Multiplicity sequence and integral dependence

Abstract: The first numerical criterion for integral dependence was proved by Rees in 1961 which states that two m-primary ideals $I \subset J$ in an equidimensional and universally catenary local ring $(R, m)$ have the same integral closure if and only if they have the same Hilbert-Samuel multiplicity. This result plays an important role in Teissier's work on the equisingularity of families of hypersurfaces with isolated singularities. For hypersurfaces with non-isolated singularities, one needs a similar numerical criterion for integral dependence of non-$m$-primary ideals. Since the Hilbert-Samuel multiplicity is no longer defined for non-$m$-primary ideals, one has to use other notions of multiplicities that can be used to check for integral dependence. A possibility is the multiplicity sequence which was introduced by Achilles and Manaresi in 1997 and has its origin in the intersection numbers of the Stuckrad-Vogel algorithm. It was conjectured that two arbitrary ideals $I \subset J$ in an equidimensional and universally catenary local ring have the same integral closure if and only if they have the same multiplicity sequence. This talk will present a recent solution of this conjecture by Polini, Trung, Ulrich and Validashti.

Mathematics

Audience: advanced learners

IIT Bombay Virtual Commutative Algebra Seminar

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