Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Projective normal flatness and Hilbert functions

Authors: U. Orbanz and L. Robbiano
Journal: Trans. Amer. Math. Soc. 283 (1984), 33-47
MSC: Primary 14B25; Secondary 13H15
MathSciNet review: 735407
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Projective normal flatness of a local ring $ R$ along an ideal $ I$ is defined to be the flatness of the morphism on the exceptional divisor induced by blowing up $ R$ with center $ I$. It is shown that most of the important properties of normal flatness have an analogue for projective normal flatness. In particular, we study the local Hilbert function in connection with projective normal flatness. If $ R/I$ is regular and $ R$ projectively normally flat along $ I$, then we obtain the same inequality for the local Hilbert functions under blowing up as in the permissible case.

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

  • [1] A. Altman and S. Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Math., vol. 146, Springer-Verlag, Berlin, Heidelberg and New York, 1970. MR 0274461 (43:224)
  • [2] B. M. Bennett, On the characteristic functions of a local ring, Ann. of Math. (2) 91 (1970), 25-87. MR 0252388 (40:5608)
  • [3] R. Hartshorne, Algebraic geometry, Springer-Verlag, Berlin, Heidelberg and New York, 1977. MR 0463157 (57:3116)
  • [4] M. Herrmann and U. Orbanz, Faserdimensionen von Aufblasungen lokaler Ringe und Aquimultiplizität, J. Math. Kyoto Univ. 20 (1980), 651-659. MR 592352 (83k:13016)
  • [5] -, Normale Flachheit und Äquimultiplizität für vollständige Durchschnitte, J. Algebra 70 (1981), 437-451. MR 623818 (83i:13010)
  • [6] -, Between equimultiplicity and normal flatness, Algebraic Geometry, Proceedings La Rabida 1981 (ed. Aroca-Buchweitz-Giusti-Merle), Lecture Notes in Mathematics, vol. 961, Springer, Berlin and New York, 1982, pp. 200-232. MR 708335 (84k:13024)
  • [7] -, On equimultiplicity, Math. Proc. Cambridge Philos. Soc. 91 (1982), 207-213. MR 641524 (83d:13031b)
  • [8] M. Herrmann, R. Schmidt and W. Vogel, Theorie der normalen Flachheit, Teubner Verlag, Leipzig, 1977. MR 0568896 (58:27959)
  • [9] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109-326. MR 0199184 (33:7333)
  • [10] J. Lipman, Equimultiplicity, reduction and blowing up, Commutative Algebra-Analytic Methods (ed. R. Draper), Lecture Notes in Pure and Appl. Math., vol. 68, Dekker, New York and Basel, 1981. MR 655801 (84a:13023)
  • [11] D. Nesselmann, Über monoidale Transformations lokaler Ringe, Math. Nachr. 88 (1979), 285-294. MR 543408 (80m:13005)
  • [12] U. Orbanz, Multiplicities and Hilbert functions under blowing up, Manuscripta Math. 36 (1981), 179-186. MR 641973 (83d:13031a)
  • [13] L. Robbiano, On normal flatness and some related topics, Commutative Algebra, Proceedings of the Trento Conference (ed. Greco-Valla), Lecture Notes in Pure and Appl. Math., vol. 84, Marcel Dekker, New York and Basel, 1983, pp. 235-251. MR 686948 (84i:13021)
  • [14] J.-P. Serre, Algèbre locale--Multiplicités, Lecture Notes in Math., vol. 11, Springer-Verlag, Berlin, Heidelberg and New York, 1965. MR 0201468 (34:1352)
  • [15] B. Singh, Effect of a permissible blowing up on the local Hilbert functions, Invent. Math. 26 (1974), 201-212. MR 0352097 (50:4584)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 14B25, 13H15

Retrieve articles in all journals with MSC: 14B25, 13H15

Additional Information

Keywords: Normal flatness, regular sequence, Hilbert function, blowing up, associated graded ring
Article copyright: © Copyright 1984 American Mathematical Society

American Mathematical Society