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

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 $\mathrm{\Delta }$ is an effective $\mathbb{Q}$-Weil divisor on $\mathrm{Spec}R$ such that $K_R+\mathrm{\Delta }$ is $\mathbb{Q}$-Cartier, then the BCM test ideal ${\tau }_{\widehat{\mathcal{B}(R)}}(\widehat{R},\widehat{\mathrm{\Delta }})$ of $(\widehat{R},\widehat{\mathrm{\Delta }})$ with respect to $\widehat{\mathcal{B}(R)}$ coincides with the multiplier ideal $\mathcal{J}(\widehat{R},\widehat{\mathrm{\Delta }})$ of $(\widehat{R},\widehat{\mathrm{\Delta }})$, where $\widehat{R}$ and $\widehat{\mathcal{B}(R)}$ are the $\mathfrak{m}$-adic completions of $R$ and $\mathcal{B}(R)$, respectively, and $\widehat{\mathrm{\Delta }}$ is the flat pullback of $\mathrm{\Delta }$ by the canonical morphism $\mathrm{Spec}\widehat{R}\to \mathrm{Spec}R$. As an application, we obtain a result on the behavior of multiplier ideals under pure ring extensions.