One-dimensional bad Noetherian domains

Olberding, Bruce

doi:10.1090/S0002-9947-2014-05921-7

One-dimensional bad Noetherian domains
HTML articles powered by AMS MathViewer

by Bruce Olberding PDF

Trans. Amer. Math. Soc. 366 (2014), 4067-4095 Request permission

Abstract:

Local Noetherian domains arising as local rings of points of varieties or in the context of algebraic number theory are analytically unramified, meaning their completion has no nontrivial nilpotent elements. However, looking elsewhere, many sources of analytically ramified local Noetherian domains have been exhibited over the last seventy-five years. We give a unified approach to a number of such examples by describing classes of DVRs which occur as the normalization of an analytically ramified local Noetherian domain, as well as some that do not occur as such a normalization. We parameterize these examples, or at least large classes of them, using the module of Kähler differentials of a relevant field extension.

References

Y. Akizuki, Einige Bemerkungen über primäre Integritätsbereiche mit Teilerkettensatz, Proc. Phys.–Math. Soc. Japan 17 (1935), 327–336.
Bruce Bennett, On the structure of non-excellent curve singularities in characteristic $p$, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 129–170. MR 318144
I. S. Cohen, On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc. 59 (1946), 54–106. MR 16094, DOI 10.1090/S0002-9947-1946-0016094-3
Ada Maria de Souza Doering and Yves Lequain, Maximally differential prime ideals, J. Algebra 101 (1986), no. 2, 403–417. MR 847167, DOI 10.1016/0021-8693(86)90201-2
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
Daniel Ferrand and Michel Raynaud, Fibres formelles d’un anneau local noethérien, Ann. Sci. École Norm. Sup. (4) 3 (1970), 295–311 (French). MR 272779
László Fuchs and Luigi Salce, Modules over non-Noetherian domains, Mathematical Surveys and Monographs, vol. 84, American Mathematical Society, Providence, RI, 2001. MR 1794715, DOI 10.1090/surv/084
K. R. Goodearl and T. H. Lenagan, Constructing bad Noetherian local domains using derivations, J. Algebra 123 (1989), no. 2, 478–495. MR 1000498, DOI 10.1016/0021-8693(89)90057-4
William Heinzer, David Lantz, and Kishor Shah, The Ratliff-Rush ideals in a Noetherian ring, Comm. Algebra 20 (1992), no. 2, 591–622. MR 1146317, DOI 10.1080/00927879208824359
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
F. J. Herrera Govantes, M. A. Olalla Acosta, M. Spivakovsky, and B. Teissier, Extending a valuation centred in a local domain to the formal completion, Proc. Lond. Math. Soc. (3) 105 (2012), no. 3, 571–621. MR 2974200, DOI 10.1112/plms/pds002
Shihoko Ishii, Jet schemes, arc spaces and the Nash problem, C. R. Math. Acad. Sci. Soc. R. Can. 29 (2007), no. 1, 1–21 (English, with English and French summaries). MR 2354631
W. Krull, Dimensionstheorie in Stellenringen, J. Reine Angew. Math. 179 (1938), 204–226.
Ernst Kunz, Kähler differentials, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1986. MR 864975, DOI 10.1007/978-3-663-14074-0
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
Yves Lequain, Differential simplicity and complete integral closure, Pacific J. Math. 36 (1971), 741–751. MR 284422
Joseph Lipman, Stable ideals and Arf rings, Amer. J. Math. 93 (1971), 649–685. MR 282969, DOI 10.2307/2373463
Eben Matlis, $1$-dimensional Cohen-Macaulay rings, Lecture Notes in Mathematics, Vol. 327, Springer-Verlag, Berlin-New York, 1973. MR 0357391
Eben Matlis, The theory of $Q$-rings, Trans. Amer. Math. Soc. 187 (1974), 147–181. MR 340241, DOI 10.1090/S0002-9947-1974-0340241-4
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
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
Bruce Olberding, On the structure of stable domains, Comm. Algebra 30 (2002), no. 2, 877–895. MR 1883031, DOI 10.1081/AGB-120013188
Bruce Olberding, Stability, duality, 2-generated ideals and a canonical decomposition of modules, Rend. Sem. Mat. Univ. Padova 106 (2001), 261–290. MR 1876223
Bruce Olberding, A counterpart to Nagata idealization, J. Algebra 365 (2012), 199–221. MR 2928459, DOI 10.1016/j.jalgebra.2012.05.002
B. Olberding, Generic formal fibers and analytically ramified stable rings, Nagoya Math. J., to appear.
Bruce Olberding, Noetherian rings without finite normalization, Progress in commutative algebra 2, Walter de Gruyter, Berlin, 2012, pp. 171–203. MR 2932595
Ana J. Reguera, Towards the singular locus of the space of arcs, Amer. J. Math. 131 (2009), no. 2, 313–350. MR 2503985, DOI 10.1353/ajm.0.0046
Judith D. Sally, Numbers of generators of ideals in local rings, Marcel Dekker, Inc., New York-Basel, 1978. MR 0485852
Judith D. Sally and Wolmer V. Vasconcelos, Stable rings, J. Pure Appl. Algebra 4 (1974), 319–336. MR 409430, DOI 10.1016/0022-4049(74)90012-7
Friedrich Karl Schmidt, Über die Erhaltung der Kettensätze der Idealtheorie bei beliebigen endlichen Körpererweiterungen, Math. Z. 41 (1936), no. 1, 443–450 (German). MR 1545632, DOI 10.1007/BF01180433
Charles A. Weibel, $K$-theory and analytic isomorphisms, Invent. Math. 61 (1980), no. 2, 177–197. MR 590161, DOI 10.1007/BF01390120
Oscar Zariski and Pierre Samuel, Commutative algebra, Volume I, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. MR 0090581