Normal \(\mathbb{Z}_ r\)-graded rings and normal cyclic covers
Let \(\bigoplus^{r-1}_{i=0}S_ i\) be a noetherian \(\mathbb{Z}_ r\)-graded domain (i.e. \(S_ iS_ j\subseteq S_ k\) with \(k=i+j \text{mod} r)\). Put \(R=S_ 0\); then each \(S_ i\) turns out to be an \(R\)-module of rank 1 and if \(S\) is normal, then each \(S_ i\) is isomorphic to a divisorial ideal of \(R\). The pair \((S,u)\), for any choice of \(u\neq 0\) in \(S_ 1\), determines a fractional divisor \(D\) of \(R\) with \(rD=\text{div}_ R(u^ r)\); conversely, starting with \(R\) and taking \(f\in R\), \(D\in\text{Div}(R)\otimes_ \mathbb{Z}\mathbb{Q}\) with \(rD=\text{div}_ R(P)\), we may reconstruct a \(\mathbb{Z}_ r\)-graded extension \(S\) of \(R\). Such extensions are the ``cyclic covers'' of \(R\). The authors study relations between properties of \(S\) and \(R\); in particular they look for conditions which make \(S\) normal, Gorenstein or regular. These results are applied to the theory of normal graded rings: if such a ring has an isolated singularity, then for all closed points \(x\in\text{Proj}(R)\), the local ring \(A_ x\) must have a regular cyclic cover; the structure of these covers is determined by the paper.
scientific article; zbMATH DE number 219906