Multiplier ideals and klt singularities via (derived) splittings

Let $X$ be a normal, excellent, noetherian scheme over $\mathrm{Spec}\mathbb{Q}$ with a dualizing complex. In this note, we find an alternate characterization of the multiplier ideal of $X$, as defined by de Fernex-Hacon, by considering maps ${\pi }_{\ast }{\omega }_Y\to {\mathcal{O}}_X$ where $\pi :Y\to X$ ranges over all regular alterations. As a corollary to this result, we give a derived splinter characterization of klt singularities, akin to the characterization of rational singularities given by Kovács and Bhatt. We also give an analogous description of the test ideal in characteristic $p>2$ as a corollary to a result of Epstein-Schwede.