Test ideals of non-principal ideals: Computations, jumping numbers, alterations and division theorems

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Given an ideal a⊆R in a (log) Q-Gorenstein F-finite ring of characteristic p>0, we study and provide a new perspective on the test ideal τ(R,at) for a real number t>0. Generalizing a number of known results from the principal case, we show how to effectively compute the test ideal and also describe τ(R,at) using (regular) alterations with a formula analogous to that of multiplier ideals in characteristic zero. We further prove that the F-jumping numbers of τ(R,at) as t varies are rational and have no limit points, including the important case where R is a formal power series ring. Additionally, we obtain a global division theorem for test 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 directly mimics the proof for multiplier ideals.