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 \(\Delta\) 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.