H}^{G} and, since σ_{2}(v) = 1, we conclude that v is not in V(I_{H}^{G}). However, v is fixed by the transposition (2, 3) and (2, 3) is not in H. Thus 2 divides [G_{v} : H_{v}].

Corollary 9. If H is normal and [G : H] = p then V(I_{H}^{G}) = ∪_{τ∈G\H} \overline{V}^{τ}.

Proof. If [G : H] = p then, for any v ∈ \overline{V}, [G_{v} : H_{v}] is either p or 1. If v ∈ V(I_{H}^{G}) then [G_{v} : H_{v}] = p and v ∈ \overline{V}^{τ} for some τ ∈ G \ H. Conversely, if v ∈ \overline{V}^{τ} for some τ ∈ G \ H then [G_{v} : H_{v}] ≠ 1. Therefore [G_{v} : H_{v}] = p and v ∈ V(I_{H}^{G}).

If U is a subspace of V then we define the annihilator U^{⊥} = {φ ∈ V^{∗} | φ(v) = 0 ∀v ∈ U}. Let (U^{⊥}) denote the ideal in R generated by U^{⊥} and define I_{R^{G}}(U) = (U^{⊥}) ∩ R^{G}. For f ∈ R we write N^{G}(f) = ∏_{τ∈G} τ(f) ∈ R^{G}.

Proposition 10. If U is a KG-submodule of V then V(I_{R^{G}}(U)) = \overline{U}.

Proof. Using Proposition 5, we see that extending the field doesn’t change the variety and so we may assume that K is algebraically closed. It is easy to see that U ⊆ V(I_{R^{G}}(U)). To complete the proof we need only show that any vector in V \ U is also in V \ V(I_{R^{G}}(U)).

Suppose v ∈ V \ U. The orbit of v is a finite set and, since U is a KG-module, this set is disjoint from U. Therefore, since K is infinite, there exists a linear functional ℓ ∈ V^{∗} such that ℓ is zero on U and non-zero on each point in the orbit of v. Let f = N^{G}(ℓ). Clearly f ∈ I_{R^{G}}(U) and it is not hard to show that f(v) is non-zero. Therefore v ∈ V \ V(I_{R^{G}}(U)) as required.

We note that G\overline{U} ⊆ V(I_{R^{G}}(U)) and so, if U is not a KG-module, V(I_{R^{G}}(U)) properly contains \overline{U}.

Corollary 11. If H is a normal subgroup of G then V(I_{R^{G}}(V^{H})) = \overline{V}^{H}.

2. Constructing regular sequences

Suppose that U is a subspace of V. We write S(U^{⊥}) for the symmetric algebra of the annihilator of U. If U is a KG-module then so is U^{⊥} and S(U^{⊥})^{G} is a subalgebra of R^{G}. As in Section 1, for f ∈ R we write N^{G}(f) = ∏_{τ∈G} τ(f) ∈ R^{G}.

Proposition 12. Suppose U ≤ V is a KG-submodule of dimension 1 and choose x ∈ V^{∗} \ U^{⊥}. Let f_{1}, …, f_{m} be a regular sequence on R^{G} whose elements lie in S(U^{⊥})^{G} ≤ R^{G}. Then f_{1}, …, f_{m}, N^{G}(x) is also a regular sequence on R^{G}.

Proof. Let A denote the G-stable algebra S(U^{⊥}). Then R = A[x] is the polynomial ring over A in the variable x. Write f = N^{G}(x) and assume that

g_{1}f_{1} + ⋯ + g_{m}f_{m} + gf = 0 (1)

with g, g_{i} ∈ R^{G}. We must show that g lies in the ideal generated by f_{1}, …, f_{m} in R^{G}. View f and g_{i} as polynomials in x. Since the leading coefficient of f is an element of K, we may apply the division algorithm to get

g_{i} = q_{i}f + r_{i}

with q_{i}, r_{i} ∈ A[x] and deg_{x}(r_{i}) < deg_{x}(f). Suppose τ ∈ G. Observe that applying τ does not change the x-degree of a polynomial. Hence deg_{x}((τ − 1)r_{i}) < deg_{x}(f).

