Maximal chains of prime ideals in integral extension domains. I

Ratliff, L. J.; McAdam, S.

doi:10.1090/S0002-9947-1976-0437513-3

Maximal chains of prime ideals in integral extension domains. I
HTML articles powered by AMS MathViewer

by L. J. Ratliff and S. McAdam PDF

Trans. Amer. Math. Soc. 224 (1976), 103-116 Request permission

Abstract:

Let (R, M) be a local domain, let k be a positive integer, and let Q be a prime ideal in ${R_k} = R[{X_1}, \ldots ,{X_k}]$ such that $M{R_k} \subset Q$. Then the following statements are equivalent: (1) There exists an integral extension domain of R which has a maximal chain of prime ideals of length n. (2) There exists a minimal prime ideal z in the completion of R such that depth $z = n$. (3) There exists a minimal prime ideal w in the completion of ${({R_k})_Q}$ such that depth $w = n + k - {\text {depth}}\;Q$. (4) There exists an integral extension domain of ${({R_k})_Q}$ which has a maximal chain of prime ideals of length $n + k - {\text {depth}}\;Q$. (5) There exists a maximal chain of prime ideals of length $n + k - {\text {depth}}\;Q$ in ${({R_k})_Q}$. (6) There exists a maximal chain of prime ideals of length $n + 1$ in $R{[{X_1}]_{(M,{X_1})}}$.

References

Stephen McAdam, Saturated chains in Noetherian rings, Indiana Univ. Math. J. 23 (1973/74), 719–728. MR 332768, DOI 10.1512/iumj.1974.23.23060
Evan G. Houston and Stephen McAdam, Chains of primes in Noetherian rings, Indiana Univ. Math. J. 24 (1974/75), 741–753. MR 360566, DOI 10.1512/iumj.1975.24.24057
Masayoshi Nagata, On the chain problem of prime ideals, Nagoya Math. J. 10 (1956), 51–64. MR 78974
Masayoshi Nagata, Note on a chain condition for prime ideals, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 32 (1959), 85–90. MR 108511, DOI 10.1215/kjm/1250776698
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
L. J. Ratliff Jr., On quasi-unmixed local domains, the altitude formula, and the chain condition for prime ideals. II, Amer. J. Math. 92 (1970), 99–144. MR 265339, DOI 10.2307/2373501
L. J. Ratliff Jr., Characterizations of catenary rings, Amer. J. Math. 93 (1971), 1070–1108. MR 297752, DOI 10.2307/2373746
Louis J. Ratliff Jr., Chain conjectures and $H$-domains, Conference on Commutative Algebra (Univ. Kansas, Lawrence, Kan., 1972), Lecture Notes in Math., Vol. 311, Springer, Berlin, 1973, pp. 222–238. MR 0337945
L. J. Ratliff Jr., Four notes on saturated chains of prime ideals, J. Algebra 39 (1976), no. 1, 75–93. MR 399072, DOI 10.1016/0021-8693(76)90062-4
Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0120249