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Given an ideal $a \subseteq R$ in a (log) $Q$-Gorenstein $F$-finite ring ofcharacteristic $p > 0$, we study and provide a new perspective on the testideal $\tau(R, a^t)$ for a real number $t > 0$. Generalizing a number of knownresults from the principal case, we show how to effectively compute the testideal and also describe $\tau(R, a^t)$ using (regular) alterations with aformula analogous to that of multiplier ideals in characteristic zero. Wefurther prove that the $F$-jumping numbers of $\tau(R, a^t)$ as $t$ varies arerational and have no limit points, including the important case where $R$ is aformal power series ring. Additionally, we obtain a global division theorem fortest ideals related to results of Ein and Lazarsfeld from characteristic zero,and also recover a new proof of Skoda's theorem for test ideals which directlymimics the proof for multiplier ideals.

Test ideals of non-principal ideals: Computations, Jumping Numbers, Alterations and Division Theorems