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The trace of the canonical module

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Abstract

The trace of the canonical module (the canonical trace) determines the non-Gorenstein locus of a local Cohen-Macaulay ring. We call a local Cohen-Macaulay ring nearly Gorenstein, if its canonical trace contains the maximal ideal. Similar definitions can be made for positively graded Cohen-Macaulay K-algebras. We study the canonical trace for tensor products and Segre products of algebras, as well as of (squarefree) Veronese subalgebras. The results are used to classify the nearly Gorenstein Hibi rings. We study connections between the class of nearly Gorenstein rings and that of almost Gorenstein rings. We show that in dimension one, the former class includes the latter.

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Acknowledgement

We gratefully acknowledge the use of the Singular ([6]) software and of the numericalsgps package ([7]) in GAP ([10]) for our computations. We thank the anonymous referees for their comments which improved the exposition and strengthened some results, in particular for kindly suggesting Theorem 6.6 and its proof.

Takayuki Hibi was partially supported by JSPS KAKENHI 19H00637.

Dumitru Stamate was supported by a fellowship at the Research Institute of the University of Bucharest (ICUB).

Author information

Authors and Affiliations

  1. Fachbereich Mathematik, Universität Duisburg-Essen, Campus Essen, 45117, Essen, Germany

    Jürgen Herzog

  2. Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka, 565-0871, Japan

    Takayuki Hibi

  3. ICUB/Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei 14, Bucharest, 010014, Romania

    Dumitru I. Stamate

Authors
  1. Jürgen Herzog
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  2. Takayuki Hibi
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  3. Dumitru I. Stamate
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Correspondence to Dumitru I. Stamate.

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Herzog, J., Hibi, T. & Stamate, D.I. The trace of the canonical module. Isr. J. Math. 233, 133–165 (2019). https://doi.org/10.1007/s11856-019-1898-y

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  • DOI: https://doi.org/10.1007/s11856-019-1898-y

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