Reduction numbers and Rees algebras of powers of an ideal

Abstract

Let I I be an ideal in a Noetherian local ring ( R , m ) (R,\mathfrak {m}) . It is shown that for n ≫ 0 n \gg 0 the reduction number r J ( I n ) {r_J}({I^n}) of I n {I^n} with respect to a minimal reduction J J is not only independent from the choice of J J but also is stable. If I I is an m \mathfrak {m} -primary ideal, we give a criterion for the Rees algebra R [ I n t ] R[{I^n}t] with n ≫ 0 n \gg 0 to be Cohen-Macaulay.