Maximal Chains of Prime Ideals in Integral Extension Domains. I

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 $R{[{X_1}]_{(M,{X_1})}}$.