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Maximal chains of prime ideals in integral extension domains. II


Author: L. J. Ratliff
Journal: Trans. Amer. Math. Soc. 224 (1976), 117-141
MSC: Primary 13A15; Secondary 13B20
DOI: https://doi.org/10.1090/S0002-9947-1976-0437514-5
MathSciNet review: 0437514
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Abstract: Four related subjects are investigated: (1) If (L, N) is a locality over a local domain (R, M) such that $ N \cap R = M$, and if there exists an integral extension domain of L which has a maximal chain of prime ideals of length n (for short, a mcpil n), then there exists an integral extension domain of R which has a mcpil $ n - {\text{trd}}\;L/R + {\text{trd}}(L/N)/(R/M)$. A refinement of the altitude inequality follows from this. (2) A condition for the converse of (1) to hold is given. (3) The class of local domains R such that there exists an integral extension domain of R which has a mcpil n if and only if there exists a mcpil n in R is studied. (4) Two new equivalences for the existence of mcpil n in an integral extension domain of a local domain are given.


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  • [1] S. Abhyankar, Resolution of singularities of embedded algebraic surfaces, Pure and Appl. Math., vol. 24, Academic Press, New York and London, 1966. MR 36 #164; erratum, 36, p. 1568. MR 0217069 (36:164)
  • [2] S. McAdam, Saturated chains in Noetherian rings, Indiana Univ. Math. J. 23 (1973/74), 719-728. MR 48 #11094. MR 0332768 (48:11094)
  • [3] S. McAdam and E. G. Houston, Chains of primes in Noetherian rings, Indiana Univ. Math. J. 24 (1975), 741-753. MR 0360566 (50:13014)
  • [4] S. McAdam and L. J. Ratliff, Jr., Semi-local taut rings (forthcoming).
  • [5] M. Nagata, On the chain problem of prime ideals, Nagoya Math. J. 10 (1956), 51-64. MR 18, 8. MR 0078974 (18:8e)
  • [6] -, Local rings, Interscience Tracts in Pure and Appl. Math., no. 13, Interscience, New York, 1962. MR 27 #5790. MR 0155856 (27:5790)
  • [7] M. E. Pettit, Jr., Properties of $ {H_i}$-rings, Ph. D. Thesis, University of California, Riverside, 1973.
  • [8] M. E. Pettit, Jr., Properties of $ {H_i}$-rings (forthcoming).
  • [9] L. J. Ratliff, Jr., On quasi-unmixed semi-local rings and the altitude formula, Amer. J. Math. 87 (1965), 278-284. MR 31 #3448. MR 0179199 (31:3448)
  • [10] -, On quasi-unmixed local domains, the altitude formula, and the chain condition for prime ideals. I, Amer. J. Math. 91 (1969), 508-528. MR 40 #136. MR 0246867 (40:136)
  • [11] -, On quasi-unmixed local domains, the altitude formula, and the chain condition for prime ideals. II, Amer. J. Math. 92 (1970), 99-144. MR 42 #249. MR 0265339 (42:249)
  • [12] -, Characterizations of catenary rings, Amer. J. Math. 93 (1971), 1070-1108. MR 45 #6804. MR 0297752 (45:6804)
  • [13] -, Catenary rings and the altitude formula, Amer. J. Math. 94 (1972), 458-466. MR 47 #221. MR 0311659 (47:221)
  • [14] -, Chain conjectures and H-domains, Conf. on Commutative Algebra (Univ. Kansas, Lawrence, Kan., 1972), Lecture Notes in Math., vol. 311, Springer-Verlag, Berlin, 1973. MR 49 #2714. MR 0337945 (49:2714)
  • [15] -, Four notes on saturated chains of prime ideals, J. Algebra 39 (1976), 75-92. MR 0399072 (53:2923)
  • [16] -, Equivalences of the chain conjectures (forthcoming).
  • [17] L. J. Ratliff, Jr. and S. McAdam, Maximal chains of prime ideals in integral extension domains. I, Trans. Amer. Math. Soc. 224 (1976), 103-116. MR 0437513 (55:10438a)
  • [18] L. J. Ratliff, Jr. and M. E. Pettit, Jr., Characterizations of $ {H_i}$-local rings and of $ {C_i}$-local rings (forthcoming).
  • [19] D. Rees, A note on form rings and ideals, Mathematika 4 (1957), 51-60. MR 19, 835. MR 0090588 (19:835c)
  • [20] -, A-transforms of local rings and a theorem on multiplicities of ideals, Proc. Cambridge Philos. Soc. 57 (1961), 8-17. MR 22 #9521. MR 0118750 (22:9521)
  • [21] O. Zariski and P. Samuel, Commutative algebra. Vol. II, University Ser. in Higher Math., Van Nostrand, Princeton, N. J., 1960. MR 22 #11006. MR 0120249 (22:11006)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1976-0437514-5
Keywords: Algebraic extension, altitude formula, altitude inequality, analytically independent, catenary ring, chain condition for prime ideals, chain conjecture, completion of local ring, first chain condition, form ring, H conjecture, $ {H_i}$-ring, Henselian local ring, integral extension, local ring, locality, maximal chain of prime ideals, Noetherian ring, quadratic transformation, quasi-unmixed local ring, Rees ring, second chain condition, subspace, transcendental extension ring, unmixed local ring, upper conjecture
Article copyright: © Copyright 1976 American Mathematical Society

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