Abstract
In this paper, we study Noetherian local rings R having a finite number of trace ideals. We proved that such rings are of dimension at most two. Furthermore, if the integral closure of R/H, where H is the zeroth local cohomology, is equi-dimensional, then the dimension of R is at most one. In the one-dimensional case, we can reduce to the situation that rings are Cohen–Macaulay. Then, we give a necessary condition to have a finite number of trace ideals in terms of the value set obtained by the canonical module. We also gave the correspondence between trace ideals of R and those of the endomorphism algebra of the maximal ideal of R when R has minimal multiplicity.
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References
Atiyah, M.F., Macdonald, I.G.: Introduction to Commutative Algebra. Addison-Wesley Series in Mathematics. Westview Press, Boulder (1969)
Arf, C.: Une interprétation algébrique de la suite des ordres de multiplicité d’une branche algébrique. Proc. Lond. Math. Soc. Ser. 2 50, 256–287 (1949)
Bass, H.: On the ubiquity of Gorenstein rings. Mathematische Zeitschrift 82, 8–28 (1963)
Bruns, W., Herzog, J.: Cohen–Macaulay Rings. Cambridge University Press, Cambridge (1993)
Chau, T.D.M., Goto, S., Kumashiro, S., Matsuoka, N.: Sally modules of canonical ideals in dimension one and 2-AGL rings. J. Algebra 521, 299–330 (2019)
Dao, H.: Reflexive modules, self-dual modules and Arf rings. arXiv:2105.12240
Dao, H., Kobayashi, T., Takahashi, R.: Trace of canonical modules, annihilator of Ext, and classes of rings close to being Gorenstein. J. Pure Appl. Algebra 225(9), 106655 (2021)
Dao, H., Maitra, S., Sridhar, P.: On reflexive and \(I\)-Ulrich modules over curve singularities. Trans. Am. Math. Soc. Ser. B 10, 355–380 (2023)
Faber, E.: Trace ideals, normalization chains, and endomorphism rings. Pure Appl. Math. Q. 16, 1001–1025 (2020)
Goto, S., Isobe, R., Kumashiro, S.: Correspondence between trace ideals and birational extensions with application to the analysis of the Gorenstein property of rings. J. Pure Appl. Algebra 224, 747–767 (2020)
Goto, S., Matsuoka, N., Phuong, T.T.: Almost Gorenstein rings. J. Algebra 379, 355–381 (2013)
Goto, S., Watanabe, K.: On graded rings I. J. Math. Soc. Jpn. 30(2), 179–213 (1978)
Herzog, J., Kumashiro, S., Stamate, D.I.: The tiny trace ideals of the canonical modules in Cohen–Macaulay rings of dimension one. J. Algebra 619, 626–642 (2023)
Herzog, J., Kunz, E.: Der kanonische Modul eines Cohen–Macaulay-Rings. Lecture Notes in Mathematics, vol. 238. Springer, Berlin (1971)
Herzog, J., Hibi, T., Stamate, D.I.: The trace of the canonical module. Isr. J. Math. 233, 133–165 (2019)
Herzog, J., Hibi, T., Stamate, D.I.: Canonical trace ideal and residue for numerical semigroup rings. Semigroup Forum 103, 550–566 (2021)
Herzog, J., Rahimbeigi, M.: On the set of trace ideals of a Noetherian ring. Beiträge zur Algebra und Geometrie 64, 41–54 (2023)
Isobe, R., Kumashiro, S.: Reflexive modules over Arf local rings, arXiv:2105.07184
Kobayashi, T., Takahashi, R.: Rings whose ideals are isomorphic to trace ideals. Mathematische Nachrichten 292(10), 2252–2261 (2019)
Kumashiro, S.: The reduction number of canonical ideals. Commun. Algebra 50(11), 4619–4635 (2022)
Kunz, E.: Über die Klassifikation numerischer Halbgruppen, Fakultät für mathematik der universität (1987)
Leuschke, G.J., Wiegand, R.: Cohen–Macaulay Representations. Mathematical Surveys and Monographs, vol. 181. American Mathematical Society, Providence (2012)
Lindo, H.: Trace ideals and centers of endomorphism rings of modules over commutative rings. J. Algebra 482, 102–130 (2017)
Lindo, H., Pande, N.: Trace ideals and the Gorenstein property. Commun. Algebra 50(10), 4116–4121 (2022)
Lipman, J.: Stable ideals and Arf rings. Am. J. Math. 93, 649–685 (1971)
Rosales, J.C., García-Sánchez, P.A.: Numerical Semigroups. Springer, Berlin (2009)
Swanson, I., Huneke, C.: Integral Closure of Ideals, Rings, and Modules. London Mathematical Society Lecture Note Series, vol. 336. Cambridge University Press, Cambridge (2006)
Acknowledgements
Question 1.1, the starting point of this paper, was asked by Jürgen Herzog to the author. The author is grateful to him. The author would like to thank Ryotaro Isobe and Kazuho Ozeki for giving useful comments to Proposition 2.3, Lemma 2.5, and Theorem 2.6. The author would also like to thank Toshinori Kobayashi for telling the author about Kunz’s coordinates (Remark 5.3). The author is also grateful to the anonymous referee for his/her careful reading and for pointing out the error in the previous version.
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The author was supported by JSPS KAKENHI under Grant No. 21K13766 and by Grant for Basic Science Research Projects from the Sumitomo Foundation (Grant No. 2200259).
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Kumashiro, S. When are Trace Ideals Finite?. Mediterr. J. Math. 20, 278 (2023). https://doi.org/10.1007/s00009-023-02481-4
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DOI: https://doi.org/10.1007/s00009-023-02481-4