Commutative Algebra Seminar

Title

Density functions for families of graded ideals.

Abstract

In the first part of the talk we discuss Hilbert-Kunz density function and its applications for positive characteristic invariants, namely Hilbert-Kunz multiplicities and (F)-thresholds.

Next, based on a joint work with Suprajo Das and Sudeshna Roy, we introduce the adic and saturated density functions. These were introduced to give numerical characterizations for integral dependence of ideals in a graded set up, where we recall that two ideals (I \subset J) in a commutative Noetherian ring are integrally dependent if they have the same integral closure.

Attempts to give a numerical characterization for ideals which might not necessarily be of finite colength led to numerical invariants like (j)-multiplicity, (\varepsilon)-multiplicity which require (even in graded set up) computing the invariant at several localizations, hence not readily amenable to computations. Also there exists a notion of multiplicity sequence which gives a numerical characterization of integral dependence.

As an application of these density functions we show that any of the multiplicities, namely, the polar multiplicities, the (\varepsilon)-multiplicities or the (j)-multiplicities of the truncated ideals (I[Y]{\geq c}) and (J[Y]{\geq c}) in (R[Y]) characterizes the integral dependence of (I) and (J). A novelty of this approach is that it does not involve localization and only requires checking computable and well-studied invariants like Hilbert-Samuel multiplicities.