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.
Categories: math.AC math.AG