The existence of minimal regular local overrings for an arbitrary domain

Johnston, Bernard

doi:10.1090/S0002-9939-1987-0891138-7

The existence of minimal regular local overrings for an arbitrary domain
HTML articles powered by AMS MathViewer

by Bernard Johnston PDF

Proc. Amer. Math. Soc. 100 (1987), 419-423 Request permission

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

Shreeram Abhyankar, On the valuations centered in a local domain, Amer. J. Math. 78 (1956), 321–348. MR 82477, DOI 10.2307/2372519
Bernard Johnston, A finiteness condition on regular local overrings of a local domain, Trans. Amer. Math. Soc. 299 (1987), no. 2, 513–524. MR 869218, DOI 10.1090/S0002-9947-1987-0869218-6
Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856
Judith Sally, Regular overrings of regular local rings, Trans. Amer. Math. Soc. 171 (1972), 291–300. MR 309929, DOI 10.1090/S0002-9947-1972-0309929-3
David L. Shannon, Monoidal transforms of regular local rings, Amer. J. Math. 95 (1973), 294–320. MR 330154, DOI 10.2307/2373787
Oscar Zariski, Reduction of the singularities of algebraic three dimensional varieties, Ann. of Math. (2) 45 (1944), 472–542. MR 11006, DOI 10.2307/1969189
Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0120249