**Spring 2016
Schedule**

Location: Minard 404 (Seminar Room)

Time: Tuesday, 10:00-10:50 a.m.

Organizer: Cătălin Ciupercă

Previous
Semesters
**9 February 2016**
Cătălin Ciupercă: 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 \(\operatorname{e}(J)\), a
recent result of Hickel shows that every element in the integral
closure of \(J\) satisfies an equation of integral dependence over \(J\) of
degree at most \(\operatorname{e}(J)\). We extend this result to equimultiple ideals \(J\)
by showing that the degree of such an equation of integral
dependence is at most \(c(J)\), which 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 \(c(J)-1\).