Nearly Gorenstein cyclic quotient singularities. (English) Zbl 1481.13012
In the paper under review, the authors classify nearly Gorenstein rings for the invariant ring \(R^G\) where \(R=\Bbbk[[ x_1,\dotsc,x_d]]\) is a formal power series ring over an algebraically closed field \(\Bbbk,\) \(G\) is a finite cyclic subgroup of \(\mathrm{GL}(d,\Bbbk)\) acting linearly on \(R\), and \(|G|\) is invertible in \(\Bbbk.\) Since \(G\) is cyclic, the invariant ring is monomial, hence the classification can be described in terms of a numerical criterion.
Nearly Gorenstein property extends the property of being Gorenstein and the authors also provide a family of examples where nearly Gorenstein property fails by defining the residue of the trace ideal of the corresponding canonical module.
Moreover, the authors provide an exhaustive list of cyclic quotient singularities for \(d=3\) and \(4\le |G|\le 7\) and for \(d=4\) and \(4\le |G|\le 6.\) The cases where \(d=2\) or \(|G|\le3\) are already covered by their main result.
Finally, the paper contains an addendum to Ding’s classification of nearly Gorenstein quotient singularities when \(d=2.\)
Nearly Gorenstein property extends the property of being Gorenstein and the authors also provide a family of examples where nearly Gorenstein property fails by defining the residue of the trace ideal of the corresponding canonical module.
Moreover, the authors provide an exhaustive list of cyclic quotient singularities for \(d=3\) and \(4\le |G|\le 7\) and for \(d=4\) and \(4\le |G|\le 6.\) The cases where \(d=2\) or \(|G|\le3\) are already covered by their main result.
Finally, the paper contains an addendum to Ding’s classification of nearly Gorenstein quotient singularities when \(d=2.\)
Reviewer: Ugur Madran (Eqaila)
MSC:
| 13A50 | Actions of groups on commutative rings; invariant theory |
| 13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |
| 14L30 | Group actions on varieties or schemes (quotients) |
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