Multiplier Modules of extended Rees algebras

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, Mustaţă 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).