论文标题
关于模块化不变理论的多项式不变环
On polynomial invariant rings in modular invariant theory
论文作者
论文摘要
令$ \ bbbk $为特征$ p> 0 $，$ v $ a有限维$ \ bbbk $ - vector-vector-vector-vector-v $ a $ g $ a有限的$ p $ - group代理$ \ bbbk $ - 在$ v $上固定。令$ s = \ sym v^*$。我们证明，$ s^g $是一个多项式戒指，并且仅当其奇异基因座的尺寸小于$ \ rank_ \ bbbk v^g $时。确认了shank-wehlau-broer的猜想，我们表明，如果$ s^g $是$ s $的直接汇总，则$ s^g $是一个多项式环，在以下情况下： \ item $ \ bbbk = \ bbf_p $和$ \ rank_ \ bbbk v^g = 4 $;或者 \ item $ | g | = p^3 $。 \ end {enumerate}为了证明上述结果，我们还表明，如果$ \ rank_ \ bbbk v^g \ geq \ geq \ rank_ \ bbbk v -2 $ 2 $，则Hilbert Ideal $ \ hilbertideal_ {g，s} $是完整的交叉点。
Let $\Bbbk$ be a field of characteristic $p>0$, $V$ a finite-dimensional $\Bbbk$-vector-space, and $G$ a finite $p$-group acting $\Bbbk$-linearly on $V$. Let $S = \Sym V^*$. We show that $S^G$ is a polynomial ring if and only if the dimension of its singular locus is less than $\rank_\Bbbk V^G$. 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: \begin{enumerate} \item $\Bbbk = \bbF_p$ and $\rank_\Bbbk V^G = 4$; or \item $|G| = p^3$. \end{enumerate} In order to prove the above result, we also show that if $\rank_\Bbbk V^G \geq \rank_\Bbbk V - 2$, then the Hilbert ideal $\hilbertIdeal_{G,S}$ is a complete intersection.
