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 , 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 . 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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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,
-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