Vol. 86, No. 2, 1980

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Good chains with bad contractions

Raymond Heitmann and Stephen Joseph McAdam

Vol. 86 (1980), No. 2, 477–489

Let R T be commutative rings with T integral over R. In the study of chains of prime ideals, it is often of interest to know about primes q qof T such that height (q∕q) < height (q′∩R∕q R). In this paper we will consider a chain of primes q1 q2 qm in T which is well behaved in that height (qm∕q1) = i=2m height (qi∕qi1), but which suffers the pathology that height (qi R∕qi1 R) > height (qi∕qi1) for each i = 2,,m. Our goal is to find a bound on how large m can be.

Our main result is that if T is generated as an R-module by n elements, then there is a bound bn such that m bn; moreover b2 = 2 and in general bn bn1n2 + bn1n3 + + bn1 + 2. Let us quickly add that we do not claim that this formula gives the best bound possible. (We rather suspect not.) If c = bn1 + 2, we also have, as part of our main result, that m height (qc∕q1) + bn1. (If m > bn1, so that qc exists.) Finally, if we have the added assumption that height (qi∕qi1) r for i = 2,,m, then m 2(r + 1)n2.

Mathematical Subject Classification
Primary: 13A17, 13A17
Secondary: 13B20
Received: 12 July 1978
Revised: 15 March 1978
Published: 1 February 1980
Raymond Heitmann
University of Texas, Austin
Austin TX
United States
Stephen Joseph McAdam