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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 pullbackof $\Delta$ by the canonical morphism $\operatorname{Spec} \hat{R}\to\operatorname{Spec} R$. As an application, we obtain a result on the behaviorof multiplier ideals under pure ring extensions.

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