Integral Closure, Multiplier Ideals and Cores

December 17 to December 21, 2006

at the

American Institute of Mathematics, Palo Alto, California

organized by

Alberto Corso, Claudia Polini, and Bernd Ulrich

Original Announcement

This workshop will be devoted to questions related to the notion of integral closure of ideals. The generalization to ideals of the basic concepts of integral extensions and integral closures of rings can be traced back to the fundamental work of Zariski and Rees in local algebra. Loosely speaking, the integral closure of an ideal I is an ideal contained in the radical of I that shares a number of finer properties with I. Determining the integral closure of I is a difficult task, which essentially amounts to finding solutions in the ring itself of special polynomial equations whose coefficients belong to higher and higher powers of I.

The aspects intimately connected to the integral closure that we are planning to focus on are: computation of the integral closure and its complexity; multiplicities and equisingularity theory; cores of ideals and Briancon-Skoda type theorems; multiplier ideals and test ideals; multiplier ideals and jet schemes. More concretely, some of the specific questions/open problems that we will address during the workshop are:

• find effective methods to compute (parts of) the integral closure of an ideal (or, more generally, of a submodule of a free module);
• find tests to detect when an ideal (or, more generally, a submodule of a free module) is integrally closed;
• find a relationship between the core and the adjoint for integrally closed and monomial ideals;
• find necessary and/or sufficient conditions for integrally closed ideals in regular local rings of dimension at least 3 to be multiplier ideals;
• further explore the relationship between test ideals and multiplier ideals.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Lecture notes

Continuous closure and variants of integral closure, by Hochster
Open problems on powers of ideals, by Huneke
Equisingularity and integral closure, by Kleiman
Local syzygies of multiplier ideals, by Lee
A vanishing theorem for finitely supported ideals in regular local rings, by Lipman
F-Pure thresholds and log canonical thresholds, by Tagaki
Square-free monomial ideals and hypergraphs, by Trung
Rees criteria, by Ulrich
On F-thresholds (ring theoretic aspects), by Watanabe