Skip to main content
SpringerLink
Log in

Moderately ramified actions in positive characteristic

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

In characteristic 2 and dimension 2, wild \({\mathbb Z}/2{{\mathbb {Z}}}\)-actions on k[[u, v]] ramified precisely at the origin were classified by Artin, who showed in particular that they induce hypersurface singularities. We introduce in this article a new class of wild quotient singularities in any characteristic \(p>0\) and dimension \(n\ge 2\) arising from certain non-linear actions of \({\mathbb {Z}}/p{\mathbb {Z}}\) on the formal power series ring \(k[[u_1,\dots ,u_n]]\). These actions are ramified precisely at the origin, and their rings of invariants in dimension 2 are hypersurface singularities, with an equation of a form similar to the form found by Artin when \(p=2\). In higher dimension, the rings of invariants are not local complete intersection in general, but remain quasi-Gorenstein. We establish several structure results for such actions and their corresponding rings of invariants.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abramovich, D.: Raynaud’s group-scheme and reduction of coverings. With an appendix by Jonathan Lubin. In: Number Theory, Analysis and Geometry, pp. 1–18. Springer, New York (2012)

  2. Altman, A., Kleiman, S.: Introduction to Grothendieck duality theory. Lecture Notes in Mathematics, vol. 146. Springer, Berlin (1970)

  3. Aoyama, Y.: Some basic results on canonical modules. J. Math. Kyoto Univ. 23, 85–94 (1983)

    MathSciNet  MATH  Google Scholar 

  4. Artin, M.: Wildly ramified \(Z/2\) actions in dimension two. Proc. Am. Math. Soc. 52, 60–64 (1975)

    MathSciNet  MATH  Google Scholar 

  5. Auslander, M., Buchsbaum, D.: Codimension and multiplicity. Ann. Math. 68, 625–657 (1958)

    MathSciNet  MATH  Google Scholar 

  6. Avramov, L.: Pseudoreflection group actions on local rings. Nagoya Math. J. 88, 161–180 (1982)

    MathSciNet  MATH  Google Scholar 

  7. Benson, D., Webb, P.: Unique factorization in invariant power series rings. J. Algebra 319(2), 702–715 (2008)

    MathSciNet  MATH  Google Scholar 

  8. Bourbaki, N.: Algebra II, Chapters 4–7. Translated from the French by P. M. Cohn and J. Howie. Elements of Mathematics (Berlin). Springer, Berlin, (1990)

  9. Bourbaki, N.: Algèbre commutative. Chapitre 10, Reprint of the 1998 original. Springer, Berlin (2007)

  10. Boutot, J.-F.: Singularités rationnelles et quotients par les groupes réductifs. Invent. Math. 88, 65–68 (1987)

    MathSciNet  MATH  Google Scholar 

  11. Braun, A.: On the Gorenstein property for modular invariants. J. Algebra 345, 81–99 (2011)

    MathSciNet  MATH  Google Scholar 

  12. Brieskorn, E.: Rationale Singularitäten komplexer Flächen. Invent. Math. 4, 336–358 (1967/1968)

  13. Campbell, H., Hughes, I.: Vector invariants of \(U_2({{\mathbb{F}}}_p)\): a proof of a conjecture of Richman. Adv. Math. 126(1), 1–20 (1997)

    MathSciNet  Google Scholar 

  14. Cartan, H.: Quotient d’un espace analytique par un groupe d’automorphismes. In: Fox, R. (ed.) Algebraic geometry and topology, pp. 90–102. Princeton University Press, Princeton (1957)

    Google Scholar 

  15. Ellingsrud, G., Skjelbred, T.: Profondeur d’anneaux d’invariants en caractéristique \(p\). Compos. Math. 41, 233–244 (1980)

    MATH  Google Scholar 

  16. Fogarty, J.: On the depth of local rings of invariants of cyclic groups. Proc. Am. Math. Soc. 83(3), 448–452 (1981)

    MathSciNet  MATH  Google Scholar 

  17. Gilmer, R.: Contracted ideals with respect to integral extensions. Duke Math. J. 34, 561–571 (1967)

    MathSciNet  MATH  Google Scholar 

  18. Giraud, J.: Cohomologie non abélienne, Die Grundlehren der mathematischen Wissenschaften, vol. 179. Springer, Berlin (1971)

    Google Scholar 

  19. Grothendieck, A.: Techniques de construction et théorèmes d’existence en géométrie algébrique. IV. Les schémas de Hilbert, Séminaire Bourbaki, vol. 6, Exp. No. 221, pp. 249–276. Soc. Math. France, Paris (1995)

  20. Grothendieck, A., Dieudonné, J.: Éléments de géométrie algébrique, Publ. Math. IHÉS 4 (Chapter 0, 1–7, and I, 1–10), 8 (II, 1–8), 11 (Chapter 0, 8–13, and III, 1–5), 17 (III, 6–7), 20 (Chapter 0, 14–23, and IV, 1), 24 (IV, 2–7), 28 (IV, 8–15), and 32 (IV, 16–21), (1960–1967)

