Multiplier Modules of extended Rees algebras
Rahul Ajit
TL;DR
This work develops a comprehensive decomposition framework for multiplier modules of extended and Rees algebras associated to an ideal $\mathfrak{a}$ over a normal local ring of characteristic zero. By combining deformation to the normal cone, canonical-module computations, and Blickle’s toric formula, it derives explicit graded decompositions: $\mathcal{J}(\omega_{\mathcal{S}},(\mathfrak{a}\cdot\mathcal{S})^\lambda)=\bigoplus_{n\ge0}\mathcal{J}(\omega_R,\mathfrak{a}^{n+1+\lambda})t^{n+1}$ and $\mathcal{J}(\omega_{\mathcal{T}},(t^{-1})^\lambda)=\bigoplus_{k\in\mathbb{Z}}\mathcal{J}(\omega_R,\mathfrak{a}^{k+\lambda})t^k$. These decompositions relate multiplier modules across $R$, $\mathcal{S}$, and $\mathcal{T}$ and enable transfer of rational singularity properties between the algebras via the associated graded $G$. The paper also provides a detailed toric-model analysis, invariance results under étale morphisms, and concrete applications showing equivalences of rationality for the Rees constructions, thereby linking multiplier theory, deformation theory, and toric geometry in the study of 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 ${J}(ω_{T}, t^{-λ})$ for the extended Rees algebra ${T} =R[\mathfrak{a} t, t^{-1}]$ for an ideal $\mathfrak{a}$ by proving a decomposition theorem for ${J}(ω_{T}, t^{-λ})$, (also see the works of Budur, Mustaţă and Saito). We compute the multiplier module ${J}(ω_{S}, (\mathfrak{a} \cdot {S})^λ)$ 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).