However,

0 = (τ − 1)g_{i} = (τ − 1)q_{i} · f + (τ − 1)r_{i},

so f divides (τ − 1)r_{i}, implying that (τ − 1)r_{i} = 0. Therefore r_{i} ∈ R^{G} and, consequently, q_{i} ∈ R^{G}. Setting h = r_{1}f_{1} + ⋯ + r_{m}f_{m}, we rewrite (1) as

(q_{1}f_{1} + ⋯ + q_{m}f_{m} + g)f + h = 0. (2)

Since f_{i} ∈ A, we have deg_{x}(h) < deg_{x}(f). Thus h = 0. Now dividing (2) by f shows that indeed g ∈ (f_{1}, …, f_{m})R^{G}.

By iteration, we now obtain the following corollary. We remind the reader that, if U is a subspace of V, then I_{R^{G}}(U) = (U^{⊥}) ∩ R^{G}.

Corollary 13. Let U = V^{G′_{p}} and m = dim(U). Suppose that f_{1}, …, f_{d} is a regular sequence for R^{G} whose elements lie in S(U^{⊥})_{+}^{G}. Then after a possible extension of K there exist g_{1}, …, g_{m} ∈ R^{G} such that:

(a) the images of g_{1}, …, g_{m} form a homogeneous system of parameters of the quotient ring R^{G}/I_{R^{G}}(U),

(b) f_{1}, …, f_{d}, g_{1}, …, g_{m} is a regular sequence for R^{G}. Moreover, the degree of the necessary extension of K is bounded above by φ(|G/G′_{p}|), where φ is the Euler totient function.

Proof. G acts on U as an abelian group of order coprime to p. Hence after adding enough roots of unity to K (which is an extension of degree ≤ φ(|G/G′_{p}|)), we obtain a basis e_{1}, …, e_{m} of U such that the one-dimensional subspaces Ke_{i} are KGsubmodules. Extend the e_{i} to a basis of V, take the dual basis x_{1}, …, x_{n} and set g_{i} = N^{G}(x_{i}) for i = 1, …, m. To see that g_{1}, …, g_{m} form a system of parameters of R^{G}/I_{R^{G}}(U), it is enough to show that they form a system of parameters of R/(U^{⊥}), since this is an integral extension of R^{G}/I_{R^{G}}(U). But writing τ(e_{i}) = a_{τ,i}e_{i} with a_{τ,i} ∈ K, we see that (τ−a_{τ,i}^{−1})x_{i} ∈ U^{⊥}, hence g_{i} ≡ a_{i}x_{i}^{|G|} mod (U^{⊥}) with a_{i} ∈ K\{0}. But, together with U^{⊥}, the x_{i} generate V^{∗}.

We show by induction on i that f_{1}, …, f_{d}, g_{1}, …, g_{i} is regular on R^{G}. In fact, by assumption, f_{1}, …, f_{d} ∈ S((Ke_{i})^{⊥}), and for j < i and τ ∈ G we have (τ(x_{j}))(e_{i}) = 0, hence also g_{j} ∈ S((Ke_{i})^{⊥}). So it follows from Proposition 12 that f_{1}, …, f_{d}, g_{1}, …, g_{i} is regular on R^{G}.

3. The proof of Theorems 1 and 2

Suppose that N is a normal subgroup of G with index p, and choose an element σ ∈ G \ N. Consider the ideal

J = (σ − 1)R^{N} ∩ R^{G} = {σ(g) − g | g ∈ R^{N}} ∩ R^{G} ⊆ R^{G}.

Lemma 14. The ideal J is non-zero and independent of the choice of σ.

Proof. To see that the ideal J is independent of the choice of σ, we first note that elements in the same coset of N produce the same ideal and thus it suffices to consider the non-trivial coset representatives σ^{i} for 1 ≤ i ≤ p − 1. Also, since N is normal, σ induces an automorphism on R^{N}. Next notice that σ^{i} −1 = (σ −1)(1+σ +⋯+σ^{i−1}) and thus (σ^{i} − 1)R^{N} ⊆ (σ − 1)R^{N}. Conversely, choosing j such that ij ≡ 1 (mod p) we have

(σ − 1)R^{N} = ((σ^{i})^{j} − 1)R^{N} = (σ^{i} − 1)(1 + σ^{i} + σ^{2i} + ⋯ + σ^{i(j−1)}) ⊆ (σ^{i} − 1)R^{N}.