  21. Hartshorne, R.: Generalised divisors on Gorenstein schemes. K-Theory 8, 287–339 (1994)

    MathSciNet  MATH  Google Scholar 

  22. Herzog, J., Kunz, E. (eds.) Der kanonische Modul eines Cohen–Macaulay-Rings, Lect. Notes Math., vol. 238. Springer, Berlin (1971)

  23. Hochster, M.: Contracted ideals from integral extensions of regular rings. Nagoya Math. J. 51, 25–43 (1973)

    MathSciNet  MATH  Google Scholar 

  24. Hochster, M., Eagon, J.: Cohen–Macaulay rings, invariant theory, and the generic perfection of determinantal loci. Am. J. Math. 93, 1020–1058 (1971)

    MathSciNet  MATH  Google Scholar 

  25. Iyama, O., Reiten, I.: Fomin–Zelevinsky mutation and tilting modules over Calabi–Yau algebras. Am. J. Math. 130, 1087–1149 (2008)

    MathSciNet  MATH  Google Scholar 

  26. Kemper, G.: On the Cohen–Macaulay property of modular invariant rings. J. Algebra 215, 330–351 (1999)

    MathSciNet  MATH  Google Scholar 

  27. Kemper, G.: Loci in quotients by finite groups, pointwise stabilizers and the Buchsbaum property. J. Reine Angew. Math. 547, 69–96 (2002)

    MathSciNet  MATH  Google Scholar 

  28. Kemper, G., Körding, E., Malle, G., Matzat, B., Vogel, D., Wiese, G.: A database of invariant rings. Exp. Math. 10(4), 537–542 (2001)

    MathSciNet  MATH  Google Scholar 

  29. Kiràly, F., Lüktebohmert, W.: Group actions of prime order on local normal rings. Algebra Number Theory 7(1), 63–74 (2013)

    MathSciNet  MATH  Google Scholar 

  30. Klopsch, B.: Automorphisms of the Nottingham group. J. Algebra 223, 37–56 (2000)

    MathSciNet  MATH  Google Scholar 

  31. Koh, J.: Degree \(p\) extensions of an unramified regular local ring of mixed characteristic \(p\). J. Algebra 99(2), 310–323 (1986)

    MathSciNet  MATH  Google Scholar 

  32. Lipman, J.: Rational singularities, with applications to algebraic surfaces and unique factorization. Inst. Hautes Études Sci. Publ. Math. 36, 195–279 (1969)

    MathSciNet  MATH  Google Scholar 

  33. Lipman, J.: Unique factorization in complete local rings. In: Algebraic Geometry (Proc. Sympos. Pure Math., vol. 29, Humboldt State Univ., Arcata, Calif., 1974), pp. 531–546. Amer. Math. Soc., Providence (1975)

  34. Lorenzini, D.: Wild quotients of products of curves. Eur. J. Math. 4, 525–554 (2018)

    MathSciNet  MATH  Google Scholar 

  35. Lorenzini, D., Schröer, S.: Discriminant groups of wild cyclic quotient singularities (in preparation)

  36. Lubin, J.: Torsion in the Nottingham group. Bull. Lond. Math. Soc. 43(3), 547–560 (2011)

    MathSciNet  MATH  Google Scholar 

  37. Majadas, J.: On tensor products of complete intersections. Bull. Lond. Math. Soc. 45(6), 1281–1284 (2013)

    MathSciNet  MATH  Google Scholar 

  38. Marrama, A.: A Purity Theorem for Torsors, ALGANT Master Thesis (2016). http://algant.eu/algant_theses.php

  39. Matsumura, H.: Commutative Algebra, 2nd edn. Benjamin/Cummings, Reading (1980)

    MATH  Google Scholar 

  40. Matsumura, H.: Commutative Ring Theory. Cambridge University Press, Cambridge (1989)

    MATH  Google Scholar 

  41. Mollier, D.: Descente de la propriété noethérienne. Bull. Sci. Math. (2) 94, 25–31 (1970)

    MathSciNet  MATH  Google Scholar 

  42. Moret-Bailly, L.: Un théorème de pureté pour les familles de courbes lisses. C. R. Acad. Sci. Paris Sér. I Math. 300(4), 489–492 (1985)

    MathSciNet  MATH  Google Scholar 

  43. Mori, S.: On affine cones associated with polarized varieties. Jpn. J. Math. (N.S.) 1(2), 301–309 (1975)

    MathSciNet  MATH  Google Scholar 

  44. Nagata, M.: On the purity of branch loci in regular local rings. Ill. J. Math. 3, 328–333 (1959)

    MathSciNet  MATH  Google Scholar 

  45. Nagata, M.: Local rings. Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers a division of Wiley, New York (1962)

