Asymptotic sequences and rees rings

Abstract

Let b 1, …, b s be an asymptotic sequence over an ideal I in a Noetherian ring R and let B i = ( b 1, …, b i ) R. Then it is shown that certain sequences closely related to these elements are asymptotic sequences over R ( R, B i ), over tI R ( R, I) and over u R ( R, I) where R ( R, J) is the Rees ring of R with respect to the ideal J. These results then imply that certain other sequences are asymptotic sequences over the corresponding ideals in the associated graded rings and in the monadic transformation rings. As an application of the results, it is shown that if R is local, then each permutation of b 1, …, b s is an asymptotic sequence over I and that b 1, …, b s are an asymptotic sequence in R.