## Nagoya Mathematical Journal

### Generic formal fibers and analytically ramified stable rings

Bruce Olberding

#### Abstract

Let $A$ be a local Noetherian domain of Krull dimension $d$. Heinzer, Rotthaus, and Sally have shown that if the generic formal fiber of $A$ has dimension $d-1$, then $A$ is birationally dominated by a 1-dimensional analytically ramified local Noetherian ring having residue field finite over the residue field of $A$. We explore further this correspondence between prime ideals in the generic formal fiber and 1-dimensional analytically ramified local rings. Our main focus is on the case where the analytically ramified local rings are stable, and we show that in this case the embedding dimension of the stable ring reflects the embedding dimension of a prime ideal maximal in the generic formal fiber, thus providing a measure of how far the generic formal fiber deviates from regularity. A number of characterizations of analytically ramified local stable domains are also given.

#### Article information

Source
Nagoya Math. J., Volume 211 (2013), 109-135.

Dates
First available in Project Euclid: 29 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1367242338

Digital Object Identifier
doi:10.1215/00277630-2148583

Mathematical Reviews number (MathSciNet)
MR3079281

Zentralblatt MATH identifier
1278.13020

#### Citation

Olberding, Bruce. Generic formal fibers and analytically ramified stable rings. Nagoya Math. J. 211 (2013), 109--135. doi:10.1215/00277630-2148583. https://projecteuclid.org/euclid.nmj/1367242338

#### References

• [1] Y. Akizuki, Einige Bemerkungen über primäre Integritätsbereiche mit Teilerkettensatz, Proc. Phys. Math. Soc. Japan 17 (1935), 327–336.
• [2] H. Bass, On the ubiquity of Gorenstein rings, Math Z. 82 (1963), 8–28.
• [3] J. A. Drozd and V. V. Kiričenko, The quasi-Bass orders (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 6 (1972), 328–370; English translation in Math. USSR-Izv. 6 (1972), 323–365.
• [4] S. El Baghdadi and S. Gabelli, Ring-theoretic properties of P$v$MDs, Comm. Algebra 35 (2007), 1607–1625.
• [5] L. Fuchs and L. Salce, Modules over non-Noetherian Domains, Math. Surveys Monogr. 84, Amer. Math. Soc., Providence, 2001.
• [6] S. Gabelli and G. Picozza, Star stable domains, J. Pure Appl. Algebra 208 (2007), 853–866.
• [7] W. Heinzer, D. Lantz, and K. Shah, The Ratliff–Rush ideals in a Noetherian ring, Comm. Algebra 20 (1992), 591–622.
• [8] W. Heinzer, C. Rotthaus, and J. D. Sally, Formal fibers and birational extensions, Nagoya Math. J. 131 (1993), 1–38.
• [9] S. Kabbaj and A. Mimouni, $t$-Class semigroups of integral domains, J. Reine Angew. Math. 612 (2007), 213–229.
• [10] W. Krull, Dimensionstheorie in Stellenringen, J. Reine Angew. Math. 179 (1938), 204–226.
• [11] J. Lipman, Stable ideals and Arf rings, Amer. J. Math. 93 (1971), 649–685.
• [12] E. Matlis, Injective modules over Noetherian rings, Pacific J. Math. 8 (1958), 511–528.
• [13] E. Matlis, Injective modules over Prüfer rings, Nagoya Math. J. 15 (1959), 57–69.
• [14] E. Matlis, Torsion-Free Modules, University of Chicago Press, Chicago, 1972.
• [15] E. Matlis, $1$-Dimensional Cohen-Macaulay Rings, Lecture Notes in Math. 327, Springer, Berlin, 1973.
• [16] H. Matsumura, Commutative Algebra, 2nd ed., Math. Lecture Note Series 56, Benjamin/Cummings, Reading, Mass., 1980.
• [17] H. Matsumura, Commutative Ring Theory, Cambridge Stud. Adv. Math. 8, Cambridge University Press, Cambridge, 1986.
• [18] H. Matsumura, “On the dimension of formal fibres of a local ring” in Algebraic Geometry and Commutative Algebra, Vol. I, Kinokuniya, Tokyo, 1988, 261–266.
• [19] A. Mimouni, Ratliff–Rush closure of ideals in integral domains, Glasg. Math. J. 51 (2009), 681–689.
• [20] R. Nagasawa, Some remarks on one-dimensional local domains, Publ. Res. Inst. Math. Sci. 11 (1975/76), 21–30.
• [21] M. Nagata, Local Rings, Interscience Tracts Pure Appl. Math. 13, Wiley, New York, 1962.
• [22] B. Olberding, On the classification of stable domains, J. Algebra 243 (2001), 177–197.
• [23] B. Olberding, Stability, duality, $2$-generated ideals and a canonical decomposition of modules, Rend. Semin. Mat. Univ. Padova 106 (2001), 261–290.
• [24] B. Olberding, “Stability of ideals and its applications” in Ideal Theoretic Methods in Commutative Algebra (Columbia, Mo., 1999), Lecture Notes Pure Appl. Math. 220, Dekker, New York, 2001, 319–341.
• [25] B. Olberding, On the structure of stable domains, Comm. Algebra 30 (2002), 877–895.
• [26] B. Olberding, A counterpart to Nagata idealization, J. Algebra 365 (2012), 199–221.
• [27] B. Olberding, One-dimensional bad Noetherian rings, to appear in Trans. Amer. Math. Soc., preprint, arXiv:1208.2913 [math.AC]
• [28] G. Picozza and F. Tartarone, Flat ideals and stability in integral domains, J. Algebra 324 (2010), 1790–1802.
• [29] J. D. Sally and W. V. Vasconcelos, Stable rings and a problem of Bass, Bull. Amer. Math. Soc. (N.S.) 79 (1973), 574–576.
• [30] J. Sally and W. Vasconcelos, Stable rings, J. Pure Appl. Algebra 4 (1974), 319–336.
• [31] F. K. Schmidt, Über die Erhaltung der Kettensätze der Idealtheorie bei beliebigen endlichen Körpererweiterungen, Math. Z. 41 (1936), 443–450.
• [32] L. Sega, Ideal class semigroups of overrings, J. Algebra 311 (2007), 702–713.
• [33] P. Zanardo, The class semigroup of local one-dimensional domains, J. Pure Appl. Algebra 212 (2008), 2259–2270.
• [34] P. Zanardo, Algebraic entropy of endomorphisms over local one-dimensional domains, J. Algebra Appl. 8 (2009), 759–777.