Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A prime ideal in a polynomial ring whose symbolic blow-up is not Noetherian


Author: Paul C. Roberts
Journal: Proc. Amer. Math. Soc. 94 (1985), 589-592
MSC: Primary 13E15; Secondary 13A17, 13B30
DOI: https://doi.org/10.1090/S0002-9939-1985-0792266-5
MathSciNet review: 792266
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ R$ be the polynomial ring $ k[X,Y,Z]$ localized at the maximal ideal $ M = (X,Y,Z)$. We construct a prime ideal $ P$ in $ R$ which is equal to the ideal of $ m$ generic lines through the origin modulo $ {M^m}$, and we show that, for suitable choice of $ m$, the symbolic blow-up of such an ideal $ P$ is not Noetherian.


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

  • [1] J. Dixmier, Solution négative du problème des invariants, d'après Nagata, Séminaire Bourbaki, Exp. 175, Secrétariat Math., Paris, 1959.
  • [2] R. Hartshorne, Algebraic geometry, Graduate Texts in Math., Springer-Verlag, Berlin and New York, 1977. MR 0463157 (57:3116)
  • [3] C. Huneke, On the finite generation of symbolic blow-ups, Math. Z. 179 (1982), 465-472. MR 652854 (83h:13029)
  • [4] M. Nagata, On the fourteenth problem of Hilbert, Proc. Internat. Congress Math. (1958), Cambridge Univ. Press, 1960. MR 0116056 (22:6851)
  • [5] D. Rees, On a problem of Zariski, Illinois J. Math. 2 (1958), 145-149. MR 0095843 (20:2341)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 13E15, 13A17, 13B30

Retrieve articles in all journals with MSC: 13E15, 13A17, 13B30


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0792266-5
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society