Good chains with bad contractions

Raymond Heitmann and Stephen Joseph McAdam

Vol. 86 (1980), No. 2, 477–489

DOI: 10.2140/pjm.1980.86.477

Abstract

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 ⊂ q′ of T such that height (q′∕q) < height (q′∩R∕q ∩R). In this paper we will consider a chain of primes q₁ ⊂ q₂ ⊂⋯ ⊂ q_m in T which is well behaved in that height (q_m∕q₁) = ∑ _i=2^m height (q_i∕q_i−1), but which suffers the pathology that height (q_i ∩ R∕q_i−1 ∩ R) > height (q_i∕q_i−1) 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 b_n such that m ≦ b_n; moreover b₂ = 2 and in general b_n ≦ b_n−1ⁿ⁻² + b_n−1ⁿ⁻³ + ⋯ + b_n−1 + 2. Let us quickly add that we do not claim that this formula gives the best bound possible. (We rather suspect not.) If c = b_n−1 + 2, we also have, as part of our main result, that m ≦ height (q_c∕q₁) + b_n−1. (If m > b_n−1, so that q_c exists.) Finally, if we have the added assumption that height (q_i∕q_i−1) ≦ r for i = 2,⋯,m, then m ≦ 2(r + 1)ⁿ⁻².

Mathematical Subject Classification

Primary: 13A17, 13A17

Secondary: 13B20

Milestones

Received: 12 July 1978

Revised: 15 March 1978

Published: 1 February 1980