ONE-DIMENSIONAL LOCAL-RINGS OF MAXIMAL AND ALMOST MAXIMAL LENGTH

Let (A, m, k) denote a one dimensional, Cohen-Macaulay, local ring with maximal ideal m and residue class field k. We assume A is a reduced, excellent local ring which has a canonical module ωA. With no loss of generality, we can also assume k is infinite. Let Ā denote the integral closure of A in its total quotient ring Q, and let b denote the conductor of A in Ā. A has maximal length if lA( A ̄ A) = lA( A b) t(A) Here t(A) denotes the Cohen-Macaulay type of A, and lA(*) denotes the length of the A-module *. A has almost maximal length if lA( A ̄ A) = lA( A b) t(A) - 1 The two main results of this paper are as follows: A has maximal length if and only if either A is Gorenstein or there exists an x ε{lunate} m and a positive integer p such that m = (x, b), andb A ̄ = xp A ̄. Let e(A) denote the multiplicity of A, Then A has almost maximal length, and 1 + t(A) = e(A) if and only if there exists a transversal x of m such that m = (x, b), and lA( b xp A ̄) = 1. Here p = min{i|xiε{lunate}b} These two theorems generalize more specific results for semigroup rings obtained in Brown and Curtis ("Numerical Semigroups of Maximal and Almost Maximal Length," Semigroup Forum, Vol. 42, Springer-Verlag, Berlin/New York, 1991, 219-235). © 1992.