A characterization of multiplier ideals via ultraproducts

TL;DR

This paper introduces ultra-tight closure using ultra-Frobenii and demonstrates its equivalence to multiplier ideals in certain rings, providing new insights into their behavior under ring extensions.

Contribution

It develops ultra-tight closure via ultra-Frobenii and proves its equivalence to multiplier ideals in normal $Q$-Gorenstein rings, linking tight closure and multiplier ideal theories.

Findings

Ultra-tight closure coincides with multiplier ideals in normal $Q$-Gorenstein rings.

The paper establishes the behavior of multiplier ideals under pure ring extensions.

Introduces a new variant of non-standard tight closure using ultra-Frobenii.

Abstract

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.