Two-dimensional vector invariant rings of Abelian p-groups

For any faithful representation V of a non-trivial p-group over a field of characteristic p>0, it is known that the ring of vector invariants of m copies of V is not Cohen–Macaulay if m 3. However, much less is known about the case m=2. In this paper we show that, if m=2 and the group is an Abelian p-group, then the ring of invariants of 2V is a complete intersection in some cases and is not Cohen–Macaulay in most cases. As a corollary we obtain that if the field is and the ring of invariants of the representation V is a polynomial ring, then the ring of invariants of 2V is either a complete intersection or not Cohen–Macaulay.