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 ofmcopies of that representation is not Cohen-Macaulay form 3. In the second section

of the paper we use Poincar´e 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(jGj ? 1). If the ring of invariants is a hypersurface, the upper

bound can be improved to [G].