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.