Characterization of completions of unique factorization domains

Heitmann, Raymond C.

doi:10.1090/S0002-9947-1993-1102888-9

Characterization of completions of unique factorization domains
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by Raymond C. Heitmann PDF

Trans. Amer. Math. Soc. 337 (1993), 379-387 Request permission

Abstract:

It is shown that a complete local ring is the completion of a unique factorization domain if and only if it is a field, a discrete valuation ring, or it has depth at least two and no element of its prime ring is a zerodivisor. It is also shown that the Normal Chain Conjecture is false and that there exist local noncatenary UFDs.

References

Nicolas Bourbaki, Elements of mathematics. Commutative algebra, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass., 1972. Translated from the French. MR 0360549
Ada Maria de Souza Doering, The depth conjecture: a counterexample, J. Algebra 77 (1982), no. 2, 443–448. MR 673127, DOI 10.1016/0021-8693(82)90265-4
Christer Lech, A method for constructing bad Noetherian local rings, Algebra, algebraic topology and their interactions (Stockholm, 1983) Lecture Notes in Math., vol. 1183, Springer, Berlin, 1986, pp. 241–247. MR 846452, DOI 10.1007/BFb0075463
Hideyuki Matsumura, Commutative algebra, W. A. Benjamin, Inc., New York, 1970. MR 0266911
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
L. J. Ratliff Jr., On quasi-unmixed local domains, the altitude formula, and the chain condition for prime ideals. I, Amer. J. Math. 91 (1969), 508–528. MR 246867, DOI 10.2307/2373524
Louis J. Ratliff Jr., Chain conjectures in ring theory, Lecture Notes in Mathematics, vol. 647, Springer, Berlin, 1978. An exposition of conjectures on catenary chains. MR 496884
L. J. Ratliff Jr., A brief history and survey of the catenary chain conjectures, Amer. Math. Monthly 88 (1981), no. 3, 169–178. MR 619563, DOI 10.2307/2320461