Thus (σ^{i} − 1)R^{N} = (σ − 1)R^{N} as required.

We now show that J is non-zero. First observe that the kernel of σ − 1 acting on R^{N} is R^{G}. Since R^{G} is not equal to R^{N}, the kernel is a proper subset and we may choose g ∈ R^{N} such that (σ − 1)g ≠ 0. However (σ − 1)^{p}g = (σ^{p} − 1)g = 0. Choose i so that (σ − 1)^{i}g ≠ 0 and (σ − 1)^{i+1}g = 0. Then (σ − 1)^{i}g is a non-zero element of J.

Proposition 15. depth_{J} (R^{G}) = min{2, ht(J)}.

Proof. Assume, by way of contradiction, that there are f_{1}, f_{2}, f_{3} ∈ J which form a regular sequence for R^{G}. By definition of J there exist g_{i} ∈ R^{N} such that σ(g_{i})−g_{i} = f_{i} for i = 1, 2, 3, and we have

0 = |\begin{matrix}f_{1} & f_{2} & f_{3} \\ f_{1} & f_{2} & f_{3} \\ g_{1} & g_{2} & g_{3} \end{matrix}| = u_{23} · f_{1} + u_{31} · f_{2} + u_{12} · f_{3} with u_{ij} = |\begin{matrix}f_{i} & f_{j} \\ g_{i} & g_{j} \end{matrix}| .

Observe that u_{ij} ∈ R^{G}, since

σ(u_{ij}) = |\begin{matrix}f_{i} & f_{j} \\ g_{i} + f_{i} & g_{j} + f_{j} \end{matrix}| = u_{ij}.

The regularity of f_{1}, f_{2}, f_{3} implies that u_{12} must lie in the ideal in R^{G} generated by f_{1} and f_{2}, so

g_{2}f_{1} − g_{1}f_{2} = u_{12} = h_{1}f_{1} + h_{2}f_{2}

with h_{1}, h_{2} ∈ R^{G}, which yields

(g_{2} − h_{1})f_{1} = (g_{1} + h_{2})f_{2}.

However f_{1}, f_{2} is a regular sequence on R^{G} and hence also a partial homogeneous system of parameters for R^{G}. Since R is an integral extension of R^{G}, any partial homogeneous system of parameters for R^{G} is a partial homogeneous system of parameters for R. Moreover R is Cohen-Macaulay and so f_{1}, f_{2} is a regular sequence on R. Therefore f_{1} and f_{2} are coprime as polynomials in R. Hence f_{1} divides g_{1} +h_{2}, so

g_{1} = q · f_{1} − h_{2} with q ∈ R.

Since f_{1} and h_{2} are invariants, we obtain

f_{1} = (σ − 1)g_{1} = f_{1} · (σ − 1)q, or (σ − 1)q = 1,

which is impossible.

This shows that depth_{J} (R^{G}) ≤ 2. Also clearly depth_{J} (R^{G}) ≤ ht(J). On the other hand, if ht(J) = 1 then there exists f ∈ J which forms a regular sequence for R^{G}, so depth_{J} (R^{G}) = 1. If ht(J) ≥ 2, then there exist homogeneous f_{1}, f_{2} ∈ J such that ht(f_{1}, f_{2}) = 2. Thus a relation g_{1}f_{1} + g_{2}f_{2} = 0 with g_{i} ∈ R^{G} implies that f_{1} divides g_{2} in R. However, since f_{1} and g_{2} are both invariant, we conclude that f_{1} divides g_{2} in R^{G}. This shows that f_{1}, f_{2} is regular for R^{G}. Hence depth_{J} (R^{G}) = 2 if ht(J) ≥ 2.

As a consequence of Proposition 15 we get the following generalization of Campbell et al. [4, Theorem 1.2]. This result also follows from Kemper [6, Theorem 2.3].

Corollary 16. If G contains a normal subgroup N of index p then the ring of vector invariants S((mV)^{∗})^{G} is not Cohen-Macaulay for m ≥ 3.

