The core of ideals

Abstract: Let I be an ideal in a Noetherian commutative ring. Among all the closures of I, the integral closure plays a central role. A reduction of I is a sub ideal with the same integral closure. We can think of reductions as simplifications of the given ideal, which carry most of the information about I itself but, in general, with fewer generators. Minimal reductions, reductions minimal with respect to inclusion, are loosely speaking the counterpart of the integral closure. However, unlike the integral closure, minimal reductions are not unique. For this reason, we consider their intersection, called the core of I. The core is related to adjoint and multiplier ideals. Motivation for studying this object comes from the Briancon-Skoda theorem. Furthermore, a better understanding of the core could lead to solving Kawamata's conjecture on the non-vanishing of sections of a certain line bundle. In this talk, I will discuss the importance of the core, its ubiquity in algebra and geometry, and some effective formulas for its computation.

Mathematics

Audience: advanced learners

IIT Bombay Virtual Commutative Algebra Seminar

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