Multiplicities of semidualizing modules

TL;DR

This paper investigates the properties of semidualizing modules over certain Cohen-Macaulay rings, establishing the equality of their Hilbert-Samuel multiplicities with those of the ring itself.

Contribution

It verifies the multiplicity equality for semidualizing modules over specific classes of Cohen-Macaulay rings, including those defined by fat point schemes and monomial ideals.

Findings

e_R(J;C) = e_R(J;R) for all semidualizing modules C and m-primary ideals J

The result applies to rings defined by fat point schemes in projective space

The result also applies to rings defined by monomial ideals

Abstract

A finitely generated module C over a commutative noetherian ring R is semidualizing if Hom_R(C,C) \cong R and Ext^i_R(C,C) = 0 for all i \geq 1. For certain local Cohen-Macaulay rings (R,m), we verify the equality of Hilbert-Samuel multiplicities e_R(J;C) = e_R(J;R) for all semidualizing R-modules C and all m-primary ideals J. The classes of rings we investigate include those that are determined by ideals defining fat point schemes in projective space or by monomial ideals.