Big Cohen-Macaulay test ideals in equal characteristic zero via ultraproducts

TL;DR

This paper demonstrates that in characteristic zero, BCM test ideals constructed via ultraproducts coincide with multiplier ideals for certain normal local domains, linking these two important concepts in algebraic geometry.

Contribution

It establishes the equality of BCM test ideals and multiplier ideals in equal characteristic zero using ultraproducts, extending the understanding of their relationship.

Findings

BCM test ideals coincide with multiplier ideals in the setting considered.

The result applies to normal local domains with effective $Q$-Weil divisors.

Provides insights into the behavior of multiplier ideals under pure ring extensions.

Abstract

Utilizing ultraproducts, Schoutens constructed a big Cohen-Macaulay algebra $\mathcal{B}(R)$ over a local domain $R$ essentially of finite type over $\mathbb{C}$. We show that if $R$ is normal and $\Delta$ is an effective $\mathbb{Q}$-Weil divisor on $\operatorname{Spec} R$ such that $K_R+\Delta$ is $\mathbb{Q}$-Cartier, then the BCM test ideal $\tau_{\hat{\mathcal{B}(R)}}(\hat{R},\hat{\Delta})$ of $(\hat{R},\hat{\Delta})$ with respect to $\hat{\mathcal{B}(R)}$ coincides with the multiplier ideal $\mathcal{J}(\hat{R},\hat{\Delta})$ of $(\hat{R},\hat{\Delta})$, where $\hat{R}$ and $\hat{\mathcal{B}(R)}$ are the $\mathfrak{m}$-adic completions of $R$ and $\mathcal{B}(R)$, respectively, and $\hat{\Delta}$ is the flat pullback of $\Delta$ by the canonical morphism $\operatorname{Spec} \hat{R}\to \operatorname{Spec} R$. As an application, we obtain a result on the behavior of multiplier ideals under pure ring extensions.