Non Cohen-Macaulay Vector Invariants and a Noether Bound for a Gorenstein Ring of Invariants

This paper contains two essentially independent results in the invariant theory of finite groups. First we prove that, for any faithful representation of a non-trivial p-group over a field of characteristic p, the ring of vector invariants of m copies of that representation is not Cohen-Macaulay for m ≥ 3. In the second section of the paper we use Poincaré series methods to produce upper bounds for the degrees of the generators for the ring of invariants as long as that ring is Gorenstein. We prove that, for a finite non-trivial group G and a faithful representation of dimension n with n > 1, if the ring of invariants is Gorenstein then the ring is generated in degrees less than or equal to n(|G| − 1). If the ring of invariants is a hypersurface, the upper bound can be improved to |G|. Introduction Let V be a vector space of dimension n over a field k, let V ∗ denote the dual of V and let k[V ] denote the symmetric algebra of V ∗. By choosing a basis, {x1, . . . , xn}, for V ∗, we can identify k[V ] with the polynomial algebra k[x1, . . . , xn]. Let G be a finite subgroup of GL(V ). The elements of G act as degree preserving algebra automorphisms on k[V ]. We denote the subring of G-invariant polynomials by k[V ]G. A homogeneous system of parameters for k[V ]G is a collection of homogeneous elements, {a1, . . . , an}, of k[V ]G such that k[V ]G is a finitely generated k[a1, . . . , an]-module. By the Noether normalization theorem k[V ]G contains a homogeneous system of parameters (see, for example, [5, Theorem 5.3.3] or [1, Theorem 2.2.7]). k[V ]G is Cohen-Macaulay if for every homogeneous system of parameters {a1, . . . , an}, k[V ]G is a free k[a1, . . . , an]-module. If the characteristic of k does not divide the order of G, then k[V ]G is always Cohen-Macaulay ([3], see [2, Section 6.4]). However, when the order of G divides the characteristic of k, k[V ]G often fails to be CohenMacaulay. Let mV denote the faithful representation of G formed by taking the direct sum of m copies of V . k[mV ]G is known as the ring of vector invariants (see [9]). In the first section of the paper we prove that if the characteristic of k is p and V is a faithful representation of a non-trivial p-group P, then k[mV ]P is not Cohen-Macaulay when m ≥ 3. When G is finite, k[V ]G is finitely generated. In [4], Noether proved that if k has characteristic zero then k[V ]G is generated by elements in degrees less than or equal to |G|. This is not true when the characteristic of k divides |G|. Any two minimal homogeneous generating sets for k[V ]G have the same number of elements of each degree. The maximum such degree is called the Noether number of the representation. We will refer to an upper Received by the editors July 23, 1997; revised February 4, 1998. This research is supported in part by the NSERC of Canada. AMS subject classification: 13A50. c ©Canadian Mathematical Society 1999. 155 156 H. E. A. Campbell, A. V. Geramita, I. P. Hughes, R. J. Shank and D. L. Wehlau bound on the Noether number as a Noether bound. If the characteristic of k does not divide |G| then, as long as n > 1 and G is non-trivial, n(|G| − 1) is a Noether bound for the representation (see [6, Corollary 2.5]). A Cohen-Macaulay ring which is isomorphic to its own canonical module is called a Gorenstein ring. If k[V ]G is a Cohen-Macaulay domain then k[V ]G is Gorenstein if and only if the Poincaré series satisfies the duality condition P ( k[V ], 1/t ) = (−1)tP ( k[V ], t ) for some integer m (see [8, Theorem 8.1]). In the second section of the paper we use Poincaré series methods to find a Noether bound when k[V ]G is Gorenstein. We show that if G is non-trivial and n > 1 then k[V ]G is generated in degrees less than or equal to n(|G| − 1). If p is the characteristic of k, G is a p-group and k[V ]G is Cohen-Macaulay, then k[V ]G is Gorenstein so this bound applies. If k[V ]G is a hypersurface then the Noether bound can be improved to |G|. 1 Vector Invariants In this section we assume that the characteristic of k is p and that P is a non-trivial psubgroup of GL(V ). We use V ∗ to denote the dual to V and (V ∗)P to denote the subspace of V ∗ fixed by P. We remind the reader that every linear action of a p-group over a field of characteristic p has a non-zero fixed point. Choose z ∈ V ∗ so that z represents a non-zero element in ( V ∗/(V ∗)P )P . There is an Fp-subspace of (V ∗)P, say W , such that the P-orbit of z is the set z + W . Choose y to be a non-zero element in W and let U be an Fp-subspace of W which is complementary to Fp y. For any g ∈ P, g(z) = z +βg y + ug for some βg ∈ Fp and ug ∈ U . For any t ∈ V ∗ define N(t ; U ) = ∏