On polynomial invariant rings in modular invariant theory

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 $S=\mathrm{Sym}{V}^{⁎}$. Confirming a conjecture of Shank-Wehlau-Broer, we show that if $S^G$ is a direct summand of S, then $S^G$ is a polynomial ring, in the following cases:

• (a)

  $k=F_p$ and ${\dim }_{k}V=4$; or

• (b)

  $\left|G\right|=p^3$.

In order to prove the above result, we also show that if ${\dim }_k{V}^G\ge {\dim }_{k}V-2$, then the Hilbert ideal $h_{G,S}$ is a complete intersection.