Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-28T10:02:40.470Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Canonical Ideals of One-Dimensional Cohen–Macaulay Local Rings

Part of: Local rings and semilocal rings Local theory

Published online by Cambridge University Press:  13 July 2015

Juan Elias
Juan Elias*
Affiliation:
Departament d'Àlgebra i Geometria, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we consider the problem of explicitly finding canonical ideals of one-dimensional Cohen–Macaulay local rings. We show that Gorenstein ideals contained in a high power of the maximal ideal are canonical ideals. In the codimension 2 case, from a Hilbert–Burch resolution, we show how to construct canonical ideals of curve singularities. Finally, we translate the problem of the analytic classification of curve singularities to the classification of local Artin Gorenstein rings with suitable length.

Keywords

canonical idealGorensteinCohen–MacaulayHilbert–Samuel function

MSC classification

Secondary: 14B05: Singularities 13H15: Multiplicity theory and related topics 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 1 , February 2016 , pp. 77 - 90
Copyright
Copyright © Edinburgh Mathematical Society 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Altman, A. and Kleiman, S., Introduction to Grothendieck duality theory, Lecture Notes in Mathematics, Volume 146 (Springer, 1970).CrossRefGoogle Scholar
2. Bertin, J. and Carbonne, P., Semi-groupes d'entiers et application aux branches, J. Alg. 49(1) (1977), 81–95 (in French).Google Scholar
3. Boij, M., Gorenstein Artin algebras and points in projective space, Bull. Lond. Math. Soc. 31(1) (1999), 11–16.CrossRefGoogle Scholar
4. Bruns, W. and Herzog, J., Cohen–Macaulay rings, revised edn, Cambridge Studies in Advanced Mathematics, Volume 39 (Cambridge University Press, 1997).Google Scholar
5. Buchsbaum, D. and Eisenbud, D., Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3, Am. J. Math. 99 (1977), 447–485.Google Scholar
6. Buchweitz, R. O. and Greuel, G. M., The Milnor number and deformations of complex curve singularities, Invent. Math. 58 (1980), 241–281.Google Scholar
7. Burch, L., On ideals of finite homological dimension in local rings, Math. Proc. Camb. Phil. Soc. 64 (1968), 941–946.Google Scholar
8. Elias, J., On the analytic equivalence of curves, Math. Proc. Camb. Phil. Soc. 100 (1986), 57–64.CrossRefGoogle Scholar
9. Elias, J., Characterization of the Hilbert–Samuel polynomials of curve singularities, Compositio Math. 74 (1990), 135–155.Google Scholar
10. Elias, J., On the deep structure of the blowing-up of curve singularities, Math. Proc. Camb. Phil. Soc. 131 (2001), 227–240.Google Scholar
11. Goto, S. and Watanabe, K., On graded rings, I, J. Math. Soc. Jpn 30(2) (1978), 179–213.Google Scholar
12. Herzog, J., Generators and relations of abelian semigroups and semigroup-rings, Manuscr. Math. 3 (1970), 153–193.Google Scholar
13. Herzog, J. and Kunz, E., Dir kanonische Modul eines Cohen–Macaulay-Rings, Lecture Notes in Mathematics, Volume 238 (Springer, 1971).Google Scholar
14. Iarrobino, A., Associated graded algebra of a Gorenstein Artin algebra, Memoirs of the American Mathematical Society, Volume 107 (American Mathematical Society, Providence, RI, 1994).Google Scholar
15. Iarrobino, A. and Kanev, V., Power sums, Gorenstein algebras, and determinantal loci, Lecture Notes in Mathematics, Volume 1721 (Springer, 1999).CrossRefGoogle Scholar
16. Matlis, E., 1-dimensional Cohen–Macaulay rings, Lecture Notes in Mathematics, Volume 327 (Springer, 1977).Google Scholar
17. Northcott, D. G., The reduction number of a one-dimensional local ring, Mathematika 6 (1959), 87–90.Google Scholar
18. Robbiano, L. and Valla, G., On the equations defining tangent cones, Math. Proc. Camb. Phil. Soc. 88 (1980).Google Scholar
19. Sally, J., Number of generators of ideals in local rings, Lecture Notes in Pure and Applied Mathematics, Volume 35 (Dekker, New York, 1978).Google Scholar
20. Serre, J. P., Groupes algébriques et corps de classes, Publications de l'institut de mathématique de l'université de Nancago, Volume 7 (Hermann, Paris, 1959).Google Scholar