Journal of Commutative Algebra

The symmetric signature of cyclic quotient singularities

Alessio Caminata and Lukas Katthan

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Abstract

The symmetric signature is an invariant of local domains which was recently introduced by Brenner and the first author in an attempt to find a replacement for the $F$-signature in characteristic zero. In the present note we compute the symmetric signature for two-dimensional cyclic quotient singularities, i.e., invariant subrings $k[\mkern -2.75mu[ u,v]\mkern -2.75mu]^G$ of rings of formal power series under the action of a cyclic group. Equivalently, these rings arise as the completions (at the irrelevant ideal) of two-dimensional normal toric rings. We show that for this class of rings the symmetric signature coincides with the $F$-signature.

Article information

Source
J. Commut. Algebra, Volume 11, Number 2 (2019), 163-174.

Dates
First available in Project Euclid: 24 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.jca/1561363356

Digital Object Identifier
doi:10.1216/JCA-2019-11-2-163

Mathematical Reviews number (MathSciNet)
MR3973135

Zentralblatt MATH identifier
07080073

Subjects
Primary: 13A35: Characteristic p methods (Frobenius endomorphism) and reduction to characteristic p; tight closure [See also 13B22]
Secondary: 13A50: Actions of groups on commutative rings; invariant theory [See also 14L24] 52B20.

Keywords
$F$-signature quotient singularity toric ring Auslander correspondence

Citation

Caminata, Alessio; Katthan, Lukas. The symmetric signature of cyclic quotient singularities. J. Commut. Algebra 11 (2019), no. 2, 163--174. doi:10.1216/JCA-2019-11-2-163. https://projecteuclid.org/euclid.jca/1561363356


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References

  • Holger Brenner and Alessio Caminata, The symmetric signature, Comm. Algebra 45 (2017), no. 9, 3730–3756.
  • Holger Brenner and Alessio Caminata, Differential symmetric signature in high dimension, to appear in Proc. Amer. Math. Soc.
  • Igor Burban and Yuriy Drozd, Maximal Cohen-Macaulay modules over surface singularities, pp. 101–166 in Trends in representation theory of algebras and related topics, Eur. Math. Soc., Zürich, 2008.
  • Winfried Bruns and Joseph Gubeladze, Polytopes, rings, and K-theory, Springer, 2009.
  • Alessio Caminata, The symmetric signature, Ph.D. thesis, Universität Osnabrück, 2016.
  • Jürgen Herzog, Ringe mit nur endlich vielen Isomorphieklassen von maximalen, unzerlegbaren Cohen-Macaulay-Moduln, Math. Ann. 233 (1978), no. 1, 21–34.
  • Craig Huneke and Graham J. Leuschke, Two theorems about maximal Cohen–Macaulay modules, Math. Ann. 324 (2002), no. 2, 391–404.
  • Mitsuyasu Hashimoto and Yusuke Nakajima, Generalized $F$-signature of invariant subrings, J. Algebra 443 (2015), 142–152.
  • Graham J. Leuschke and Roger Wiegand, Cohen-Macaulay representations, Mathematical Surveys and Monographs 181, Amer. Math. Soc., Providence, RI, 2012.
  • Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics 227, Springer, 2005.
  • Kevin Tucker, $F$-signature exists, Invent. Math. 190 (2012), no. 3, 743–765.
  • Keiichi Watanabe, Certain invariant subrings are Gorenstein, I, Osaka J. Math. 11 (1974), 1–8.
  • Keiichi Watanabe, Certain invariant subrings are Gorenstein, II, Osaka J. Math. 11 (1974), 379–388.
  • Keiichi Watanabe and Kenichi Yoshida, Hilbert-Kunz multiplicity and an inequality between multiplicity and colength, J. Algebra 230 (2000), no. 1, 295–317.
  • Keiichi Watanabe and Kenichi Yoshida, Minimal relative Hilbert-Kunz multiplicity, Illinois J. Math. 48 (2004), no. 1, 273–294.
  • Yuji Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, London Mathematical Society Lecture Note Series 146, Cambridge University Press, 1990.