Unmixed 2-dimensional local domains

Stephen Joseph McAdam

Vol. 68 (1977), No. 1, 153–160

DOI: 10.2140/pjm.1977.68.153

Abstract

Let b,c be a system of parameters in a 2-dimensional local (Noetherian) domain (R,M). For n ≧ 0, the chain (bⁿ : 1) ⊂ (bⁿ : c) ⊂ (bⁿ : c²) ⊂⋯ becomes stable. Thus define a function S(b,c,−) by letting S(b,c,n) be the least integer k ≧ 0 such that (bⁿ : c^k) = (bⁿ : c^k+1). Ratliff has shown that R is unmixed if and only if S(b,c,−) is bounded. This paper shows that if R is unmixed then for any 0≠d ∈ M there is an integer ∗d ≧ 0 such that for any system of parameters b,c and any i ≧ 0,S(b,c,∗b + i) = ∗c.

Mathematical Subject Classification 2000

Primary: 13H99

Milestones

Received: 1 June 1976

Published: 1 January 1977