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

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.