Projective normal flatness and Hilbert functions

Orbanz, U.; Robbiano, L.

doi:10.1090/S0002-9947-1984-0735407-1

Projective normal flatness and Hilbert functions
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by U. Orbanz and L. Robbiano PDF

Trans. Amer. Math. Soc. 283 (1984), 33-47 Request permission

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

Allen Altman and Steven Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Mathematics, Vol. 146, Springer-Verlag, Berlin-New York, 1970. MR 0274461
Bruce Michael Bennett, On the characteristic functions of a local ring, Ann. of Math. (2) 91 (1970), 25–87. MR 252388, DOI 10.2307/1970601
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
M. Herrmann and U. Orbanz, Faserdimensionen von Aufblasungen lokaler Ringe und Äquimultiplizität, J. Math. Kyoto Univ. 20 (1980), no. 4, 651–659 (German). MR 592352, DOI 10.1215/kjm/1250522164
M. Herrmann and U. Orbanz, Normale Flachheit und Äquimultiplizität für vollständige Durchschnitte, J. Algebra 70 (1981), no. 2, 437–451 (German). MR 623818, DOI 10.1016/0021-8693(81)90228-3
M. Herrmann and U. Orbanz, Between equimultiplicity and normal flatness, Algebraic geometry (La Rábida, 1981) Lecture Notes in Math., vol. 961, Springer, Berlin-New York, 1982, pp. 200–233. MR 708335
Ulrich Orbanz, Multiplicities and Hilbert functions under blowing up, Manuscripta Math. 36 (1981/82), no. 2, 179–186. MR 641973, DOI 10.1007/BF01170133
Manfred Herrmann, Rolf Schmidt, and Wolfgang Vogel, Theorie der normalen Flachheit, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1977 (German). Mit englischer und russischer Zusammenfassung; Mit einem Geleitwort von Wolfgang Gröbner; Teubner-Texte zur Mathematik. MR 0568896
Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; ibid. (2) 79 (1964), 205–326. MR 0199184, DOI 10.2307/1970547
Joseph Lipman, Equimultiplicity, reduction, and blowing up, Commutative algebra (Fairfax, Va., 1979) Lecture Notes in Pure and Appl. Math., vol. 68, Dekker, New York, 1982, pp. 111–147. MR 655801
Dieter Nesselmann, Über monoidale Transformationen lokaler Ringe, Math. Nachr. 88 (1979), 285–294 (German). MR 543408, DOI 10.1002/mana.19790880121
Ulrich Orbanz, Multiplicities and Hilbert functions under blowing up, Manuscripta Math. 36 (1981/82), no. 2, 179–186. MR 641973, DOI 10.1007/BF01170133
Lorenzo Robbiano, On normal flatness and some related topics, Commutative algebra (Trento, 1981) Lecture Notes in Pure and Appl. Math., vol. 84, Dekker, New York, 1983, pp. 235–251. MR 686948
Jean-Pierre Serre, Algèbre locale. Multiplicités, Lecture Notes in Mathematics, vol. 11, Springer-Verlag, Berlin-New York, 1965 (French). Cours au Collège de France, 1957–1958, rédigé par Pierre Gabriel; Seconde édition, 1965. MR 0201468
Balwant Singh, Effect of a permissible blowing-up on the local Hilbert functions, Invent. Math. 26 (1974), 201–212. MR 352097, DOI 10.1007/BF01418949