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Filtrations and Noetherian symbolic blow-up rings

Author: Peter Schenzel
Journal: Proc. Amer. Math. Soc. 102 (1988), 817-822
MSC: Primary 13E05; Secondary 13B99, 13C13
MathSciNet review: 934849
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Abstract: For a one-dimensional prime ideal in a local Noetherian ring it is characterized when the symbolic blow-up ring is an algebra of finite type. More generally, for a filtration of ideals of a local Noetherian ring there is a necessary and sufficient condition for the corresponding Rees ring to be a Noetherian ring. Applications concern asymptotic prime divisors and the analytic spread.

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

  • [E] S. Eliahou, Courbes monomiales et algebre de Rees symbolique, Thèse, Université de Genève, 1983.
  • [H] C. Huneke, On the finite generation of symbolic blow-ups, Math. Z. 179 (1982), 465-472. MR 652854 (83h:13029)
  • [M] H. Matsumura, Commutative algebra, 2nd ed., Benjamin, New York, 1980. MR 575344 (82i:13003)
  • [N] M. Nagata, Lectures on the fourteenth problem of Hilbert, Tata Inst. Fund. Res., Lectures on Math., No. 31, Bombay, 1965. MR 0215828 (35:6663)
  • [NR] D. G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Cambridge Philos. Soc. 50 (1954), 145-158. MR 0059889 (15:596a)
  • [O] A. Ooishi, Noetherian property of symbolic Rees algebras, Hiroshima Math. J. 15 (1985), 581-584. MR 813574 (87c:13020)
  • [Re] D. Rees, On a problem of Zariski, Illinois J. Math. 2 (1958), 145-149. MR 0095843 (20:2341)
  • [Ro] P. C. Roberts, A prime ideal in a polynomial ring whose symbolic blow-up is not Noetherian, Proc. Amer. Math. Soc. 94 (1985), 589-592. MR 792266 (86k:13017)
  • [S$ _{1}$] P. Schenzel, Finiteness of relative Rees rings and asymptotic prime divisors, Math. Nachr. 129 (1986), 123-148. MR 864628 (87m:13008)
  • [S$ _{2}$] -, Examples of Noetherian symbolic blow-up rings, Rev. Roumaine Math. Pures Appl. (to appear). MR 950134 (89j:13015)

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Keywords: Noetherian filtration, Veronesean subring, asymptotic prime divisor, analytic spread, symbolic power, symbolic blow-up ring, Rees ring, Hilbert's fourteenth problem
Article copyright: © Copyright 1988 American Mathematical Society

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