Proof. By Lemma 14, there exists a homogeneous g ∈ S(V^{∗})^{N} such that f = (σ − 1)g ∈ S(V^{∗})^{G}\{0}. Let g_{i}, …, g_{m} be the elements corresponding to g in S((mV)^{∗})^{N}. Define f_{i} = (σ − 1)g_{i}. The sequence f_{1}, …, f_{m} is a partial homogeneous system of parameters for S((mV)^{∗})^{G} lying inside J. From Proposition 15, depth_{J} (R^{G}) ≤ 2. Therefore, if m ≥ 3, we have a partial homogeneous of parameters which fails to be a regular sequence so R^{G} is not Cohen-Macaulay.

We remind the reader that I_{N}^{G} denotes the image of the relative transfer and that I_{R^{G}}(U) = (U^{⊥}) ∩ R^{G}.

Proposition 17.

\sqrt{J} = \sqrt{I_{N}^{G}} = ⋂_{τ∈G\N} I_{R^{G}}(V^{τ}).

Proof. Let J = ∩_{τ∈G\N}I_{R^{G}}(V^{τ}). Note that J is a radical ideal. Thus by Proposition 6 it is sufficient to show that V(J) = V(I_{N}^{G}) = V(J).

Suppose that f ∈ J and v ∈ \overline{V}^{τ} for some τ ∈ G \ N. Then, using Lemma 14, f = (τ −1)(h) for some h ∈ R^{N}. Thus f(v) = h(τ^{−1}(v))−h(v) = 0. Hence v ∈ V(J) and V(J) ⊆ V(J).

Since [G : N] = p, we may use Corollary 9 to get

V(I_{N}^{G}) = ∪_{τ∈G\N} \overline{V}^{τ} ⊆ ∪_{τ∈G\N}V(I_{R^{G}}(V^{τ})) = V(∩_{τ∈G\N}I_{R^{G}}(V^{τ})) = V(J).

Suppose f ∈ I_{N}^{G}. Then for some h ∈ R^{N}, f = T r_{N}^{G}(h) = ∑_{i=0}^{p−1} σ^{i}(h) = (σ − 1)^{p−1}(h). Thus f ∈ J and I_{N}^{G} ⊆ J. Therefore V(J) ⊆ V(I_{N}^{G}) ⊆ V(J).

Now we consider the ideal

I = I_{R^{G}}(V^{G′_{p}}) ⊆ R^{G}

and bring the hypothesis that G is shallow into play.

Proposition 18. Suppose that G is shallow with respect to N. Then \sqrt{I} = \sqrt{J} and

depth_{I}(R^{G}) = min{2, ht(I)}.

Proof. We first prove that \sqrt{I} = \sqrt{J}. By Proposition 6 it is sufficient to show that the corresponding varieties are equal. From Corollary 11 we know that V(I) = \overline{V}^{G′_{p}}. From Proposition 17 and Corollary 9 we know that V(J) = ∪_{τ∈G\N} \overline{V}^{τ}.

Let ρ be an arbitrary element in G \ N. The order of ρ is kp^{m} for some k relatively prime to p and some m ≥ 0. However, [G : N] = p so m > 0 and ρ^{k} ∉ N. Note that G′_{p} contains all of the p-Sylow subgroups of G and, therefore, ρ^{k} ∈ G′_{p}.

Hence \overline{V}^{G′_{p}} ⊆ \overline{V}^{ρ^{k}}. Since G is shallow with respect to N and ρ^{k} ∉ N, we have dim(V^{ρ^{k}}) ≤ dim(V^{G′_{p}}). Therefore \overline{V}^{ρ^{k}} = \overline{V}^{G′_{p}}. However \overline{V}^{ρ} ⊆ \overline{V}^{ρ^{k}} ⊆ ∪_{τ∈G\N} \overline{V}^{τ}. Hence \overline{V}^{ρ} ⊆ \overline{V}^{G′_{p}} ⊆ ∪_{τ∈G\H} \overline{V}^{τ}. Since this is true for an arbitrary ρ ∈ G \ N, the varieties are equal.

Using Proposition 15, and the fact that an ideal and its radical share a common depth and height, gives depth_{I}(R^{G}) = min{2, ht(I)}.

We can now give a proof of the theorems stated in the Introduction.

