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)

 
 

 

Test elements in excellent rings with an application to the uniform Artin-Rees property


Author: Ian M. Aberbach
Journal: Proc. Amer. Math. Soc. 118 (1993), 355-363
MSC: Primary 13A35; Secondary 13D25, 13F40
DOI: https://doi.org/10.1090/S0002-9939-1993-1129869-9
MathSciNet review: 1129869
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Working in positive prime characteristic throughout, we show that excellent rings of dimension $ 2$ or smaller have completely stable test elements and use this to show that excellent domains of dimension $ 3$ have the uniform Artin-Rees property.


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

  • [BE] D. Buchsbaum and D. Eisenbud, What makes a complex exact?, J. Algebra 25 (1973), 259-268. MR 0314819 (47:3369)
  • [DO] A. J. Duncan and L. O'Carroll, A full uniform Artin-Rees theorem, J. Reine Angew. Math. 394 (1989), 203-207. MR 977443 (90c:13011)
  • [EN] J. A. Eagon and D. G. Northcott, Ideals defined by matrices and a certain complex associated with them, Proc. Roy. Soc. London Ser. A 269 (1962), 188-204. MR 0142592 (26:161)
  • [Gr] A. Grothendieck, Éléments de géométrie algébrique $ {\operatorname{IV} _2}$, Inst. Hautes Études Sci. Publ. Math. 20 (1964).
  • [Ho] M. Hochster, Cyclic purity versus purity in excellent Noetherian rings, Trans. Amer. Math. Soc. 231 (1977), 463-488. MR 0463152 (57:3111)
  • [HH1] M. Hochster and C. Huneke, Tight closure, invariant theory, and the Briançon-Skoda Theorem, J. Amer. Math. Soc. 3 (1990), 31-116. MR 1017784 (91g:13010)
  • [HH2] -, $ F$-regularity, test elements and smooth base change rings, preprint.
  • [HH3] -, Tight closure and strong $ F$-regularity, Numéro consacré ou colloque en l'honneur de P. Samuel, Mém. Soc. Math. de France (N.S), no. 38 (1989), 119-133.
  • [HH4] -, Phantom homology, preprint.
  • [Hu] C. Huneke, Uniform bounds in Noetherian rings, Invent. Math. (to appear). MR 1135470 (93b:13027)
  • [Ma] H. Matsumura, Commutative algebra, Benjamin/Cummings, Cambridge, New York, New Rochelle, Melbourne, Sydney, 1980. MR 575344 (82i:13003)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 13A35, 13D25, 13F40

Retrieve articles in all journals with MSC: 13A35, 13D25, 13F40


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1129869-9
Keywords: Tight closure, test elements, uniform Artin-Rees
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society