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Torsion in differentials and Berger’s conjecture


Let \((R,{\mathfrak {m}},\mathbb {k})\) be an equicharacteristic one-dimensional complete local domain over an algebraically closed field \(\mathbb {k}\) of characteristic 0. R. Berger conjectured that R is regular if and only if the universally finite module of differentials \(\Omega _R\) is a torsion-free R module. We give new cases of this conjecture by extending works of Güttes (Arch Math 54:499–510, 1990) and Cortiñas et al. (Math Z 228:569–588, 1998). This is obtained by constructing a new subring S of \({\text {Hom}}_R({\mathfrak {m}},{\mathfrak {m}})\) and constructing enough torsion in \(\Omega _S\), enabling us to pull back a nontrivial torsion to \(\Omega _R\).

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  1. 1.

    Bassein, R.: On smoothable curve singularities: local methods. Math. Ann. 230, 273–277 (1977)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Berger, R.: Differentialmoduln eindimensionaler lokaler Ringe. Math. Z. 81, 326–354 (1963)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Berger, R.W.: On the torsion of the differential module of a curve singularity. Arch. Math. (Basel) 50, 526–533 (1988)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Berger, R.W.: Report on the Torsion of Differential Module of an Algebraic Curve, in Algebraic Geometry and its Applications, pp. 285–303. Springer (1994)

  5. 5.

    Brown, W.C., Herzog, J.: One-dimensional local rings of maximal and almost maximal length. J. Algebra 151, 332–347 (1992)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bruns, W., Herzog, J.: Cohen-Macaulay Rings, 2nd edn. Cambridge Studies in Advanced Mathematics, Cambridge University Press (1998)

  7. 7.

    Buchweitz, R.-O., Greuel, G.-M.: The Milnor number and deformations of complex curve singularities. Invent. Math. 58, 241–281 (1980)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Cortiñas, G., Geller, S.C., Weibel, C.A.: The Artinian Berger conjecture. Math. Z. 228, 569–588 (1998)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Cortiñas, G., Krongold, F.: Artinian algebras and differential forms. Commun. Algebra 27, 1711–1716 (1999)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Eisenbud, D.: Commutative algebra, vol. : With a view toward algebraic geometry. In: 150 of Graduate Texts in Mathematics. Springer-Verlag, New York (1995)

  11. 11.

    Güttes, K.: Zum Torsionsproblem bei Kurvensingularitäten. Arch. Math. 54, 499–510 (1990)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Herzog, J.: Ein Cohen-Macaulay-Kriterium mit Anwendungen auf den Konormalenmodul und den Differentialmodul. Math. Z. 163, 149–162 (1978)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Herzog, J., Kunz, E.: Die Wertehalbgruppe eines lokalen Rings der Dimension 1, in Die Wertehalbgruppe eines lokalen Rings der Dimension I, Springer, pp. 3–43 (1971)

  14. 14.

    Herzog, J., Waldi, R.: Differentials of linked curve singularities. Arch. Math. 42, 335–343 (1984)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Herzog, J., Waldi, R.: Cotangent functors of curve singularities. Manuscripta Math. 55, 307–341 (1986)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Hübl, R.: A note on the torsion of differential forms. Arch. Math. (Basel) 54, 142–145 (1990)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Isogawa, S.: On Berger’s conjecture about one-dimensional local rings. Arch. Math. (Basel) 57, 432–437 (1991)

  18. 18.

    Koch, J.: Über die Torsion des Differentialmoduls von Kurvensingularitäten. Regensburger Mathematische Schriften [Regensburg Mathematical Publications], vol. 5. Universität Regensburg, Fachbereich Mathematik, Regensburg (1983)

  19. 19.

    Kunz, E.: The value-semigroup of a one-dimensional Gorenstein ring. Proc. Am. Math. Soc. 25, 748–751 (1970)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Kunz, E.: Kähler differentials. Advanced Lectures in Mathematics. Friedr. Vieweg & Sohn, Braunschweig (1986)

    Book  Google Scholar 

  21. 21.

    Maitra, S.: Partial Trace Ideals and Berger’s Conjecture, arXiv preprint arXiv:2003.11648. (2020)

  22. 22.

    Pohl, T.: Torsion des Differentialmoduls von Kurvensingularitäten mit maximaler Hilbertfunktion. Arch. Math. 52, 53–60 (1989)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Pohl, T.: Differential modules with maximal torsion. Arch. Math. (Basel) 57, 438–445 (1991)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Scheja, G.: Differentialmoduln lokaler analytischer Algebren. Inst. Univ. Fribourg, Univ. Fribourg, Switzerland, Schriftenreihe Math (1970)

  25. 25.

    Swanson, I., Huneke, C.: Integral closure of ideals, rings, and modules. London Mathematical Society Lecture Note Series, vol. 336. Cambridge University Press, Cambridge (2006)

  26. 26.

    Ulrich, B.: Torsion des Differentialmoduls und Kotangentenmodul von Kurvensingularitäten. Arch. Math. (Basel) 36, 510–523 (1981)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Yoshino, Y.: Torsion of the differential modules and the value semigroups of one dimensional local rings. Math. Rep. 9, 83–96 (1986)

    MathSciNet  MATH  Google Scholar 

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Correspondence to Vivek Mukundan.

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This paper is dedicated to Jürgen Herzog, whose fundamental research in commutative algebra has inspired researchers for 50 years.

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Huneke, C., Maitra, S. & Mukundan, V. Torsion in differentials and Berger’s conjecture. Res Math Sci 8, 60 (2021).

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  • Module of differentials
  • Berger Conjecture
  • Reduced curves

Mathematics Subject Classification

  • Primary: 13N05
  • Secondary: 13H10