Reduction numbers of equimultiple ideals

Abstract:

Let $(A, \mathfrak {M})$ be an unmixed local ring containing a field. If $J$ is an $\mathfrak {M}$-primary ideal with Hilbert-Samuel multiplicity $e_A(J)$, a recent result of Hickel shows that every element in the integral closure $\overline {J}$ satisfies an equation of integral dependence over $J$ of degree at most $\operatorname {e}_A(J)$. We extend this result to equimultiple ideals $J$ by showing that the degree of such an equation of integral dependence is at most $c_q(J)$, where $c_q(J)$ is one of the elements of the so-called multiplicity sequence introduced by Achilles and Manaresi. As a consequence, if the characteristic of the field contained in $A$ is zero, it follows that the reduction number of an equimultiple ideal $J$ with respect to any minimal reduction is at most $c_q(J)-1$.