Multiplier Modules of extended Rees algebras

Abstract

Given a local ring $(R,\mathfrak{m})$ and an ideal $\mathfrak{a}$ of positive height, we give a way of computing multiplier module $J(\ddot{I}{‰}_T,t^{-\hat{I}»})$ for the extended Rees algebra $T=R[\mathfrak{a}t,t^{-1}]$ for an ideal $\mathfrak{a}$ by proving a decomposition theorem for $J(\ddot{I}{‰}_T,t^{-\hat{I}»})$, (also see the works of Budur, Mustaţă and Saito). We compute the multiplier module $J(\ddot{I}{‰}_S,(\mathfrak{a}\cdot S{)}^{\hat{I}}»)$ 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).

20 pages, Dedicated to Prof. Karen E. Smith, on the occasion of her 60th birthday!