Algebra & Discrete Mathematics Seminar
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 $$c(J)-1$$.