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One dimensional local rings of maximal and almost maximal length

Journal of Algebra
Publication Date
DOI: 10.1016/0021-8693(92)90118-6
  • Mathematics


Abstract 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 ω A . 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 l A( A ̄ A ) = l A( A b ) t(A) Here t( A) denotes the Cohen-Macaulay type of A, and l A(∗) denotes the length of the A-module ∗. A has almost maximal length if l A( A ̄ A ) = l A( 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 m and a positive integer p such that m = (x, b), andb A ̄ = x p 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 l A( b x p A ̄ ) = 1 . Here p = min{i¦x iϵb} 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).

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