Two criteria for quasihomogeneity

Maitra, Sarasij; Mukundan, Vivek

doi:10.1090/proc/16773

Two criteria for quasihomogeneity
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by Sarasij Maitra and Vivek Mukundan;

Proc. Amer. Math. Soc. 152 (2024), 2369-2375

DOI: https://doi.org/10.1090/proc/16773

Published electronically: April 12, 2024

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Abstract:

Let $(R,\mathfrak {m}_R,\mathbb {k})$ be a one-dimensional complete local reduced $\mathbb {k}$-algebra over a field of characteristic zero. The ring $R$ is said to be quasihomogeneous if there exists a surjection $\Omega _R\twoheadrightarrow \mathfrak {m}$ where $\Omega _R$ denotes the module of differentials. We present two characterizations of quasihomogeniety of $R$ in the case when $R$ is a domain. The first one on the valuation semigroup of $R$ and the other on the trace ideal of the module $\Omega _R$.

References

Robert W. Berger, Report on the torsion of the differential module of an algebraic curve, Algebraic geometry and its applications (West Lafayette, IN, 1990) Springer, New York, 1994, pp. 285–303. MR 1272036
Jürgen Herzog, The module of differentials, Lecture notes from Workshop on Commutative Algebra and its Relation to Combinatorics and Computer Algebra (Trieste, 1994), 1994.
Craig Huneke, Sarasij Maitra, and Vivek Mukundan, Torsion in differentials and Berger’s conjecture, Res. Math. Sci. 8 (2021), no. 4, Paper No. 60, 15. MR 4323642, DOI 10.1007/s40687-021-00295-y
Toshinori Kobayashi and Ryo Takahashi, Rings whose ideals are isomorphic to trace ideals, Math. Nachr. 292 (2019), no. 10, 2252–2261. MR 4019663, DOI 10.1002/mana.201800309
Ernst Kunz, Kähler differentials, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1986. MR 864975, DOI 10.1007/978-3-663-14074-0
Ernst Kunz and Walter Ruppert, Quasihomogene Singularitäten algebraischer Kurven, Manuscripta Math. 22 (1977), no. 1, 47–61 (German, with English summary). MR 463180, DOI 10.1007/BF01182066
Haydee Lindo, Trace ideals and centers of endomorphism rings of modules over commutative rings, J. Algebra 482 (2017), 102–130. MR 3646286, DOI 10.1016/j.jalgebra.2016.10.026
Sarasij Maitra, Partial trace ideals and Berger’s conjecture, J. Algebra 598 (2022), 1–23. MR 4379271, DOI 10.1016/j.jalgebra.2022.01.018
Sarasij Maitra and Vivek Mukundan, Valuations and nonzero torsion in module of differentials, Bull. Sci. Math. 187 (2023), Paper No. 103287, 21. MR 4601799, DOI 10.1016/j.bulsci.2023.103287
Kyoji Saito, Quasihomogene isolierte Singularitäten von Hyperflächen, Invent. Math. 14 (1971), 123–142 (German). MR 294699, DOI 10.1007/BF01405360
Günter Scheja, Differentialmoduln lokaler analytischer Algebren, Schriftenreihe Math, Inst. Univ. Fribourg, Univ. Fribourg, Switzerland, 1970.
Craig Huneke and Irena Swanson, Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006. MR 2266432
Rolf Waldi, Deformation von Gorenstein-Singularitäten der Kodimension $3$, Math. Ann. 242 (1979), no. 3, 201–208 (German). MR 545214, DOI 10.1007/BF01420726
Oscar Zariski, Characterization of plane algebroid curves whose module of differentials has maximum torsion, Proc. Nat. Acad. Sci. U.S.A. 56 (1966), 781–786. MR 202716, DOI 10.1073/pnas.56.3.781