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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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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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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