Proof of Theorem 1 and Theorem 2. First we remark that the depth of R^{G} does not change if the ground field K is extended. This follows from the characterization of depth in terms of Ext-functors (see, for example, Benson [1, Theorem 4.4.3]).

We first prove Theorem 2. As above, we write U = V^{G′_{p}}. Let d = min{2, dim(U^{⊥})}, then we can assume that K is large enough that there exist x_{1}, …, x_{d} ∈ U^{⊥} such that the ideal generated by f_{i} := N^{G}(x_{i}) (i = 1, …, d) has height d. In fact, the x_{i} can be obtained by “Dade’s construction” (see Stanley [10, page 483]). It follows that the f_{i} lie in S(U^{⊥}), and they form a regular sequence since d ≤ 2 (see the proof of Proposition 15). Now after a further extension of K, Corollary 13 yields g_{1}, …, g_{m} with m = dim(U), such that f_{1}, …, f_{d}, g_{1}, …, g_{m} are a regular sequence on R^{G} and the images of g_{1}, …, g_{m} form a homogeneous system of parameters for R^{G}/I. Hence

depth(R^{G}) ≥ m + d = min{m + 2, dim(V)}

since d = min{2, dim(V) − m}. Observe that up to now we have not used the hypothesis that G is shallow, so we have proved Theorem 2.

We now show that if G is shallow, f_{1}, …, f_{d}, g_{1}, …, g_{m} is a maximal regular sequence. From this the equality depth(R^{G}) = m + d follows. In fact, by Proposition 18 we have that depth_{I}(R^{G}) = d. Assume that there exists f ∈ R_{+}^{G} such that f_{1}, …, f_{d}, g_{1}, …, g_{m}, f is a regular sequence for R^{G}. Since the images of g_{1}, …, g_{m} form a homogeneous system of parameters for R^{G}/I, there is a k ∈ ℕ such that f^{k} ∈ (g_{1}, …, g_{m})R^{G} + I. But f_{1}, …, f_{d}, g_{1}, …, g_{m}, f^{k} is also regular, and this is not changed if we add an element from (g_{1}, …, g_{m})R^{G} to f^{k}, so we may assume that f^{k} ∈ I. But then f_{1}, …, f_{d}, f^{k} ∈ I is regular for R^{G}, in contradiction to depth_{I}(R^{G}) = d. This completes the proof.

4. Some remarks on the Buchsbaum property

For graded or local rings, the Cohen-Macaulay property can be weakened to obtain the so-called Buchsbaum property. We concentrate on the graded case, so let A = ⊕_{i=0}^{∞}A_{i} be a finitely generated graded algebra over a field K = A_{0} and let A_{+} = ⊕_{i=1}^{∞}A_{i}. We begin with a few definitions. We refer the reader to Stückrad and Vogel [11] for a more detailed discussion.

A sequence a_{1}, …, a_{r} in A_{+} is called weakly regular if: for every i ≤ r and m ∈ A, m · a_{i} ∈ (a_{1}, … a_{i−1}) implies m · A_{+} ⊆ (a_{1}, … a_{i−1}). A is Buchsbaum if every system of parameters is a weakly regular sequence. A system of parameters a_{1}, …, a_{n} is said the measure the depth of A if a_{1}, …, a_{d} is a regular sequence for d = depth(A).

Remark 19. Suppose that K is the finite field with q = p^{s} elements. Then GL_{n}(K) is finite and R^{GL_{n}(K)} is generated by the Dickson invariants d_{1}, …, d_{n} (with deg(d_{i}) = q^{n} − q^{n−i}) (see Smith [9, Section 8.1]). For any subgroup G ⊆ GL_{n}(K), the Dickson invariants form a homogeneous system of parameters for R^{G}. Landweber and Stong [7] conjectured that d_{1}, …, d_{n} measure the depth of R^{G}. This conjecture has recently been proved by Bourguiba and Zarati [2].

Proposition 20. If A is Buchsbaum, then every system of parameters measures the depth.

