Parametric decomposition of powers of ideals versus regularity of sequences

摘要

Let Q = (a(1), a(2), ... , a(s)) (not subset of or equal to A) be an ideal in a Noetherian local ring A. Then the sequence a(1), a(2), ... , a(s) is A-regular if every a(i) is a non-zerodivisor in A and if Q(n) = boolean AND(alpha)(alpha(1)(alpha1), alpha(2)(alpha2), ... , alpha(s)(alphas)) for all integers n greater than or equal to 1, where alpha = (alpha(1), alpha(2), ... , alpha(s)) runs over the elements of the set A(s,n) = {(alpha(1), alpha(2), ... , alpha(s)) is an element of Z(s) alpha(i) greater than or equal to 1 for all 1 less than or equal to i less than or equal to s and Sigma(i=1)(s) alpha(i) = s + n - 1}. [References: 3]