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Graded Bourbaki ideals of graded modules

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Abstract

In this paper we study graded Bourbaki ideals. It is a well-known fact that for torsionfree modules over Noetherian normal domains, Bourbaki sequences exist. We give criteria in terms of certain attached matrices for a homomorphism of modules to induce a Bourbaki sequence. Special attention is given to graded Bourbaki sequences. In the second part of the paper, we apply these results to the Koszul cycles of the residue class field and determine particular Bourbaki ideals explicitly. We also obtain in a special case the relationship between the structure of the Rees algebra of a Koszul cycle and the Rees algebra of its Bourbaki ideal.

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Acknowledgements

We gratefully acknowledge the use of the computer algebra software CoCoA [1] and Singular [6] for our computations. The authors would like to thank Ernst Kunz and Bernd Ulrich for discussions around Lemma 4.3. We thank an anonymous referee for suggestions which clarified the presentation. This paper was written while the second author visited Essen for three months and Bucharest for two weeks. During the visits, Professor Herzog and Professor Stamate were extremely kind to him every day and he would like to express sincere gratitude for their hospitality. The third author is grateful to Professor Herzog and the Department of Mathematics of the Universität Duisburg-Essen for being excellent hosts, one more time.

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Authors and Affiliations

  1. Fachbereich Mathematik, Universität Duisburg-Essen, Fakultät für Mathematik, 45117, Essen, Germany

    Jürgen Herzog

  2. Department of Mathematics and Informatics, Graduate School of Science and Engineering, Chiba University, Yayoi-cho 1-33, Inage-ku, Chiba, 263-8522, Japan

    Shinya Kumashiro

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

    Dumitru I. Stamate

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  1. Jürgen Herzog
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  2. Shinya Kumashiro
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  3. Dumitru I. Stamate
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Correspondence to Shinya Kumashiro.

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S. Kumashiro was supported by JSPS KAKENHI Grant Number JP19J10579 and JSPS Overseas Challenge Program for Young Researchers. D. I. Stamate was partly supported by the University of Bucharest, Faculty of Mathematics and Computer Science through the 2019 Mobility Fund.

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Herzog, J., Kumashiro, S. & Stamate, D.I. Graded Bourbaki ideals of graded modules. Math. Z. 299, 1303–1330 (2021). https://doi.org/10.1007/s00209-021-02724-8

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  • DOI: https://doi.org/10.1007/s00209-021-02724-8

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