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)



Formal fibers and complete homomorphic images

Authors: William Heinzer and Christel Rotthaus
Journal: Proc. Amer. Math. Soc. 120 (1994), 359-369
MSC: Primary 13F40; Secondary 13J15
MathSciNet review: 1189544
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ (R,{\mathbf{m}})$ be an excellent normal local Henselian domain, and suppose that $ {\mathbf{q}}$ is a prime ideal in $ R$ of height $ > 1$. We show that, if $ R/{\mathbf{q}}$ is not complete, then there are infinitely many height one prime ideals $ {\mathbf{p}} \subseteq {\mathbf{q}}\hat R$ of $ \hat R$ with $ {\mathbf{p}} \cap R = 0$; in particular, the dimension of the generic formal fiber of $ R$ is at least one. This result may in fact indicate that a much stronger relationship between maximal ideals in the formal fibers of an excellent Henselian local ring and its complete homomorphic images is possibly satisfied. The second half of the paper is concerned with a property of excellent normal local Henselian domains $ R$ with zero-dimensional formal fibers. We show that for such an $ R$ one has the following good property with respect to intersection: for any field $ L$ such that $ \mathcal{Q}(R) \subseteq L \subseteq \mathcal{Q}(\hat R)$, the ring $ L \cap \hat R$ is a local Noetherian domain which has completion $ \hat R$.

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

  • [AN] M. André, Cinq exposés sur la desingularisation (in preparation).
  • [A] M. Artin, Algebraic approximation of structures over complete local rings, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 23–58. MR 0268188
  • [BKKN] R. Berger, R. Kiehl, E. Kunz, and H.-J. Nastold, Differentialrechnung in der analytischen Geometrie, Lecture Notes in Mathematics, No. 38, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224870
  • [C] I. S. Cohen, Lengths of prime ideal chains, Amer. J. Math. 76 (1954), 654–668. MR 0062116
  • [G] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math. 24 (1965), 231 (French). MR 0199181
  • [HR] W. Heinzer and C. Rotthaus, On a certain class of algebraically independent elements (in preparation).
  • [HRS] W. Heinzer, C. Rotthaus, and J. Sally, Formal fibers and birational extensions (in preparation).
  • [M1] 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
  • [M2] -, On the dimension of formal fibers in a local ring, Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata, Kinokuniya, Tokyo, 1988.
  • [M3] Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
  • [N] Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers a division of John Wiley & Sons New York-London, 1962. MR 0155856
  • [Ni] Jun-ichi Nishimura, Note on Krull domains, J. Math. Kyoto Univ. 15 (1975), no. 2, 397–400. MR 0384783
  • [P1] Dorin Popescu, General Néron desingularization, Nagoya Math. J. 100 (1985), 97–126. MR 818160
  • [P2] Dorin Popescu, General Néron desingularization and approximation, Nagoya Math. J. 104 (1986), 85–115. MR 868439
  • [R1] Christel Rotthaus, On the approximation property of excellent rings, Invent. Math. 88 (1987), no. 1, 39–63. MR 877005, 10.1007/BF01405090
  • [R2] Christel Rotthaus, Rings with approximation property, Math. Ann. 287 (1990), no. 3, 455–466. MR 1060686, 10.1007/BF01446905
  • [R3] Christel Rotthaus, On rings with low-dimensional formal fibres, J. Pure Appl. Algebra 71 (1991), no. 2-3, 287–296. MR 1117639, 10.1016/0022-4049(91)90152-R
  • [V] Paolo Valabrega, On two-dimensional regular local rings and a lifting problem, Ann. Scuola Norm. Sup. Pisa (3) 27 (1973), 787–807 (1974). MR 0364228

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 13F40, 13J15

Retrieve articles in all journals with MSC: 13F40, 13J15

Additional Information

Keywords: Generic formal fiber, approximation property, excellent local Henselian domain, complete homomorphic image, Noetherian intermediate ring
Article copyright: © Copyright 1994 American Mathematical Society