A characterization of locally Macaulay completions

Craig Huneke

Vol. 94 (1981), No. 1, 131–138

DOI: 10.2140/pjm.1981.94.131

Abstract

The purpose of this note is to prove the following theorem.

Theorem 1.1. Let (R,m) be a Noetherian local ring of dimension d ≧ 1 and depth d − 1. By R denote the completion of R in the m-adic topology. Then the following are equivalent:

1. R is equidimensional and satisfies Serre’s property S_d−1
2. H_m^d−1(R) has finite length
3. There exists an N > 0 such that if x₁,⋯,x_d is a sequence of elements R with ht(x_i1,⋯,x_ij) = j for all j-elements subsets of {1,⋯,n}, 1 ≦ j ≦ n, and if m_i ≧ N, 1 ≦ i ≦ d, then x₁^m₁,⋯,x_d^m_d is an unconditioned d-sequence.

Mathematical Subject Classification 2000

Primary: 13H99

Milestones

Received: 18 October 1979

Published: 1 May 1981