Multiplier Modules of extended Rees algebras

TL;DR

This paper develops methods to compute multiplier modules for extended Rees algebras of ideals in local rings, providing decomposition theorems that relate these modules to properties like rational singularities.

Contribution

It introduces a decomposition theorem for multiplier modules of extended Rees algebras, enabling explicit calculations and insights into their singularity properties.

Findings

Decomposition theorem for multiplier modules of extended Rees algebras

Explicit computation methods for multiplier modules of Rees and extended Rees algebras

Connections established between multiplier modules and rational singularities

Abstract

Given a local ring $(R, \mathfrak{m})$ and an ideal $\mathfrak{a}$ of positive height, we give a way of computing multiplier module ${J}(\omega_{{T}}, t^{-\lambda})$ for the extended Rees algebra ${T} =R[\mathfrak{a} t, t^{-1}]$ for an ideal $\mathfrak{a}$ by proving a decomposition theorem for ${J}(\omega_{{T}}, t^{-\lambda})$, (also see the works of Budur, Musta\c{t}\u{a} and Saito). We compute the multiplier module ${J}(\omega_{{S}}, (\mathfrak{a} \cdot {S})^{\lambda})$ for the Rees algebra ${S} =R[\mathfrak{a} t]$ as well (also see the works of Hyry and Kotal-Kummini). We use these decompositions to understand relationships between associated graded rings, Rees and extended Rees algebras having rational singularities (also see the works of Hara, Watanabe, and Yoshida).