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)



The existence of minimal regular local overrings for an arbitrary domain

Author: Bernard Johnston
Journal: Proc. Amer. Math. Soc. 100 (1987), 419-423
MSC: Primary 13H05
MathSciNet review: 891138
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that the set of regular local rings of dimension $ n$ containing an integral domain $ D$, having the same quotient field as $ D$, and ordered by containment satisfies the descending chain condition. The set of regular local rings of dimension $ n$ contained in a regular local ring of dimension $ m,n > m$, need not satisfy the ascending chain condition, as is shown by example.

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

  • [1] S. Abhyankar, On the valuations centered in a local domain, Amer. J. Math. 78 (1956), 321-348. MR 0082477 (18:556b)
  • [2] B. Johnston, A finiteness condition on regular local overrings of a local domain, Trans. Amer. Math. Soc. 299 (1987), 513-524. MR 869218 (87k:13035)
  • [3] H. Matsumura, Commutative algebra (2nd ed.), Benjamin/Cummings, Reading, Mass., 1980. MR 575344 (82i:13003)
  • [4] M. Nagata, Local rings, Interscience, New York, 1962. MR 0155856 (27:5790)
  • [5] J. Sally, Regular overrings of regular local rings, Trans. Amer. Math. Soc. 171 (1972), 291-300. MR 0309929 (46:9033)
  • [6] D. Shannon, Monoidal transforms of regular local rings, Amer. J. Math. 95 (1973), 294-320. MR 0330154 (48:8492)
  • [7] O. Zariski, Reductions of the singularities of algebraic three-dimensional varieties, Ann. of Math. (2) 45 (1944), 472-542. MR 0011006 (6:102f)
  • [8] O. Zariski and P. Samuel, Commutative algebra, Vol. II, Van Nostrand, Princeton, N. J., 1960. MR 0120249 (22:11006)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 13H05

Retrieve articles in all journals with MSC: 13H05

Additional Information

Keywords: Birational, regular local ring, quadratic transform, descending chain condition
Article copyright: © Copyright 1987 American Mathematical Society

American Mathematical Society