A characterization of multiplier ideals via ultraproducts

In this paper, using ultra-Frobenii, we introduce a variant of Schoutens' non-standard tight closure, ultra-tight closure, on ideals of a local domain $R$ essentially of finite type over $\mathbb{C}$. We prove that the ultra-test ideal $\tau_{\rm u}(R,\mathfrak{a}^t)$, the annihilator ideal of all ultra-tight closure relations of $R$, coincides with the multiplier ideal $\mathcal{J}(\operatorname{Spec} R,\mathfrak{a}^t)$ if $R$ is normal $\mathbb{Q}$-Gorenstein. As an application, we study a behavior of multiplier ideals under pure ring extensions.