Mathematics Seminar
Date: Thursday, 20 February 2025
Time: 3:30 to 4:30 PM
Venue: Seminar Hall
A density function for the epsilon multiplicity and applications to integral closures
Suprajo Das
IIT Madras.
20-02-25

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Abstract

Suppose that I is an ideal in a Noetherian local ring (R,mathfrak{m}) of dimension d. Ulrich and Validashti defines the varepsilon-multiplicity of I to be

varepsilon(I):=limsup_{n to infty} dfrac {lambda_Rleft(H^0_{mathfrak{m}} left (R/I^nright)right)}{n^d/d!}

The varepsilon-multiplicity can be seen as a generalization of the classical Hilbert-Samuel multiplicity. Cutkosky showed that the `limsup' in the definition of varepsilon-multiplicity can be replaced by a limit if the local ring R is analytically unramified. An example due to Cutkosky-H`a-Srinivasan-Theodorescu shows that this limit can be an irrational number even in a regular local ring. Throughout this talk, we shall restrict ourselves to homogeneous ideals in a standard graded domain over a field. Inspired by Trivedi's approach to Hilbert-Kunz multiplicity via density functions, we shall introduce a real valued compactly supported continuous function, the so-called varepsilon-density function, whose integral gives the varepsilon-multiplicity. If time permits, we shall produce some explicit examples, and give applications in the context of integral closures.