Proof. Suppose that A is Buchsbaum and assume, by way of contradiction, that there is a system of parameters a_{1}, …, a_{n} which does not measure the depth. Let k < d := depth(A) be maximal such that a_{1}, …, a_{k} is regular. Every regular sequence of length less than d can be extended to a regular sequence of length d. Therefore there exists f ∈ A_{+} such that a_{1}, …, a_{k}, f is regular. Since a_{1}, …, a_{k+1} is not regular, there exists g ∉ (a_{1}, …, a_{k}) such that ga_{k+1} ∈ (a_{1}, …, a_{k}). However, since A is Buchsbaum, a_{1}, …, a_{k+1} is weakly regular. Therefore g · A_{+} ⊆ (a_{1}, …, a_{k}). In particular g · f ∈ (a_{1}, …, a_{k}). By the regularity of a_{1}, …, a_{k}, f, this implies g ∈ (a_{1}, …, a_{k}) and gives the required contradiction.

This shows that the Buchsbaum property is useful. However, returning to the situation of Section 3 where A = R^{G} is a ring of invariants, we can use Corollary 13 to prove the following.

Theorem 21. If G is a p-group, then R^{G} is Cohen-Macaulay if and only if it is Buchsbaum.

Proof. If R^{G} is Cohen-Macaulay, it is clearly Buchsbaum. To prove the converse, assume that R^{G} is Buchsbaum. Let U = V^{G′_{p}} and consider the subalgebra S(U^{⊥}). Observe that G′_{p} = G stabilizes U^{⊥}, and that there exists a basis x_{1}, …, x_{d} of U^{⊥} such that the action of G on U^{⊥} takes the form of upper triangular matrices. Setting f_{i} = N^{G}(x_{i}), we obtain ht(f_{1}, …, f_{d}) = d. Let k be maximal such that the subsequence f_{1}, …, f_{k} is regular. By Corollary 13 there exist g_{1}, …, g_{m} ∈ R^{G} with m = dim(U) such that f_{1}, …, f_{k}, g_{1}, …, g_{m} is a regular sequence, and since G is a p-group, m is positive, so depth(R^{G}) ≥ k + 1. Now assume k < d. Then f_{1}, …, f_{k+1} is not regular and can be extended to a homogeneous system of parameters which does not measure the depth. This contradicts Proposition 20. Hence we must have k = d, and since m + d = dim(V), it follows that R^{G} is Cohen-Macaulay.

In all examples, known to us, of rings of invariants which are not Cohen-Macaulay, there are homogeneous systems of parameters which fail to measure the depth. This prompts us to make the following conjecture.

Conjecture 22. For rings of invariants of finite linear groups acting on polynomial algebras, the Cohen-Macaulay property and the Buchsbaum property are equivalent.

References

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

[2] Dorra Bourguiba, Said Zarati, Depth and the Steenrod algebra, Invent. math. 128 (1997), 589– 602.

[3] Winfried Bruns, Jürgen Herzog, Cohen-Macaulay Rings, Cambridge University 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 1996.

[5] Geir Ellingsrud, Tor Skjelbred, Profondeur d’anneaux d’invariants en caractéristique p, Compos. Math. 41 (1980), 233–244.

[6] Gregor Kemper, On the Cohen-Macaulay Property of Modular Invariant Rings, IWR Preprint 97-38, Heidelberg 1997.

[7] Peter S. Landweber, Robert E. Stong, The Depth of Rings of Invariants over Finite Fields, in: Proc. New York Number Theory Seminar, 1984, Lecture Notes in Math. 1240, Springer-Verlag, New York 1987.

[8] J.-P. Serre, Algebraic Groups and Class Fields, Springer-Verlag, Heidelberg 1988.

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

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

[11] J. Stückrad, W. Vogel, Buchsbaum Rings and Applications, Springer-Verlag, Heidelberg 1986.

[12] O. Zariski, P. Samuel, Commutative Algebra, vol. 2, Springer-Verlag, Heidelberg 1960.

Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, K7L 3N6, Canada

E-mail address: eddy@mast.queensu.ca, hughesi@mast.queensu.ca,
shank@mast.queensu.ca, wehlau@mast.queensu.ca

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

E-mail address: Gregor.Kemper@iwr.uni-heidelberg.de

Department of Mathematics and Computer Science, Royal Military College, Kingston, Ontario, K7K 5L0, Canada E-mail address: wehlau@rmc.ca