Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Mather-Yau theorem in positive characteristic

Authors: Gert-Martin Greuel and Thuy Huong Pham
Journal: J. Algebraic Geom. 26 (2017), 347-355
Published electronically: September 23, 2016
Full-text PDF

Abstract | References | Additional Information

Abstract: The well-known Mather-Yau theorem says that the isomorphism type of the local ring of an isolated complex hypersurface singularity is determined by its Tjurina algebra. It is also well known that this result is wrong as stated for power series $ f$ in $ K[[{\bf x}]]$ over fields $ K$ of positive characteristic. In this note we show that, however, also in positive characteristic the isomorphism type of an isolated hypersurface singularity $ f$ is determined by an Artinian algebra, namely by a ``higher Tjurina algebra'' for sufficiently high index, for which we give an effective bound. We prove also a similar version for the ``higher Milnor algebra" considered as a $ K[[f]]$-algebra.

References [Enhancements On Off] (What's this?)

  • [1] Yousra Boubakri, Gert-Martin Greuel, and Thomas Markwig, Invariants of hypersurface singularities in positive characteristic, Rev. Mat. Complut. 25 (2012), no. 1, 61-85. MR 2876917,
  • [2] W. Decker, G.-M. Greuel, G. Pfister, and H. Schönemann, Singular 3-1-5 -- A computer algebra system for polynomial computations, Center for Computer Algebra, University of Kaiserslautern, 2012,
  • [3] G.-M. Greuel, C. Lossen, and E. Shustin, Introduction to singularities and deformations, Springer Monographs in Mathematics, Springer, Berlin, 2007. MR 2290112
  • [4] Gert-Martin Greuel and Gerhard Pfister, A Singular introduction to commutative algebra, Second, extended edition, with contributions by Olaf Bachmann, Christoph Lossen and Hans Schönemann. With 1 CD-ROM (Windows, Macintosh and UNIX). Springer, Berlin, 2008. MR 2363237
  • [5] John N. Mather and Stephen S.-T. Yau, Criterion for biholomorphic equivalence of isolated hypersurface singularities, Proc. Nat. Acad. Sci. U.S.A. 78 (1981), no. 10, Phys. Sci., 5946-5947. MR 773820,
  • [6] John N. Mather and Stephen S. T. Yau, Classification of isolated hypersurface singularities by their moduli algebras, Invent. Math. 69 (1982), no. 2, 243-251. MR 674404,
  • [7] A. N. Šošitaĭšvili, Functions with isomorphic Jacobian ideals, Funkcional. Anal. i Priložen. 10 (1976), no. 2, 57-62 (Russian). MR 0417442
  • [8] Stephen S.-T. Yau, Milnor algebras and equivalence relations among holomorphic functions, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 2, 235-239. MR 707965,
  • [9] Stephen S.-T. Yau, Criteria for right-left equivalence and right equivalence of holomorphic functions with isolated critical points, Complex analysis of several variables (Madison, Wis., 1982) Proc. Sympos. Pure Math., vol. 41, Amer. Math. Soc., Providence, RI, 1984, pp. 291-297. MR 740890,

Additional Information

Gert-Martin Greuel
Affiliation: Fachbereich Mathematik, Universität Kaiserslautern, Erwin-Schrödinger Str., 67663 Kaiserslautern, Germany

Thuy Huong Pham
Affiliation: Fachbereich Mathematik, Universität Kaiserslautern, Erwin-Schrödinger Str., 67663 Kaiserslautern, Germany
Address at time of publication: Department of Mathematics, Quy Nhon University, 170 An Duong Vuong Street, Quy Nhon City, Vietnam

Received by editor(s): April 16, 2014
Received by editor(s) in revised form: July 31, 2014, and August 6, 2014
Published electronically: September 23, 2016
Additional Notes: The second author was supported by DAAD (Germany).
Article copyright: © Copyright 2016 University Press, Inc.

American Mathematical Society