  46. Peskin, B.: Quotient-singularities and wild \(p\)-cyclic actions. J. Algebra 81, 72–99 (1983)

    MathSciNet  MATH  Google Scholar 

  47. Platte, E., Storch, U.: Invariante reguläre Differentialformen auf Gorenstein-Algebren. Math. Z. 157, 1–11 (1977)

    MathSciNet  MATH  Google Scholar 

  48. Raynaud, M.: Spécialisation des revêtements en caractéristique \(p>0\). Ann. Sci. École Norm. Sup. (4) 32(1), 87–126 (1999)

    MathSciNet  MATH  Google Scholar 

  49. Revêtements étales et groupe fondamental (French), Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1). Dirigé par Alexandre Grothendieck. Augmenté de deux exposés de Michèle Raynaud. Lecture Notes in Mathematics, vol. 224. Springer, Berlin (1971)

  50. Richman, D.: On vector invariants over finite fields. Adv. Math. 81, 30–65 (1990)

    MathSciNet  MATH  Google Scholar 

  51. Roberts, P.: Abelian extensions of regular local rings. Proc. Am. Math. Soc. 78(3), 307–310 (1980)

    MathSciNet  MATH  Google Scholar 

  52. Romagny, M.: Effective model of a finite group action. arXiv:math/0601639v1

  53. Romagny, M.: Effective models of group schemes. J. Algebr. Geom. 21, 643–682 (2012)

    MathSciNet  MATH  Google Scholar 

  54. Rotman, J.: An Introduction to Homological Algebra, 2nd edn. Springer, New York (2009)

    MATH  Google Scholar 

  55. Samuel, P.: Classes de diviseurs et dérivées logarithmiques. Topology 3, 81–96 (1964)

    MathSciNet  MATH  Google Scholar 

  56. Samuel, P.: Groupes finis d’automorphismes des anneaux de séries formelles. Bull. Sci. Math. 90, 97–101 (1966)

    MathSciNet  MATH  Google Scholar 

  57. Serre, J.-P.: Algèbre locale. Multiplicités. Springer, Berlin (1965)

    MATH  Google Scholar 

  58. Serre, J.-P.: Groupes finis d’automorphismes d’anneaux locaux réguliers, 1968 Colloque d’Algèbre (Paris, 1967), Exp. 8, Secrétariat mathématique, Paris

  59. Serre, J.-P.: Linear representations of finite groups, Translated from the second French edition by Leonard L. Scott. Graduate Texts in Mathematics, vol. 42. Springer, New York (1977)

  60. Shank, R., Wehlau, D.: Computing modular invariants of p-groups. J. Symb. Comput. 34, 307–327 (2002)

    MathSciNet  MATH  Google Scholar 

  61. Sjödin, G.: On filtered modules and their associated graded modules. Math. Scand. 33(1973), 229–249 (1974)

    MathSciNet  MATH  Google Scholar 

  62. Tate, J., Oort, F.: Group schemes of prime order. Ann. Sci. École Norm. Sup. 3, 1–21 (1970)

    MathSciNet  MATH  Google Scholar 

  63. Wahl, J.: Equations defining rational singularities. Ann. Sci. École Norm. Sup. (4) 10(2), 231–263 (1977)

    MathSciNet  MATH  Google Scholar 

  64. Watanabe, K.: Certain invariant subrings are Gorenstein I. Osaka J. Math. 11, 1–8 (1974)

    MathSciNet  MATH  Google Scholar 

  65. Watanabe, K.: Certain invariant subrings are Gorenstein II. Osaka J. Math. 11, 379–388 (1974)

    MathSciNet  MATH  Google Scholar 

  66. Waterhouse, W.: Divisor classes in pseudo Galois extensions. Pac. J. Math. 36, 541–548 (1971)

    MathSciNet  MATH  Google Scholar 

  67. Yuan, S.: Reflexive modules and algebra class groups over noetherian integrally closed domains. J. Algebra 32, 405–417 (1974)

    MathSciNet  MATH  Google Scholar 

  68. Zariski, O., Samuel, P.: Commutative Algebra, vol. II, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton (1960)

Download references

Acknowledgements

The authors thank the referee for a careful reading of the manuscript and for useful comments. The authors gratefully acknowledges funding support from the research training group Algebra, Algebraic Geometry, and Number Theory at the University of Georgia, from the National Science Foundation RTG grant DMS-1344994, from the Simons Collaboration Grant 245522, and the research training group GRK 2240: Algebro-geometric Methods in Algebra, Arithmetic and Topology of the Deutsche Forschungsgemeinschaft.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Georgia, Athens, GA, 30602, USA

    Dino Lorenzini

  2. Mathematisches Institut, Heinrich-Heine-Universität, 40204, Düsseldorf, Germany

    Stefan Schröer

Authors
  1. Dino Lorenzini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stefan Schröer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dino Lorenzini.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lorenzini, D., Schröer, S. Moderately ramified actions in positive characteristic. Math. Z. 295, 1095–1142 (2020). https://doi.org/10.1007/s00209-019-02408-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-019-02408-4

Mathematics Subject Classification

Search

Navigation