On polynomial invariant rings in modular invariant theory

Abstract

Let k be a field of characteristic p>0, V a finite-dimensional k-vector-space, and G a finite p-group acting k-linearly on V. Let S=SymV⁎. Confirming a conjecture of Shank-Wehlau-Broer, we show that if SG is a direct summand of S, then SG is a polynomial ring, in the following cases:(a)k=Fp and dimk⁡V=4; or(b)|G|=p3. In order to prove the above result, we also show that if dimk⁡VG≥dimk⁡V−2, then the Hilbert ideal hG,S is a complete intersection.