Approximating discrete valuation rings by regular local rings

Heinzer, William; Rotthaus, Christel; Wiegand, Sylvia

doi:10.1090/S0002-9939-00-05492-7

Approximating discrete valuation rings by regular local rings
HTML articles powered by AMS MathViewer

by William Heinzer, Christel Rotthaus and Sylvia Wiegand

Proc. Amer. Math. Soc. 129 (2001), 37-43

DOI: https://doi.org/10.1090/S0002-9939-00-05492-7

Published electronically: July 27, 2000

PDF | Request permission

Abstract:

Let $k$ be a field of characteristic zero and let $(V,\mathbf {n})$ be a discrete rank-one valuation domain containing $k$ with $V/\mathbf {n}= k$. Assume that the fraction field $L$ of $V$ has finite transcendence degree $s$ over $k$. For every positive integer $d \le s$, we prove that $V$ can be realized as a directed union of regular local $k$-subalgebras of $V$ of dimension $d$.

References

Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
Shreeram S. Abhyankar and William J. Heinzer, Ramification in infinite integral extensions, J. Algebra 170 (1994), no. 3, 861–879. MR 1305267, DOI 10.1006/jabr.1994.1367
Steven Dale Cutkosky, Local factorization of birational maps, Adv. Math. 132 (1997), no. 2, 167–315. MR 1491444, DOI 10.1006/aima.1997.1675
—, Local factorization and monomialization of morphisms, to appear.
William Heinzer, Christel Rotthaus, and Judith D. Sally, Formal fibers and birational extensions, Nagoya Math. J. 131 (1993), 1–38. MR 1238631, DOI 10.1017/S0027763000004529
William Heinzer, Christel Rotthaus, and Sylvia Wiegand, Noetherian rings between a semilocal domain and its completion, J. Algebra 198 (1997), no. 2, 627–655. MR 1489916, DOI 10.1006/jabr.1997.7169
William Heinzer, Christel Rotthaus, and Sylvia Wiegand, Building Noetherian domains inside an ideal-adic completion, Abelian groups, module theory, and topology (Padua, 1997) Lecture Notes in Pure and Appl. Math., vol. 201, Dekker, New York, 1998, pp. 279–287. MR 1651173
—, Noetherian domains inside a homomorphic image of a completion, J. Algebra 215 (1999), 666-681.
Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
Masayoshi Nagata, An example of a normal local ring which is analytically reducible, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 31 (1958), 83–85. MR 97395, DOI 10.1215/kjm/1250776950
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
Christel Rotthaus, Homomorphic images of regular local rings, Comm. Algebra 24 (1996), no. 2, 445–476. MR 1373487, DOI 10.1080/00927879608825580
David L. Shannon, Monoidal transforms of regular local rings, Amer. J. Math. 95 (1973), 294–320. MR 330154, DOI 10.2307/2373787
Leonard Eugene Dickson, New First Course in the Theory of Equations, John Wiley & Sons, Inc., New York, 1939. MR 0000002
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
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