Local cohomology modules of invariant rings

Abstract

Let $K$ be a field and let $R$ be a regular domain containing $K$. Let $G$ be a finite subgroup of the group of automorphisms of $R$. We assume that $|G|$ is invertible in $K$. Let $R^G$ be the ring of invariants of $G$. Let $I$ be an ideal in $R^G$. Fix $i\ge 0$. If $R^G$ is Gorenstein then, \begin{enumerate} \item $injdi{m}_{R^G}{H}_I^i(R^G)\le \dim Supp{H}_I^i(R^G).$ \item $H_{\mathfrak{m}}^j(H_I^i(R^G))$ is injective, where $\mathfrak{m}$ is any maximal ideal of $R^G$. \item ${\mu }_j(P,H_I^i(R^G))={\mu }_j(P^{\prime },H_{IR}^i(R))$ where $P^{\prime }$ is any prime in $R$ lying above $P$. \end{enumerate} We also prove that if $P$ is a prime ideal in $R^G$ with $R_P^G$ \textit{not Gorenstein} then either the bass numbers ${\mu }_j(P,H_I^i(R^G))$ is zero for all $j$ or there exists $c$ such that ${\mu }_j(P,H_I^i(R^G))=0$ for $j<c$ and ${\mu }_j(P,H_I^i(R^G))>0$ for all $j\ge c$.

Some of the results in the previous version was already known by work of Núñez-Betancourt. Those results have been removed in this version. Also some new results are added