Local cohomology associated to the radical of a group action on a noetherian algebra
Abstract
An arbitrary group action on an algebra $R$ results in an ideal $\mathfrak{r}$ of $R$. This ideal $\mathfrak{r}$ fits into the classical radical theory, and will be called the radical of the group action. If $R$ is a noetherian algebra with finite GKdimension and $G$ is a finite group, then the difference between the GKdimensionsof $R$ and that of $R/\mathfrak{r}$ is called the pertinency of the group action. We provide some methods to find elements of the radical, which helps to calculate the pertinency of some special group actions. The $\mathfrak{r}$adic local cohomology of $R$ is related to the singularities of the invariant subalgebra $R^G$. We establish an equivalence between the quotient category of the invariant $R^G$ and that of the skew group ring $R*G$ through the torsion theory associated to the radical $\mathfrak{r}$. With the help of the equivalence, we show that the invariant subalgebra $R^G$ will inherit certain CohenMacaulay property from $R$.
 Publication:

arXiv eprints
 Pub Date:
 December 2017
 DOI:
 10.48550/arXiv.1712.00657
 arXiv:
 arXiv:1712.00657
 Bibcode:
 2017arXiv171200657H
 Keywords:

 Mathematics  Rings and Algebras;
 Primary 16D90;
 16E65;
 Secondary 16B50
 EPrint:
 26 pages