Test Modules of Extended Rees Algebras

Given a reduced, local ring $R$ and an ideal $\mathfrak{a}$ of positive height, we give a decomposition of the test module, $\tau ({\omega }_T,t^{-\lambda })$, of the extended Rees algebra, $T=R[\mathfrak{a}t,t^{-1}]$. In particular, the degree zero component of this test module is $\tau (R,{\mathfrak{a}}^{\lambda })$, thereby reducing the computation of test modules for non-principal ideals to the much easier case of principal ideals. Additionally, we apply our decomposition to generalize results on the $F$-rationality of Rees and extended Rees algebras [HWY02a, Conjecture 4.1],[KK21] as well as give simplified proofs of discreteness and rationality of F-jumping numbers, among other applications.