Levelness versus nearly Gorensteinness of homogeneous rings. (English) Zbl 07784744
Summary: Levelness and nearly Gorensteinness are well-studied properties of graded rings as a generalized notion of Gorensteinness. In this paper, we compare the strength of these properties. For any Cohen-Macaulay homogeneous affine semigroup ring \(R\), we give a necessary condition for \(R\) to be non-Gorenstein and nearly Gorenstein in terms of the \(h\)-vector of \(R\) and we show that if \(R\) is nearly Gorenstein with Cohen-Macaulay type 2, then it is level. We also show that if Cohen-Macaulay type is more than 2, there are 2-dimensional counterexamples. Moreover, we characterize nearly Gorensteinness of Stanley-Reisner rings of low-dimensional simplicial complexes.
MSC:
| 13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |
| 13M05 | Structure of finite commutative rings |
Keywords:
level; nearly Gorenstein; homogeneous domain; affine semigroup rings; Stanley-Reisner ringsSoftware:
Macaulay2References:
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